Pressure-induced metal-insulator transition on
solid Tellurium and liquid Iodine
Scuola Dottorale in Scienze Astronomiche, Chimiche,Fisiche, Matematiche
e della Terra "Vito Volterra"
Dottorato di Ricerca in Fisica – XXIII Ciclo
Candidate
Daniele Chermisi
ID number 1187743
Thesis Advisor
Prof. Paolo Postorino
A thesis submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
December 2010
Thesis not yet defended
Daniele Chermisi. Pressure-induced metal-insulator transition on solid Tellurium
and liquid Iodine.
Ph.D. thesis. Sapienza – University of Rome
© 2010 ISBN: 000000000-0
version: 13 December 2010
website: http://www.phys.uniroma1.it/gr/HPS/HPS_file/Page353.htm
email: [email protected]
Dedicated to
my family
Acknowledgments
It was not easy to develop a PhD thesis that includes both an experimental and
a theoretical part. I do not think that this work would have been possible without
the help and suggestions of many people. First I would like to thank my advisor
Prof. Paolo Postorino for introducing me to the high pressures and spectroscopic
techniques. I also thank him for the freedom he gave me in these three years that
allowed me to combine the theoretical and the experimental approaches.
I would like to sincerely thank Prof. Sandro Scandolo for its willingness, the enlightening conversations and valuable suggestions on ab-initio calculations. I would
also like to thank Prof. Leonardo Guidoni for his essential support in carrying out
ab-initio molecular dynamics simulations. They are also grateful and indebted to
all members of Leonardo’s research group for their great patience and availability. I
would especially thanks Daniele Varsano, Daniele Bovi and Maria Montagna. I am
also very grateful to Valentina Migliorati for the support (even moral!) she has given
me during the ab-initio simulations analysis and to Dario Rocca for the support on
rigid dimers molecular dynamics simulations.
Thanks also to my lab mates (Sara, Matteo, Sara and Carlo) for having patiently
endured me during these 3 years. I warmly thank also my family for their love above
all in the most challenging times.
v
Contents
Introduction
ix
1 Insulator to metal transition in solid Tellurium
1.1 Charge density waves and Peierls distortion . . . . . . . . . . . . . .
1.1.1 Instability of the one dimensional free electron gas . . . . . .
1.1.2 One dimensional free electron gas and el-ph coupling . . . . .
1.2 Dynamical properties in Modulated crystals . . . . . . . . . . . . . .
1.2.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Approximate solutions of the equations of motion . . . . . . .
1.3 Modulated spring model . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Perturbative approach . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Commensurate approximation . . . . . . . . . . . . . . . . .
1.3.3 Some useful examples . . . . . . . . . . . . . . . . . . . . . .
1.4 Tellurium under pressure: structural and electronic phase transition.
1
2
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3
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11
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14
16
2 Experimental Setup
2.1 High pressure technique . . . . . . . . . . . .
2.1.1 The high pressure diamond anvil cell .
2.1.2 In situ pressure measurement . . . . .
2.2 Raman scattering setup . . . . . . . . . . . .
2.2.1 High pressure and temperature setup
2.3 Infrared setup . . . . . . . . . . . . . . . . . .
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3 Density functional theory
3.1 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . .
3.2 The Kohn-Sham equations . . . . . . . . . . . . . . . . .
3.3 Approximation for the exchange-correlation energy . . .
3.4 The band gap problem . . . . . . . . . . . . . . . . . . .
3.5 Plane waves and pseudopotentials . . . . . . . . . . . . .
3.6 Density functional perturbation theory . . . . . . . . . .
3.7 Short notes on classical molecular dynamics simulations
3.7.1 Microcanonical and canonical ensemble . . . . .
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4 Measurements and ab-initio calculations on Te at high pressure
4.1 High pressure Raman measurements and analysis on solid Te . . . .
4.2 High pressure infrared measurements and analysis on solid Te . . . .
4.3 Ab-initio calculations on Trigonal Tellurium . . . . . . . . . . . . . .
4.4 Ab-initio calculations on Te-II high pressure phase. . . . . . . . . . .
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5 Insulator to metal transition in liquid Iodine
5.1 Insulator to metal transition in molten elements . . . . . . . . . .
5.2 Pressure induced transitions in solid Iodine . . . . . . . . . . . .
5.3 Pressure induced transitions in liquid Iodine . . . . . . . . . . . .
5.3.1 Electronic properties of liquid Iodine under pressure . . .
5.3.2 Structural properties of liquid Iodine under high pressure
5.4 High pressure and temperature Raman experiment . . . . . . . .
5.5 Ab-initio molecular dynamics simulations on liquid iodine . . . .
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89
6 Conclusions
6.1 Solid Tellurium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Liquid Iodine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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viii
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Introduction
A pressure induced metal-insulator transition (MIT) was, for the first time, theoretically predicted in the case of dense solid hydrogen by Wigner and Huntington
in the far thirties. This suggestive hypothesis was further investigated and a discontinuous transition to a state showing metallic conductivity was predicted when
the lattice constant becomes lower than a given threshold value. Since these first
studies, a strong theoretical effort has been devoted to study pressure-induced MIT
at first in hydrogen and than in other simple elemental systems as well as in complex
materials. At present, theoretical models of the MIT, even in simple systems, are
much more complex than the early predictions and often a full description of the
microscopic mechanisms underlying the transitions is still lacking. In recent years
ab-initio techniques such as the Density Functional Theory (DFT) have been successfully applied to the study of high pressure phases in simple elemental systems
but also and in more complex materials and at present these appear to be the most
promising techniques.
On the other hand with the development of high-pressure experimental techniques, the behavior of materials in the GPa pressure range (0-50 GPa) and in particular the occurrence of a MIT can now be investigated with increasing accuracy,
precision, and sensitivity. In particular spectroscopic techniques (Raman and Infrared) coupled with the use of diamond anvil cells provide crucial and often unique
information on the properties of materials at high pressure. Indeed the phonon spectrum elucidates the changes in bonding properties of compressed crystals, glasses,
and melts, allows the identification of the phase transitions and provides information on crystal symmetry. Moreover Infrared measurements can also provide direct
information on the pressure-induced changes in electronic structure, identifying precursor phenomena of the MIT and monitoring the charge delocalization process.
In molecular systems the study of the vibrational properties can give useful
insight on the mechanism leading to the transition to a metallic state. Since the
dynamical properties are determined by the Born-Oppenheimer surface shape, i.e,
by the electron ground state properties, change in the transport properties can remarkably affect the frequencies and intensities of the observed vibrational spectrum.
For instance, in the case of MIT in solid iodine, the careful analysis of the pressure
dependence of the Raman spectrum combined with the results of ab-initio calculations has shown the key role of the intra- / inter-molecular charge transfer in the
delocalization process which drives the system to metallic phase at high pressure.
As well as solid Iodine, only a combination of ab-initio molecular dynamics simulation and experimental results of liquid Tellurium allowed a deeper understanding
of the local structure and of the change in the bonding properties underlying the
onset of a metallic behavior in liquid Te under pressure.
ix
These examples clearly show the potentiality of a combined experimental / abinitio calculations approach. In particular the ab-initio approach can give access
to properties not experimentally observable at high pressure using a diamond anvil
cell such as for instance the low-frequency phonon modes. Moreover ab-initio calculations can be used to extend the analysis to the microscopic properties such as
the phonon mode eigenvectors, the Raman tensors and the electron charge density
in solid systems, or can be used to obtain useful information on local configurations
in molten elements, thus giving more insight into the mechanism at the origin of
the observed experimental behavior both in solid and liquid systems.
The aim of the present PhD thesis is the study of the high pressure phases and
in particular of the MIT occurring in solid Tellurium and liquid Iodine using the
above discussed approach that includes both a Raman and Infrared spectroscopic
investigation as well as theoretical studies in the framework of DFT. These systems show rather different chemical-physical properties albeit the phenomenologies
related to the MIT show several common aspects.
Focusing on the solid phase, it is well known that iodine at ambient conditions is
a molecular semiconductor and that the MIT occurs within a pressure range (P=16
GPa T=300 K) where the system is still molecular. On the other hand, at ambient
pressure solid Tellurium consists of parallel spiral chains of Te atom bounded by
strong covalent bonding, and with the chains bonded each other by weaker van
der Waals interaction. On these bases also Tellurium can be thus seen as a sort
of a molecular solid. Moreover, even if the MIT in Tellurium involves a structural
transition (P=4 GPa T=300), the high pressure metallic phase preserves a structure
with parallel linear chains with high anisotropic bonding, i.e, the system still shows
a molecular-like behavior.
In the liquid phase both Tellurium and Iodine show a local structure which is
somehow reminiscent of that of the solid phase with long and short covalent bonds
coexisting in the liquid. Moreover both the systems show a pattern of structural
and electronic transitions in the liquid phase very similar to that observed in the
solid even if, in the liquid, they occur at much lower pressures. Namely solid Iodine
enters a metallic molecular phase at 16 GPa and a metallic atomic phase at 30 GPa
whereas the same two transitions occur a 3 GPa (T=900 K) and 4 GPa (T=1000 K)
in the melt. We notice that also in liquid metallic Tellurium remarkable interaction
among broken Te chains are observed.
The present thesis work is thus focused on investigating those regions of the
phase diagram around the MIT which are still not well experimentally investigated
and not definitely understood. In particular the high pressure region of the phase
diagram of solid Tellurium (0-15 GPa at room temperature) and the high pressure,
high temperature region of liquid Iodine (0-3 GPa and 300-800 K). It is worth to
notice that from both theoretical and experimental point of view the study of high
pressure liquid iodine is a much more complex problem then that of high pressure
solid Te. As a matter of fact, the experimental study of a high pressure liquid
also implies high temperature to avoid compression-induced solidification and the
disordered configuration of the liquid makes ab-initio calculations much more long
and complex.
As to solid Tellurium, the research has been mainly focused on investigating
the microscopic mechanisms driving the charge delocalization under pressure and
the nature of the high pressure structural phases. In particular one of the goals
x
was the experimental/theoretical confirm of the insurgence of incommensurately
modulated phases. Since, at the best of our knowledge, Tellurium is the only systems
where all the expected optical modes are due to a crystal lattice modulation a
Raman investigation can provide a direct check of the recently proposed structural
phase. Moreover DFT calculations combined with experimental findings can give
information on the phase stability under pressure and provide a microscopic picture
of the electronic modification occurring moving towards the metallic phase.
The presence of an incommensurately modulated phase in solid iodine at high
pressure points out that there is a transient state during the molecular dissociation
related to strong charge transfer interactions. Several experimental findings suggest
that these interactions are much more effective in modifying the electronic properties of liquid iodine. Since the Raman-active intramolecular vibration is particular
sensitive to the extent of the charge transfer, Raman spectroscopy is an ideal tool
to study the modifications induced in the liquid on approaching the MIT.
It is finally worth to mention that in solid Iodine and Tellurium the metallization
transition is obtained through a progressive band gap closure on increasing the
pressure. The experimental evidences that the MIT in liquid Iodine is obtained
by a rather abrupt band gap closure strogly suggest that a microscopic mechanism
different from that of the solid phase is at work in the melt. To this respect ab-initio
molecular dynamics simulations was expected to provide valuable information about
the origin of the threshold mechanism.
xi
Chapter 1
Insulator to metal transition in
solid Tellurium
In solid state physics the electrical conductivity of many material is related to
the transport of electrons. From the quantum mechanical point of view, electrons
in crystalline materials are arranged in states which are quite closely spaced in
energy, forming an almost continuous distribution of energies, called energy band.
An intuitive explanation is that as atoms are brought together, the bands begin
to emerge from the starting atomic states, the lower-energy states are occupied,
lowering the energy and further pulling the system together. Energy bands may
be often separated by regions in energy for which no wavelike electron orbitals
exist. Such forbidden regions are called energy gaps, or band gaps, and result
from the interaction of the conduction electron waves with the ion cores of the
crystal. In this band picture, the insulator and metallic states of a systems can
be understood in term of the filling of the electron bands: the systems behaves
as an insulator if the energy bands are either filled or empty, which means that
no electrons can move in an electric field, or as a metal if one or more bands are
partly filled. In other words, the Fermi level, which is highest occupied electron
state at T = 0 K, lies in a band gap in insulators while the level is inside a band for
metals. Metals are characterized by a low resistivity of order of 10−2 ÷ 10−6 Ω cm
at room temperature, i.e., high charge mobility. On the other hand, insulators have
resistivity of order 103 ÷ 1017 Ω cm at room temperature, i.e., low charge mobility.
The distinction between metals and insulator strictly yields only at absolute zero.
At finite temperature the distinction becomes more qualitative and the family of
semiconductor materials with conduction band (empty at T=0) slightly filled at
T �= 0, may be introduced. Semiconductors are generally classified by their electrical
resistivity at room temperature, with values in the range of 102 to 109 Ω cm, and
by its strong temperature dependence.
Electrons in a solid are mainly influenced by two different causes: the ions
potential and the electron-electron interaction. Thereby the electronic band structure of a given system is determined by the strength and the extent of coupling
of these interactions. In systems like the Mott and Charge Transfer insulators the
electron-electron interaction play a key role in determining the insulating behavior.
However, even in the absence of strong electron-electron coupling, the interaction
between electrons and the periodic lattice potential gives rise to a variety of very
1
2
1. Insulator to metal transition in solid Tellurium
interesting behaviors. This is the case of the charge density wave (CDW) systems,
where a static lattice deformation occurs leading to a charge localization.
1.1
Charge density waves and Peierls distortion
1.1.1
Instability of the one dimensional free electron gas
The CDW is a charge density redistribution due to an electron instability related
to a particular topology of the Fermi surface. To understand this phenomena is
very useful to examine the�one-dimensional free electron gas [34]. In presence of an
external potential
� ind V (r) = V (q) exp(iqr)dq there will be an induced charge density
ind
ρ (r) = ρ (q) exp(iqr)dq. According to the Lindhard linear response theory,
the response of the electron gas is described as
ρind (q) = χ(q, T )V (q)
(1.1)
where the wavevector-dependent electron response function χ(q, T ) is given by
χ(q, T ) =
1 � f (Ek ) − f (Ek+q )
V
Ek − Ek+q
(1.2)
k
where V denotes the volume of the system, f (Ek ) the Fermi function and Ek the
electronic energy at wavevector k. At T = 0, f (Ek ) can take values of only zero
or unity. The right-hand side terms of equation 1.2 have non-zero values only
when Ek < EF < Ek+q or Ek+q < EF < Ek , denoting with EF the Fermi energy.
Hence the right-hand side of equation 1.2 has a large value for a pair of electrons
with one just below the Fermi level and the other just above it. Figure 1.1 shows
χ(q, T ) at T = 0 as a function of q for free-electron gases of one, two and three
dimensions (1D,2D and 3D). We notice that only the 1D response function χ(q, T )
diverges at q = 2kF , and this point has several important consequences. Equation
Figure 1.1. Response functions of one-, two- and three-dimensional free-electron gases
[34].
1.1 implies that an external perturbation leads to a divergent charge redistribution;
this suggests, through self-consistency, that at T = 0 the electron gas itself is
1.1 Charge density waves and Peierls distortion
3
unstable with respect to the formation of a periodically varying electron charge or
electron spin density. The divergence of the response function at q = 2kF is due
to the particular topology of the Fermi surface, sometimes called “perfect nesting”.
Looking at Equation 1.2, the most significant contributions to the integral come
from pairs of states - one full, one empty - which differ by q = 2kF and have the
same energy, thus giving a divergent contribution to χ(q, T ). However, in higher
dimensions the number of such states is significantly reduced leading to the removal
of the singularity at q = 2kF (see Figure 1.1). At finite temperature the divergence
is removed too and the χ(q, T ) has a peak at q = 2kF (see Figure 1.2).
Figure 1.2. The response function of a one-dimensional free electron gas at various
temperatures. The highest curve is for the lowest temperature [34].
1.1.2
One dimensional free electron gas and el-ph coupling
In order to describe the transition to a charge density wave ground state let us
consider a one-dimensional free electron gas coupled to the underlying chain of ions
through electron-phonon coupling. The Hamiltonian of the system is written as
H = Hel + Hph + Hel−ph
(1.3)
In second quantization the the Hamiltonian of a free electron gas Hql is given by
�
Hel =
�k a†k ak
(1.4)
k
with a†k and ak being the creation and annihilation operators for the electron states
with energy �k = (h̄k)2 /(2m). The lattice vibration are described by:
Hph =
�
q
1
h̄ωq (b†k bk + )
2
(1.5)
4
1. Insulator to metal transition in solid Tellurium
with b†q and bq being the creation and annihilation operators for phonons of wavevectors q, and ωq being the normal mode frequencies. The coupled of the electron
with the phonon can be described by the term [28]
�
Hel−ph =
gq (b†−q + bq )a†k+q ak
(1.6)
kq
�
where gq is the coupling constant. Since ρq = k a†k+q ak is the q th Fourier component of the electron density, and introducing the normal coordinates Qq
Qq =
�
h̄
2M ωq
�1/2
(b†−q + bq )
(1.7)
the Hel−ph term can be written as
Hel−ph =
�
gq Qq
q
�
2M ωq
h̄
�1/2
ρq .
(1.8)
Assuming gq independent of q (gq = q), the equation of motion for the normal
coordinates Qq are
�
�
2ωq 1/2
2
Q̈q = −ωq Qq − g
ρq
(1.9)
M h̄
The fluctuation density ρq can be written, using the Equation 1.1 as
ρq = χ(q, T )g
�
2M ωq
h̄
�1/2
Qq .
With this mean field approximation the equation of motion reads
�
�
2g 2 ωq
2
Q̈q = − ωq +
χ(q, T ) Qq .
M h̄
(1.10)
(1.11)
The electron-phonon-coupling causes the renormalization of the frequency of the
acoustic phonon coupled to the charge, with a new phonon frequency ωren,q given
by
2g 2 ωq
2
ωren,q
= ωq2 +
χ(q, T ).
(1.12)
h̄
We notice that the renormalization term in Equation 1.12 reaches its maximum
value when q = 2kF , i.e, when χ(q, T ) has its minimum (negative) value as seen
in Section 1.1.1. This (strongly) renormalization of the phonon spectrum, due to
the screening of the conduction electron, is generally referred as Kohn anomaly.
Moreover, decreasing the temperature in 1D electron gas, the decrease of the χ(q =
kF , T ) (see Figure 1.2) leads to zero phonon frequency in Formula 1.12 thus to a
frozen-in distortion at a certain value of temperature TCDW (see Figure 1.3).
The lattice displacement is given by
< u(x) >= ∆u cos(2kF + Φ)
with
∆u =
�
2h̄
N M ω2kF
�1/2
|∆|
g
(1.13)
(1.14)
1.1 Charge density waves and Peierls distortion
5
Figure 1.3. Acoustic phonon dispersion relation of a one-dimensional metal at various
temperatures above the mean field transition temperature TCDW [34].
Figure 1.4. Changes in electronic energy levels near k = kF from the frozen lattice
distortion: in this region, the free electron curve is approximated with a straight line
[34].
and
|∆|eiΦ = g(< b2kF > + < b†−2kF >).
(1.15)
The frozen in lattice distortion opens gap in the dispersion relation near the Fermi
wavevectors kF , as seen in Figure 1.4. This will lower the electronic energy according
to a term
�
�
∆
CDW
0
2
∆Eel = ∆Eel
− Eel ∼ ∆ log
< 0, for ∆ < 2�F ,
(1.16)
2�F
while it will increase the elastic energy according to
CDW
0
∆Elat = ∆Elat
− Elat
=
h̄ω2kF ∆2
> 0.
2g 2
(1.17)
In the ∆ � �F case ∆Eel + ∆lat < 0, implying that the CDW ground state has,
in 1D, lower energy than the normal state. In the weak coupling limit the CDW
6
1. Insulator to metal transition in solid Tellurium
ground state is characterized by a periodic charge density variation according to
�
�
∆
ρ(x) = ρ0 1 +
cos(2kF x + φ)
(1.18)
h̄vF kF λ
where ρ0 is the (constant) electronic density in the metallic state, and vF = h̄kmF .
The dispersion relation, the electronic density, and the equilibrium lattice positions
are shown, both above TCDW and at T = 0 in Figure 1.5 in the half-filled band case.
Generally speaking the onset of a Peierls distorted phase or in other words of a
Figure 1.5.
The single particle band, electron density, and lattice distortion in the
metallic state above TCDW and in the charge density wave state at T = 0. The figure
is appropriate for a half-filled band [34].
CDW state, besides the gap opening at the Fermi level, induces lattice modulation
whose consequences will be discussed in the next Section.
1.2 Dynamical properties in Modulated crystals
1.2
7
Dynamical properties in Modulated crystals
The content of this section is mainly taken from the works of T. Janssen on incommensurate crystal phases [42, 43, 52, 68].
In the previous sections we have seen that, under certain conditions, a static displacements of the atoms can lead to the stabilization of the system. We can regard a
crystal with displacive modulation as a imaginary three dimensional crystal Λ with
s atoms at position r j in the unit cell such that the actual positions of the atoms
are [42, 52]
�
Rn,j = n + r j +
f j (q)exp(iq · n).
(1.19)
q
Here n labels the cells of the lattice Λ. For a unidimensional modulation the vector
function f (x) has periodicity 2π and the vectors q may be written as
q = ma∗4
(1.20)
The modulation vector may be expressed in terms of the basis vectors a∗1 , a∗2 , a∗3 of
the reciprocal lattice Λ∗ :
3
�
a∗4 =
σi a∗i .
(1.21)
i=1
The incommensurability of the modulation means that at least one of the factor σi
is irrational. The crystal described by the 1.19 can be embedded in a unique way
in a (3+1)-dimensional ‘superspace’ generated by a1 , a2 , a3 (the basis vectors of Λ)
and a vector b1 1 in 1-dimensional lattice D in a vector space VI perpendicular to
position space. To achieve this it is necessary to construct a lattice Σ generated by
the following basis of (3+1) vectors:
ai = (ai, −σi b1 ) ,
i = 1, 2, 3
a4 = (0, b1 )
The points of the super-crystal generated by these vectors are given by
�
�
�
n + rj +
f (q)exp(iq · n + ib∗ τ ), τ
(1.22)
(1.23)
q
where τ is an arbitrary vector of the internal space. The projection of the spectrum
of 1.23 on position space (τ = 0) is exactly the spectrum of the three dimensional
modulated crystal 1.19 The basis reciprocal to 1.23 is
a∗i = (a∗i , 0) i = 1, 2, 3
�
�
�
∗
∗ ∗
a4 =
σi ai , b1 = (a∗4 , b∗1 ).
(1.24)
q = ma∗4 ↔ b∗ = mb∗1 .
(1.26)
(1.25)
i
In view of the incommensurability there is a one to one correspondence between
points of Σ∗ and points of its projection. This means that there is a one to one
correspondence between the vectors q and those of the internal space:
1
In the following, vector belonging to a 1-dimensional lattice will not be shown in bold.
8
1. Insulator to metal transition in solid Tellurium
We denote this by ∆∗b∗ = q The vectors in 1.23 satisfy this relation. In direct
space it can be seen that the intersection of the supercrystal 1.23 with position space
(τ = 0) gives the modulated crystal 1.19. Here there is a one-to-one correspondence
between points of Σ and points of its projection on internal space
n=
3
�
i=1
ni ai ↔ ∆n =
3
�
ni σi b1 .2
(1.27)
i=1
By construction the supercrystal has translation symmetry Σ. Hence its symmetry group is a (3+1)-dimensional space group, a so-called superspace group. Its
elements are pairs (gE , gI ), where gE is a three-dimensional Euclidean transformation (in the Seitz notation gE = {RE |vE }) and and gI is a 1-dimensional Euclidean
transformation {RI |vI } of the internal space. The elements gE form an ordinary
three-dimensional space group GE , called the basic space group. The pairs (RE ,
RI ) form a (3 + 1)-dimensional point group.
The method of obtaining the representations of superspace groups is completely analogous to that for three-dimensional space groups. The representations of the translation subgroup are one-dimensional and are characterised by a (3+1)-dimensional
vector k inside the unit cell of Σ∗ :
k = (kE , kI ) .
1.2.1
(1.28)
Equation of motion
Given the equilibrium position (from Equation 1.19), we can write the displacements
from equilibrium positions of the superlattice as:
�
�
�
n + rj +
f (b∗ )exp(i∆∗b∗ · n + ib∗ τ ) + un,j,τ , τ .
(1.29)
Expanding the potential energy Φ around the equilibrium position up to second
order gives:
�
Φ(τ ) =
un,j,τ · Φ(2) (n, j, n� , j � ; τ ) · un� ,j� ,τ .
(1.30)
n,j,n� ,j �
Note that the potential still depends on the ‘phase’ of the modulation. The equation
of motion for the displacementes un,j,τ becomes
�
mj ün,j,τ (t) = −
Φ(2) (n, j, n� , j � ; τ )un� ,j � ,τ (t).
(1.31)
n� ,j
The solutions are superpositions of normal modes with frequency ω: un,j,τ (t) =
un,j,τ exp(−iωt) with
�
mj ω 2 un,j,τ =
Φ(2) (n, j, n� , j � ; τ )un� ,j � ,τ .
(1.32)
n� ,j
2
According to 1.26, each b∗ of the 1-dimensional reciprocal lattice of D corresponds to a multiple
of the modulation vector in the 3-dimensional reciprocal space. Similarly the relation 1.27 means
that each b of the 1-dimensional lattice D corresponds to an element of the 3-dimensional lattice
Λ. Thus for the scalar product of 1-dimensional vectors in real and reciprocal spaces it holds
∆n · b∗ = n · ∆∗ b∗ .
1.2 Dynamical properties in Modulated crystals
9
It can been demonstrated that the displacements may be characterized by vectors
k = k1 a∗1 +k2 a∗2 +k3 a∗3 inside the Brillouin zone of the basic space group. Therefore,
the normal modes can be chosen in the form:
−1/2
un,j,τ = mj
exp(−ik · n)Uj (τ + ∆n)
(1.33)
where U (τ ) is a function of lattice periodicity D:
Uj (τ ) = Uj (τ + b),
b ∈ D.
(1.34)
Using the translation symmetry of the matrix Φ(2) , the substitution of Equation
1.33 in 1.32 gives the eigenvalues equation:
�
1/2 1/2
ω 2 Uj (τ ) =
exp(−ik · n)mj mj � Φ(n, j, j � ; τ ) · U j (τ + ∆n).
(1.35)
n,j
If the modulation were commensurate the number of coupled equation in 1.35 would
be finite, since ∆n is commensurable with b. Then number of solutions is then three
times the number of atoms in the unit cell of the superstructure. In the case of the
incommensurate crystals there are an infinite number of atoms inside the unic cell,
therefore the equation 1.35 has an infinite number of solution for every k.
For an incommensurate crystal the number of atoms is infinite, but the solutions
with vector k are in one-to-one correspondence with the solutions of the same equation with vector k − ∆∗b∗ for an arbitrary b∗ ∈ D∗ . Indeed consider for b∗ ∈ D∗
∗
the function U �j (τ ) = U k
j (τ )exp(−ib τ ). This function is a solution of 1.35 with the
same frequency and vector k − ∆∗b∗, since it holds that
∗ ∗
�
exp(−ik · n)U k
j (τ + ∆n) = exp(−i(k − ∆ b ) · n)U j (τ + ∆n).
(1.36)
The one-to-one correspondence between solution at k and k − ∆∗b∗ means that
the dispersion relation is repeated with distance ∆∗b∗. It is worth to notice that,
if k is equal to the modulation vector a∗4 and m = 1 in 1.26, the periodicity of
the dispersion relation implies that the modes at Γ = k − ∆∗b∗ have the same
frequencies of the mode at the modulation vector.
Similarly to the case for displacements in the ordinary three dimensional crystal,
the displacements fields with a given wavevector k carry a representation of the
group of k. The character of this representation is
�
�
�
χ(gE , gI ) = χ(RE )
exp i(k + ∆∗b∗) · u(j) − ib∗ vI
(1.37)
b∗ ,RI b∗ =b∗ ,j,j � =j
where u(j) = RE rj + vE − rj� . For b∗ = 0 this character is the character for the
corresponding unmodulated crystal. The dimension of the representation, belonging
to k and b∗ = 0 is 3s (being 3s the number of the atoms in the unit cell).
One has to consider also some sort of pseudo-zone centres, defined again by the
reciprocal vectors of the structure. For a simple static distortion of wavevector a∗4 ,
these vectors is expressed by the formula
k = ha∗1 + ka∗2 + la∗3 + ma∗4
(1.38)
where a∗i , i = 1, 2, 3 are the three reciprocal basis vectors of the undistorted lattice,
h,k,l, and m are integers. Thus, it may be considered that the satellite spots
with m �= 0 define additional, secondary, pseudo-Brillouin zone centers, with, in
particular, ‘satellite’ acoustic branches going to zero energy at these points.
10
1. Insulator to metal transition in solid Tellurium
1.2.2
Approximate solutions of the equations of motion
Assuming that the effects of the modulation are small, the force constant matrix
can be expanded in power series
Φ(n, j, j � , τ ) = Φ0 (n, j, j � ) + �Φ1 (n, j, j � ; τ ).
(1.39)
Keeping only the zero order of this expansion the equation of motion 1.35 reduces
to
�
1/2 1/2
ω 2 Uj (τ ) =
exp(−ik · n)mj mj � Φ0 (n, j, j � ) · U j (τ + ∆n)
(1.40)
n,j
where
Φ(n, j, j � )
doesn’t depend on τ . Solutions are
∗
∗
ν
ν
UB
(0) = AB
exp(−iB ∗ τ )
j
j
where AB
j
∗ν
(1.41)
satisfies the equation
�
�
�
∗ν
∗ν
ω02 AB
=
exp i(k + ∆∗B ∗ · n) Φ0 (n, j, j � )AB
j
j�
(1.42)
n,j �
This is exactly the equation for the polarization vector with wavevector k + ∆∗B ∗
and frequency ω0 = ω(k + ∆∗B ∗, ν) for a periodic crystal, labeling the different
phonon branches with ν. Assuming that Φ1 can be considered as a perturbation,
the eigenvalue and the eigenfunctions can then be expanded in a power series:
ω 2 = ω02 + �ω12
UB
j
∗τ
�
∗
τ
= UB
(0) +
j
B� ∗ ν �
where U B
j�
�∗ τ
�∗ τ
(0)
(1.43)
(0) are solutions of the Equation 1.41. At first order
(ω12 )B
∗ν
= 0
�B ∗ ν |Φ1 | B �∗ ν � �
(ω02 )B ∗ ν − (ω02 )B �∗ ν �
c(B ∗ ν, B �∗ ν � ) =
where
�
c(B ∗ ν, B �∗ ν � )U B
j�
∗
�∗ �
B ν |Φ1 | B ν
�
1
=
Ω
�
dτ
�
∗
(1.44)
ν
AB
(k)Φ1 (n, j, j � , τ )AB
j
j�
�∗ ν �
n,j,j �
�
∗
�∗
(k)
�
(1.45)
exp i(B − B )τ − i(k + ∆ B ) · n .
At second order
(ω22 )B
∗ν
=
� |�B ∗ ν |Φ1 | B �∗ ν � �|2
(ω02 )B ∗ ν − (ω02 )B �∗ ν �
�∗ �
∗
∗
(1.46)
B ν
The deviation from zeroth-order solutions, and hence from the behavior of the
vibrations of an ordinary crystal, occurs for the case of degeneracy of eigenvalues. In
this case, applying the perturbation theory for the degenerated states, approximate
solutions are obtained from a diagonalization of the matrix
�
�
∗
�
(ω02 )B ν
�B ∗ ν |Φ1 | B �∗ ν � � ��
�
∗
� �B �∗ ν � |Φ1 | B ∗ ν�
�.
(ω02 )B ν
1.3 Modulated spring model
11
This degeneracy occurs if ω(k+∆∗B ∗, ν) = ω(k+∆∗B �∗, ν), which may be happen
if k = 12 ∆∗b∗ for some b∗ ∈ D∗ . Then ω(k, ν) = ω(k + ∆∗b∗, ν) . This implies that
for k = 12 ∆∗b∗ there may be a gap in the energy spectrum if the matrix element
�0ν |Φ1 | b∗ ν� does not vanish.
1.3
Modulated spring model
To study the effects of the modulation on dynamical properties let us focus on a
1-dim chain of identical atoms with nearest-neighbor interactions with the spring
constants varying periodically along the chain according to [52]
�
�
��
2π
αn = α 1 − �cos
na + ϕ
(1.47)
Tmod
where � is the strength of the modulation, Tmod is the spatial periodicity of the
modulation, the distance between the undistorted atoms is a and ϕ is the phase of
the modulation with respect to the chain framework. The equations of motion for
the displacements from the equilibrium positions are
m
d2 un
= −αn+1 (un − un+1 ) − αn (un − un−1 )
dt2
(1.48)
If the ratio of the period of the modulation Tmod to the lattice spacing a is a rational
number, the 1.48 reduces to a set of N equation in N unknowns. Otherwise if the
ratio is irrational, the modulation is incommensurate and the 1.48 consists of an
infinite set of coupled equations.
1.3.1
Perturbative approach
In the ∆α � α case is possible to obtain approximate solutions to the equation of
motion using the perturbative approach described in Section 1.2.2. The Equation
1.33 becomes
un,ϕ = m−1/2 exp(−ikan)U (ϕ + qn).
(1.49)
2π
denoting with q = Tmod
the modulation wavevector. Using this equation in the
equation of motion 1.48, after replacing ϕ + qn by ν, one gets
ω 2 U (ν) =
�
α�
2U (ν) − exp(ika)U (ν − q) − exp(−ika)U (ν + q) −
m
∆α �
−
cos(ν)U (ν) − exp(ika)cos(ν)U (ν − q) + cos(ν + q)U (ν) (1.50)
m
�
− exp(−ika)cos(ν + q)U (ν + q)
At zeroth-order (∆α = 0) this equation becomes
ω 2 U (ν) =
�
α�
2U (ν) − exp(ika)U (ν − q) − exp(−ika)U (ν + q)
m
(1.51)
According to 1.41 solutions have the form
U l (ν) = Al exp(−ilν).
(1.52)
12
1. Insulator to metal transition in solid Tellurium
This yields
ω02 (k, l)
�
�
�
�
��
α
4α
2 a(k + lq)
=
2 − 2 cos(a(k + lq)) =
sin
m
m
2
(1.53)
The solutions at wavevector k are in one-to-one correspondence with the solutions
at k + lq with arbitrary l, since ω02 (k, l) = ω02 (k + l� q, l − l� ). This property is generally true for an incommensurate crystal as seen in Section 1.2.1, thus it is not a
consequence of either the model chosen or of the perturbative approach used here.
Equation 1.53 is exactly the dispersion relation for the unmodulated lattice with
wavevector k + lq and frequency ω0 (k + lq), as seen for the approximate solutions to
the equation of motion in a generic modulated crystal (see Section 1.2.2). Recalling
the general results described in Section 1.2.2, deviations from the behavior of vibrations of the unmodulated chain occurs for the case of degeneracy of eigenvalues.
Following the results of Section 1.2.2, degeneracy occurs if k = 12 q. Indeed for this
value of k it holds that ω02 ( 12 q, 0) = ω02 ( 12 q, −1). The other degeneracy at first order
(l ± 1) occurs for k = πa − 12 q, where ω02 ( πa − 12 q, 0) = ω02 ( πa − 12 q, 1) (see Figure 1.6).
The approximate solutions for these eigenvalues are
l=0
l = ±1
1/2
( 2k
M)
Γl=±1
0
q
2
0
q
�π
q
a-2
�
π
a
Figure 1.6. Eigenvalues of the equation of motion at zeroth order 1.53 for l = 0 (black
line) l = ±1 (blue lines) and in all other cases (green dashed lines). The crossing regions
of the black and blue lines represent the occurrence of the degeneracy of eigenvalues, i.e,
the regions where the larger deviations from the unmodulated lattice is expected. The
relation between the frequency of the dispersion relation at k = q and the frequency of
the extra optical modes at Γ activated by the modulation is emphasized by the black
horizontal dashed line.
ω =
2
ω02
�
∆α
∗ 1±
2
�
(1.54)
It is worth to notice that at first order (l ± 1) the optical mode at Γ has the same
frequency of the mode at q in the dispersion relation (see Figure 1.6), since 1.53
becomes
�
�
α
ω(0, ±l) =
2 − 2 cos(qa) .
(1.55)
m
1.3 Modulated spring model
1.3.2
13
Commensurate approximation
The problem of incommensurate modulation can be treated using commensurate
approximation as each irrational number may be approximated by a rational one
with a chosen precision. For rational periods (Tmod = Nba with b an integer number),
the set of equations 1.48 may be closed off to give a finite set of N equation in N
unknowns, as j takes the values 1,2,3,. . .,N. We consider now the normal mode
vibrations of the chain by substituting
un = U0n exp{i(nka − ωt)}
(1.56)
into the equations of motion. The problem is considered for a given value of the
modulation amplitude �. The system of Equations 1.48 is now
(αn + αn+1 − mω 2 )U0n − αn exp(−ika)U0n−1 − αn+1 exp(ika)U0n+1 = 0
(1.57)
There are N equations of this form. The problem then reduces to solving the secular
equation given by the tridiagonal determinant, plus extra non-zero elements at the
upper right and bottom left. The αn are given by the equation 1.47 and they take N
different values between the two extremes α{1 − �cos[2πN + ϕ]} and α{1 − �cos(ϕ)},
incrementing by 2π
N the argument of the cosine. Note that αn+N = αn (in particular
α0 = αN ). Abbreviating 1.57
γn∗ U0n−1 + βn U0n + γn+1 U0n+1 = 0
where βn = αn + αn+1 − mω 2 and γn
is
�
� β1 γ2 0
� ∗
� γ β2 γ3
� 2
� 0 γ ∗ β3
3
�
� ..
..
..
� .
.
.
�
� γ1 0 0
(1.58)
= −αn expika . The (N × N) secular equation
�
...
0
γ1∗ ��
...
0
0 ��
...
0
0 �� = 0
�
..
. βN −1 0 ��
∗
. . . γN
βN �
The secular equation is of degree N in βn , that is of degree N in ω 2 , and the solutions
are N branches of frequency with N-1 gaps in between.
14
1. Insulator to metal transition in solid Tellurium
1.3.3
Some useful examples
Let us examine two special cases that will be very helpful in the comprehension
of the dynamical properties of solid Te under high pressure. The secular equation
1.3.2 for the simple case Tmod = 3a (N=3, b=1) is
�
�
� β1 γ2 γ1∗ �
� ∗
�
� γ β2 γ3 � = 0
� 2
�
� γ1 γ ∗ β3 �
3
It gives
− w3 + 6 w2 +
9 �2 w
�3 cos (3 k)
− 9w +
+
4
2
3 �2 cos (3 k)
�3 3 �2
− 2 cos (3 k) −
−
+ 2 = 0 (1.59)
2
2
2
The solution of this equation are 3 branch with 2 gap. The folding of the monoatomic
chain dispersion curve produces one optical vibration at Γ. The modulation splits
this optical mode in two different modes, one infrared (antisymmetric) and the other
Raman active (symmetric). The dispersion of the modulated chain with q = 2π
3a is
shown in figure 1.7. The modulation opens gaps in the dispersion curves located at
k = 2q and k = πa − 2q . Moreover it leads to new modes at Γ at same frequency of
the mode at k = q along the dispersion curve, as seen in Section 1.2.1
1
2
( 2k
M)
0
unmodulated linear chain
modulated linear chain
folding
0
q
2
π q
a-2
π
a
Figure 1.7. Dispersion of the modulated linear chain for Tmod = 3a and ∆α
α = 0.1 (blue
dots). The black line is the dispersion relation of the unodulated linear chain, while the
red line represents the folding of the dispersion relation related to the modulation. As
consequence of the folding the frequency at Γ are the same of the dispersion relation at
q
q
π
π
k = q (in the special case of q = 2π
3a , q = a − 2 ). Only the gap at k = a − 2 is clearly
observable.
In order to better understand the response to the system to an inelastic x-ray or
neutron scattering experiment it is necessary to consider a value of the modulation
which lead to a large number of phonon branches, i.e, q = 2π3
10a (the same of Figure 1.6). In this case 10 phonon branches and 9 gaps are expected. The intensity
1.3 Modulated spring model
15
of the peaks in inelastic x-ray or neutron spectrum at given transfer momentum
k [8] can be calculated using the first order approximation to the inelastic cross
section. The results in the modulated linear chain for q = 2π3
10a are swhon in Figure 1.8. Among the nine gaps, only the main gap at k = πa − q is observable. The
very weak structures along the green line related to the modulation are two order
of magnitude less intense than the main peaks along the blue line. All other peaks
200
1
2
( 2k
M)
180
160
140
120
100
80
60
40
l=0
l=1
0
0
q
2
π q
a-2
20
0
π
a
Figure 1.8.
Contour plot of the expected intensity in an inelastic x-ray or neutron
∆α
experiment for a modulate linear chain with q = 2π3
10a and α = 0.1. The blu and
green line represent the zeroth-order solution (Formula 1.53) obtained applying the
perturbative approach for l = 0 and l = −1, respectively.
(l �= 0 and l �= −1) arising from the modulation have intensities too much weak to
be detected. Therefore the main features of an inelastic x-ray or neutron spectrum
are very similar to the spectrum of the unmodulated system, except for some weak
structures in the sectrum and the small gaps in the dispersion relation.
16
1.4
1. Insulator to metal transition in solid Tellurium
Tellurium under pressure: structural and electronic
phase transition.
Recent advances in x-ray diffraction techniques at high pressure have led to the
discovery of a variety of new phenomena such as structural and electronic transitions
and the onset of incommensurate phases in several materials [21, 29, 37, 48, 51, 55,
61]. In much elemental systems (e.g. I, Br, P and VI group elements S, Se, Te) the
occurrence of incommensurate phases is preceded by an insulator to metal transition
[1, 25], indicating that the electronic degrees of freedom play an important role in
the stabilization of these phases.
At ambient pressure and 300 K Te is a semiconductor with a narrow gap of 0.33
eV [9, 86]. Under pressure Te shows a transition from a semiconducting to metal
phase together with a structural transition at 4 GPa [38]. As we will see in the
Figure 1.9. Figure a): resistance isotherms for Te showing the Te-I Te-II phase change.
RII is the resistance of TeII at the temperature in question just above the transition
region. Figure b): pressure variation of the reflectivity spectra for Te under pressure.
The zero lines of the reflectivity are shifted by 0.1 for each different pressure.
next section the electronic structure plays a fundamental role in stabilizing the
crystal structure at ambient pressure, so the transition to a metallic state involves
a structural transition too. At the transition a jump of the resistivity at least of
one order of magnitude [9] is observed (Figure 1.9) and the reflectivity shows a
clear increase of the spectral weight at low frequency [86] (see Figure 1.9). While
the ambient pressure structure has been known for a long time [23], the P-T phase
diagram has recently been revised [21, 38].
At ambient temperature and over the 0-4 GPa pressure range the new and the old
phase transition schemes predict the same semiconducting phase Te-I. The structure
is trigonal (space group P32 21) and consists of infinite helical chain with three atoms
per turn in the Wyckoff positions 3b which are determined by the chain radius
1.4 Tellurium under pressure: structural and electronic phase transition.
17
(internal parameter µ) according
x1 = (µ, 0, 1/6)
x2 = (0, µ, 5/6)
x3 = (−µ, −µ, 1/2).
The chains are arranged on an hexagonal lattice and are parallel to the c-axis as
shown in figure 1.10.
According the Space Group Theory the irreducible representation of the optical
Figure 1.10. Crystal structure of trigonal Te together with a schematic representation of
the symmetry adapted modes. They lie in the ab plane in the case of the A1 , A2 and
E1 modes, while are parallel to the c axes in the E2 mode.
normal mode is Γ = A1 + A2 + 2E. Five of these six modes are Raman active;
the A1 mode and the two doubly-degenerated E modes, whereas the A2 mode and
the E modes are infrared active. We notice that the A2 and the A1 normal modes
belong uniquely to a particular irreducible representation of the crystal point group,
therefore their directions of vibration are fixed by symmetry since normal modes
coincide with the symmetry adapted coordinates. We notice that the A2 and the
A1 normal modes belong uniquely to a particular irreducible representation of the
crystal point group, their directions of vibration are fixed by symmetry since in this
case the normal modes coincide with the symmetry adapted coordinates.
The atomic motions of the A1 mode are a symmetric breathing in planes perpendicular to the c-axis. In other words they correspond to the variation of the helical
radius maintaining the symmetry of the crystal. The A2 mode is a rigid chain rotation around the c-axis leaving both bond length and bond angle unchanged. The
symmetry adapted coordinates for the E modes correspond to a axial helix compression and to a asymmetric stretching of the chain along the c-axis. The normal
modes will be a linear combination of these coordinates with weight determined
from the values of the force constants. A schematic representation of the symmetry
adapted modes is show in Figure 1.10.
Each atom is tightly bounded to two neighbors in a chain with covalent like bonds,
18
1. Insulator to metal transition in solid Tellurium
while adjacent chain are linked by much weaker van der Waals type [4]. It is interesting to note that the structure of this phase is closely related to the electronic
states. The atomic configuration is 5 S2 5 p4 . The s bands are completely filled
and nonbonding, while the p bonds, directed along three orthogonal axes, would
lead to a cubic structure. However, the p bands is two-third filled, so the lattice
is unstable against a Peierls distortion dividing the six nearest neighbor bonds into
three long and three short bonds. More exactly, the trigonal structure can be seen
as a distortion of a rhombohedral one obtained from the first when the nearest
neighbor and the next nearest-neighbor distances become equal, that is the internal
parameters µ = 13 [77]. As a consequence of the high symmetry, the electronic
states are highly degenerate and the density of states is nonzero near the Fermi
level. The distortion of the rhombohedral structure splits these degeneracies and
opens an electronic gap (0.33 eV) lowering the total energy of the system. Density
functional theory (DFT) calculations show that the total energy minimum corresponds to the maximum value of the gap [77], therefore the structural stability of
this phase and the value of the internal parameter µ are strongly influenced by the
splitting of the conduction and valence bands near the Fermi level. The pressure,
forcing the structure to be more rhombohedral like, can be used to gradually reduce
the stabilizing effect of the Peierls mechanism.
The distortion and the anisotropic bounding are related to the strong anisotropy
of the electronic properties at ambient pressure, as it could be seen from infrared
spectra. In particular, the optical response and the optical gap of the system at
ambient pressure are strongly dependent on the incident light polarization [80] (see
Figure 1.11).
On increasing the pressure above 4 GPa the old and the new transition scheme
Table 1.1. Comparison among the predicted phases at ambient temperature according
the two different phase diagrams [4, 36–38, 41, 64].
Old phase diagram
New phase diagram
P-range (GPa)
0-4
System
Te-I
P-range (GPa)
0-4
System
Te-I
4-7
Monoclinic
4-4.5
4.5-7
Te-I + Te-II
Te-II + Te-III
7-11
11-27
Orthorhombic
Rhombohedral
7-29
Te-III
> 27
Te-IV
>29
Te-IV
predict different phases [36–38] (see Table 1.1). Following the old scheme from the
Te-I phase Te enters a monoclinic phase above 4 GPa when the internal parameter
µ is still significantly different from 1/3 [41]. In this structure the atoms are four
coordinated to form a two dimensional structure which could be seen as an intermediate step in the progression with pressure from a one dimensional chain structure
to a denser packing three dimensional structure. In spite of the metallic character,
covalent like bonds with strong directional properties still remain in this phase [4].
The Γ for this phase is Γ = 5A + 4B, and, being the system non-centrosymmetric,
1.4 Tellurium under pressure: structural and electronic phase transition.
19
Figure 1.11. Optical absorption spectra for Te at ambient pressure and room temperature
�
�
for light polarized E⊥c
and E�c.
A and B modes are Raman and infrared active.
On further increasing the pressure the following orthorhombic phase, stable between 7 GPa and 11 GPa, can be obtained from the monoclinic one when the
angle becomes β = 90◦ . In this phase the optical active modes are reduced to
Γ = Ag + B1g + B3g + B1u + B2u + B3u , among them the gerade are Raman active. On further increasing pressure above 11 GPa, Te enters a rhombohedral phase
(space group R3̄m) with atom in Wyckoff position 1a (0 0 0). The lattice constant
and the angle between the unit cell vectors correspond to the nearest neighbor
distance and to the bond angle, respectively [4, 64]. At 27 GPa, when the rhombohedral angle between the axes is � 109.47, the structure become body centered
cubic (space group Im3̄m) [64]. In both the cubic and the rhombohedral phase,
Raman active modes are not expected. This means that for pressure greater than
11 GPa no Raman spectrum is expected following the ikd transition scheme.
The first doubts about this sequence of phases started to appear in literature in 2001
when ab-initio calculations suggested a high pressure phase different from that one
experimentally observed for Se. It is worth to notice that this result is particular
relevant for the high pressure Te since, at that time, several experimental findings
indicated the same pattern of transition for Se and Te under pressure [62]. Band
structure calculations proved an instability of the electronic structure of the orthorhombic phase, caused by an unsuitability of the atomic positions. This means
that the symmetry used in the calculations causes a frustration of the atomic system
through the electronic structure. Relaxing the atomic positions without imposing
any symmetry, authors found a monoclinic structure with energy per atom 5 mRyd
20
1. Insulator to metal transition in solid Tellurium
lower than the orthorombic one [30, 31]. Recently, powder and single crystal x-ray
experiments showed a different pattern for the phase transitions in crystalline Te
which partly confirmed these theoretical previsions.
The new sequence of phases proposed thus predicts that at 4 GPa trigonal Te (Te-I)
goes into a triclinic structure (Te-II) (space group I 1̄). This phase transition involves a discontinuity in the volume of about 5 %. In Ref. [38] the authors observed
a mixture of the Te-I, Te-II and of a new incommensurate modulated body-centered
monoclinic structure (Te-III) at 4.5 GPa, and a single Te-III phase only above 8
GPa. On increase the pressure above 29.2 GPa this phase transforms into the bodycentered cubic (Te-IV). Differently from previous experimental finding a distinct
rhombohedral phase is not observed at ambient temperature, and it is reported
only at high temperature [36]. In Ref. [37] the modulation wavevector is reported
to lies along the (010) direction in the reciprocal space. It is interesting to note that
the new phase proposed, instead of the orthorhombic one reported in the range of
7-11 GPa, is monoclinic in agreement with considerations on the instability related
to the band Jahn-Teller effect [12, 30].
This new high pressure scenario has a strong impact to the expected Raman spectrum. The irreducible representation for the Te-II phase is Γ = 3Ag +3Au , while the
undistorted Te-III, with only one atom per unit cell, has not optical mode. However, the modulation introduces new selection rules, allowing light to scatter not
only with phonons at k = 0 but also at k = ±qinc (see Section 1.2.1). Using the superspace group theory (equations and formalism are from Ref. [42]), the calculated
character of the representation is χ(E, 2y , my , I) = (6, 2, 0, 0), where E, 2y , my and
I denote the point group operations of the three-dimensional space-group elements.
This representation is reducible into Γ = 2 Ag + 1 Bg + 2 Au + 1 Bu , where the
two Ag modes and the Bg mode are Raman active. Therefore, in the Te-III phase
Table 1.2. Comparison among the expected number of phonon modes in the two phases.
In the coexistence regions, the ΓRaman reported are the optical modes due to new phase.
Old phase diagram
P-range (GPa)
0-4
ΓRaman
A1 + 2E
4-7
5A + 4B
7-11
11-27
> 27
Ag + B1g + B2g
∅
∅
New phase diagram
P-range (GPa)
0-4
4-4.5
4.5-7
System
Te-I
Te-I + Te-II
Te-II + Te-III
ΓRaman
A1 + 2E
3Ag
2Ag + 1Bg
7-29
Te-III
2Ag + 1Bg
>29
Te-IV
∅
Raman response is entirely due to the incommensurate modulation.
Although the incommensurate nature of this phase does not allow a direct approach
with DFT, it can be successfully used to study the mechanisms that underlie the
formation of such structures [22]. For instance, DFT calculations allow to know
the Fermi surface topology and to estimate the nesting level. A perfect nesting can
be associated to a divergence of the susceptibility which implies that an external
perturbation leads to a divergent charge redistribution as seen in Section 1.1.1. A
normal mode vibration introduces a fluctuation of the ionic charge density which
produces a softening of the phonon mode proportional to the electronic suscep-
1.4 Tellurium under pressure: structural and electronic phase transition.
21
tibility in presence of electron-phonon coupling. Thus, a divergence or simply a
peak of the susceptibility at some q can produce a frozen-in lattice distortion if the
normal mode softens down to zero frequency at that q. Therefore, in presence of
electron-phonon coupling the system is unstable with respect to the formation of a
periodically varying electron charge density.
In Ref [55] DFT calculation on the body centered monoclinic (bcm) structure, i.e,
the unmodulated structure underlying that of Te-III structure, have been compared
with the results of an inelastic X-ray scattering (IXS) experiment on the modulated
Te-III structure. Calculation shows that the Kohn anomaly (see Section 1.1.2) in
the dispersion curve along the direction of the modulation q = (0ξ0) is much more
pronounced in the TA(1) mode (see right panel of Figure 1.12). We want to stress
that, while theoretical ad experimental results agree very well in the LA mode, the
calculated frequencies for the TA(1) mode become negative at and around the nesting wave vector, indicating a greater inaccuracy of the calculated frequency for this
mode than for the other. In any case, the directions of the TA(1) mode eigenvectors
reproduce very well the displacements of the static modulation in Te-III. Therefore
the DFT results suggest the presence of a dynamical instability at a wave vector
near the modulation vector. Unfortunately, the authors were not able to experimentally determine the TA branches.
Despite this, it remains to be determined how the nesting of the Fermi surface
Figure 1.12. Left panel: Phonon dispersion relation of Te-III at 8.5 GPa for momentum
transfers �q = (0ξ0) parallel to the modulation vector. The solid symbols represent the
results for the LA phonon dispersion branch, and small open symbols mark additional
weak features observed in the IXS spectra [55]. The solid line is a guide to the eye;
the dashed line shows a sine-type dispersion relation and the dash-dotted line ∆ shows
the difference between the sine-type dispersion and the observed LA dispersion. Right
panel: Calculated phonon dispersion curves of the bcm structure, i.e, the unmodulated
structure underlying that of Te-III structure (solid symbols). Phonons with imaginary
frequencies are displayed with negative energies. The experimental results for the LA
phonon branch in Te-III at 8.5 GPa are also shown by small open symbols.
produces an instability of the electron gas. Indeed, in real systems even in presence
of strong nesting, the Peierls instability leading to the formation of charge density
waves (i.e., divergence of the electronic susceptibility) is exponentially depressed.
Chapter 2
Experimental Setup
A brief description of the experimental techniques and the relative apparatus employed in the present work is given in this chapter. Special emphasis is placed on
the diamond anvil cell for high pressure generation.
2.1
High pressure technique
Many different types of devices are available to produce high pressures. They can
mainly be divided into two broad families: those based on a piston-cylinder vessel and those in which the sample is compressed between anvils. Diamond Anvil
Cell (DAC) is the most largely diffused device for generating high static pressures.
Compared to piston cylinder devices, the diamond anvil cell is three or four orders
of magnitude less massive and generates pressure one to two orders of magnitude
higher than previous devices. Pressure of several tenths of GPa can be applied and
with particular technical implementations the Mbar range is reachable. Moreover,
the diamond transparency to the electromagnetic radiation over a wide frequency
range make this instrument a simple but powerfull tool for the spectroscopic investigation of matter under extreme condictions.
2.1.1
The high pressure diamond anvil cell
The principle of operation of the diamond anvil cell (DAC) is sketched in figure 2.1.
Two opposite cone-shaped single crystal diamond anvils with small flat top faces
(culets) squeeze a metallic plate, which has a small hole in the center [45] (figure
2.1). Together with the sample, a proper hydrostatic medium must be placed inside
the gasket in order to avoid pressure gradients in the sample chamber, at least in the
cases in which the sample is not hydrostatic itself. On applying an external force of
moderate intensity, a large pressure can be reached inside the hole, due to the ratio
between the surface of the external and the internal anvil faces, which multiplies
the external load by a large factor (∼ 103 ). The metallic plate acts as a gasket,
containing the sample and preventing the direct anvils contact (and the consequent
breakage). The friction between the diamond surfaces and the gasket results in a
coercing strain toward the hole, thus compensating the internal pressure.
The accessible pressure range using a DAC depends on several factors such as
the dimensions of the diamond and of the hole, and the material of the gasket.
23
24
2. Experimental Setup
Figure 2.1. Schematic representation of a diamond anvil cell (DAC).
Diamonds
Diamond is suitable for anvils due to its hardness and its trasparency to electromagnetic radiation, which makes the DAC well-adapted for spectroscopic studies,
such as Raman and infrared spectroscopy and X-ray diffraction [44, 45]. On the
basis of impurities concentrations, diamonds are classified as Type Ia, employed in
diffraction experiments, or IIa, characterized by a lower impurity level employed for
both Raman and IR spectroscopy, showing a low photoluminescence and an almost
flat transmittance from the far infrared to the visible region (see figure 2.2), apart
from the 1900-2600 cm−1 region where the presence of two strong multi-phononic
absorptions makes the diamonds opaque.
The diamond anvils are cut from natural, gem quality stones to have an octagonal or hexadecagonal shape, which is preferred, for they withstand the stress
gradients better. The inner face (culet) has typically a 100-800 µm diameter, while
the external face (table) diameter is 2-4 mm large. Diamond thickness is about 1-3
mm. The culet must be perfectly parallel to the table in order to prevent dangerous
strains. Great care must be taken in centering and aligning the two anvils to avoid
premature failure of diamonds at high pressure.
Gasket
The capability of containing extremely high pressure originates from the anvil-gasket
friction and its small thickness [26]. The gasket material can be either molybdenum
(for measurements in 0-5 GPa range), or stainless steal (0-10 GPa), or rhenium (050 GPa). In order to avoid large deformation and instability of the hole, the gasket
must be precompressed to a thickness only slightly greater than the final thickness,
which depends on the maximum pressure that is planned to be applied. The higher
the pressure the thinner the gasket and the smaller the sample volume. For example,
with diamond tips of 400 µm in diameter, and a maximum pressure of 40 GPa, we
have preindented the stainless steel gaskets to a thickness of about 50 µm, while
with tips of 800 µm and maximum pressure of 15 GPa, the indented thickness was
50 µm. Typically a 200µm thick foil is compressed between the diamonds until
2.1 High pressure technique
25
Figure 2.2. Type IIa diamond transmittance and reflectance in the far and mid infrared
region, from Ref. [24].
about 50-100µm thickness is reached (indentation). The outflown gasket materials
provides a massive support which assists the pressure containment. Moreover the
diamonds imprint allows to set the gasket during the loading procedure in the same
orientation as it had during indentation.
Hole
After indentation a hole is drilled in the centre of the gasket. The radius of the hole
is around a third of the radius of the flat anvil culet. In the case of our 400 µm
diameter culets we have drilled holes of 125 µm; 250 µm for the 800 µm diamond
tips. Holes are drilled with an electro-erosion machine. After careful alignment of
the preindented gasket by use of a microscope, wires of tungsten of several diameters are used to produce fine holes of the appropriate size. The wire serving as
cathode, the conducting gasket as anode, electrical sparks are repeatedly produced.
The erosion process lasts for a few minutes. Under working condition the cylindrical
sample chamber dimensions are about 100-200 µm diameter and 50µm height.
In the next section the characteristic properties of the two different cells used in the
experiments performed in the present work are summerized: the BETSA membrane
DAC, employed mainly for the Raman investigations and the opposing plate DAC,
employed for the mainly for infrared Reflectivity and Transmittance measurements.
The BETSA membrane DAC
The BETSA (France) membrane DAC, show in figure 2.1.1, is equipped with IIa
diamonds with 800µm culets diameter. The large culet diameter allows to use a
gasket with a hole diameter up to 350 µm. Using gasket made of stainless steel
(AISI 316) the 0-15 GPa pressure range can be explored. The anvils are placed on
two tungsten carbide basal plates, mounted on two tungsten carbide hemispheres.
26
2. Experimental Setup
Figure 2.3. Sketch of the piston-cylinder BETSA DAC.
The anvil alignment is guaranteed by the basal plates translation and tilt. To
make diamond faces parallel and centered to each others, they are delicately put in
contact. The parallelism is verified looking with a microscope at the interference
fringes generated illuminating with a withe light source. Fringes arise in the region
where the culet are separated by a distance multiple of λ/2 and disappear when
they are perfectly parallel. This technique ensures parallelism with an error of the
order of 10−3 rad. The mechanical stability of the piston-cylinder matching of the
two block of the cell guarantees the conservation of the alignment during the use.
The basal plates and the alignment system have optical access in form of cones.
The optical aperture is 45 degree. Force is applied using a metallic membrane,
screwed to the body (cylinder) of the cell. On increasing the helium pressure into
the membrane by means of a capillary, the piston sliding within the cylinder pushes
the upper anvil against the bottom one.
D’anvils opposing plates DAC
The opposing plates DAC producted by D’Anvils (Israel) is a compact and simple
screw driven pressure cell. Its reduced dimensions (diameter ∼ 25 mm, height ∼
12 mm in working conditions) make this cell very suitable for using in a microscopy
setup. This DAC is equipped with IIa diamonds, with 400µm culet diameters. Using
a stainless steel (AISI 316) gasket with a 150µm hole diameter pressure up to 30
GPa can be reached. A sketch of the cell is reported in figure 2.4.
Diamond anvils (5 in Fig 2.4) are fixed with glue to tungsten carbide backing
plates (3 and 6), with a 1.2 mm hole diameter and 60◦ angular opening. The upper
backing plate is inserted in a tight way in the steel platen (2) and three lateral
adjustment screws in the lower platen (8) allow to obtain the mutual centering
of the culets. Two pins (7) assure the maintenance of the centering and help in
mounting the gasket (4). Pressure is achieved by gentle tightening the tension
2.1 High pressure technique
27
Figure 2.4. Sketch of the D’anvils opposing-plate cell. Diamonds anvils (5) are glued to
the basal plates (3 and 6), mounted on two steel platens (2 and 8). (4) gasket; (7) steel
pins (7); (1) tension screw.
screws (1). Since the cell mechanism does not pin down the tilting of the platen
during compression, it is important to follow a careful procedure during pressurizing.
Before any operation such indenting or pressurizing the two anvils must be put
in contact. Parallelism is achieved by gently tightening the tension screws till the
disappearance of the Newton interference rings. The cell thickness (i.e. the distance
between the two platens) is then measured at three reference points at the platen
circumference, 120◦ apart. During indenting or pressurizing, the cell thickness must
be measured at the same three points. Keeping flat within 5 µm the difference
between the reference and the actual cell thickness at the three points ensures the
preserving of parallelism within 10−3 rad.
2.1.2
In situ pressure measurement
Sample pressure in the DAC is evaluated putting a pressure gauge in the gasket hole
next to the sample. This gauge can be a crystal of well-known equation of state, but
a faster and more convenient method is to follow the pressure shift of fluorescence
lines of a luminescent compound. The most commonly used gauge is ruby, adopted
also in this thesis for pressure determination. Ruby (Al2 O3 with Cr+3 impurities)
shows two sharp and intense fluorescence lines R1 and R2 at 692.7 nm and 694.25
nm, respectively, at ambient temperature and pressure conditions (Figure 2.5).
R1 and R2 lines exhibit a strong dependence on the interatomic distances and
thus on pressure, showing a shift toward longer wavelength. Owing to the strong
intensities of the signal, very small ruby sphere (∼ 5µm) can be used loaded in the
DAC together with the sample and used as pressure gauge. Ruby fluorescence is
typically excited by a laser (both He-Ne or argon ion lasers are proper to use) and
analyzed by a monochromator. The pressure calibration is typically performed by
means of the more intense R2 line whose pressure dependence was established and
improved during the years by extrapolating state-equations and non hydrostatic
28
2. Experimental Setup
Figure 2.5. R1 and R2 fluorescence lines of ruby (Al2 O3 with Cr+3 impurities) at ambient
pressure.
data from shock-wave experiments [58, 66]. The room temperature calibration is
reported in figure 2.6 from Ref. [58]. In the 0-30 GPa range the linear extrapolation
P = α∆λ
(2.1)
with α = 2.74GP a/nm ca be adopted (see dashed line in figure 2.6). In prefect
hydrostatic conditions the ruby fluorescence technique ensure pressure indetermination of ± 0.1 GPa in the 0-10 GPa range and ± 0.5 GPa for pressure up to
30 GPa. However pressure gradients on the sample may be larger, depending on
the pressure transmitting medium used. In the experiments presented in this work,
pressure errors has been overestimated, proportionally to the R1 line width.
Figure 2.6. Room temperature pressure calibration of the shift of the R2 ruby fluorescence
line [58]. Dashed line is the linear extrapolation in the 0-30 GPa range.
2.2 Raman scattering setup
29
Figure 2.7. Optical scheme of the micro-Raman spectrometer employed in the present
work.
2.2
Raman scattering setup
Raman spectroscopy is a useful technique in order to obtain information about
lattice dynamic, and about the modifications of the phonon spectrum induced by
doping, pressure, or temperature. As mentioned above, diamonds are very suitable
for this type of spectroscopy owing to their large transparency in the visible light
region. The only disadvantage is the presence of the strong one-phonon Raman
contribution at around 1300 cm−1 which prevents the spectrum reliability in the
1200-1500 cm−1 range.
In this work, Raman spectra have been collected in back-scattering geometry
by means of a commercial LABRAM Infinity micro-spectrometer by Jobin Yvon
equipped with an adjustable optical notch filter, an optical microscope and a cooled
low noise multichannel charge coupled device (CCD) detector (1024 X 256 pixel).
A 16mW He-Ne laser (λ=632.8 nm) has been used. If necessary, the incident power
can be attenuated by filters placed in a filter wheel. The laser beam is reflected by
the notch filter toward the microscope. Optical objectives with 10x to 50x magnifications can be used to focus the beam onto the sample and collect the backscattered
light. The latter is sent again to the notch-filter, which in transmission rejects the
elastically scattered light from sample and optics (see figure 2.7). The low frequency
cutoff within this setup is at about 100 cm−1 . Before entering the monochromator,
30
2. Experimental Setup
Figure 2.8. A small Te crystal loaded in the gasket hole with ruby chips into a NaCl pellet
for high pressure Raman measurements. Culet diameter is 400 µm. The picture has
been taken in reflection, on the left, and transmission configuration, on the right.
the light is focused into an adjustable pinhole confocal with the sample, in order to
reduce the scattering volume along the optical axis. The monocromator is equipped
with two gratings (600 and 1800 lines/mm), that can be remotely rotated in order
to select the spectral range. Photons are detected by a CCD device and analyzed
by a computer. Frequency calibration is performed using the known emission lines
from a Ne lamp.
The backscattering geometry does not allow to collect spectra at very low Raman
shift. However this setup has the great advantage that a confocal microscope can be
used, thus allowing to easily collect Raman spectra from very small samples. The
laser spot diameter is about 5 µm using the 20x objective and about 2 µm using
the 50x. The pinhole aperture can be adjusted so to obtain a scattering volume of
only a few µm across. Moreover focusing and aligning can be set by means of a
camera. These characteristic of the Raman setup allow measurements as a function
of pressure using a DAC.
Both the BETSA and the D’Anvils DACs described in section 2.1.1 were employed for Raman measurements. A 22 mm focal length 20x objective was used in
order to fit the angular aperture and the working distance of the cells. The pinhole
aperture was adjusted in order to reduced the diamond fluorescence background
contribution. 1800 lines/mm grating was used, obtaining a resolution of about 3
cm−1 . Stainless steel gasket with about 250 µm (BETSA) and 150µm (D’Anvils)
diameter holes were used respectively. The sample, in form of finely milled powder
or of a small single crystal, was placed on top of a pre-sintered NaCl pellet into
the gasket hole. This procedure ensures good hydrostatic conditions and the high
thermal conductivity of diamond prevents strong heating of the sample impinged by
the laser spot. A small ruby chip was also placed inside the gasket hole for pressure
determination. Ruby fluorescence was directly measured by means of the Raman
spectrometer. In figure 2.8 a picture of a sample loaded in the gasket hole is shown.
2.2.1
High pressure and temperature setup
In order to pressurize liquid samples at high temperature a High-Temperature Diamond Anvil Cell (HTDAC) properly designed and constructed by LOTO Instruments (Florence Italy) for high temperature work has been employed. The anvils
2.2 Raman scattering setup
31
(low fluorescence IIA diamonds, 600 µm culet diameter) were glued to their WC
seats and externally covered with a graphite-glue mixture in order to avoid the presence in Raman spectra of black body radiation emitted by the heated coils. Mica
sheets, ceramic spacers and ceramic fibre materials have been used to thermally
insulate the mechanical body of the cell from the high-temperature core. A small
heater made of few platinum-rhodium coils (7-8 mm diameter) has been indeed located closely around the diamonds. The HTDAC was kept under vacuum in a steel
vessel equipped with a thin quartz window to avoid damage of the diamonds and
reduce heat dissipation.
A sheathed thermocouple was kept in thermal contact with the gasket and used
only to drive the thermoregulator since its reading can differ of more than 60 K
from the real sample temperature. The latter was obtained by means of a new
specifically developed differential ruby fluorescence technique. This technique yields
a simultaneous and accurate temperature and pressure gauge (±5K and ±0.1GP a)
using the measured frequency shift of the ruby fluorescence lines of in principle only
two ruby nanospheres located one inside and the other outside the gasket hole and
the known temperature and pressure calibration curves of the ruby florescence line
[70].
This technique neglects temperature gradients across the diamond itself and
yields, from the fluorescence spectra ∆v of ruby nanospheres placed in the gasket
hole and on the external surface of the diamond (at zero pressure), a simultaneous
temperature and pressure measurement. Existing temperature and pressure calibration curves of the ruby florescence line up to 600◦ C and 150Kbar have been used
[70]:
∆v = α∆T + β∆P
(2.2)
so it is possible to estimate
∆T
=
∆P
=
∆ν̃ (out)
−α
∆ν̃ (in) + α∆T
−β
with an error, due to the indetermination on ν̃, equal to ±7◦ C and ±1.3Kbar.
Zero pressure heating on a LiF sample allows us to evaluate 10K to be, at
the highest temperature investigated, an upper limit for the thermal gradient between the opposite faces of the diamond. Owing to the difficulty in driving independently pressure and temperature on the sample we have performed several
pressure-temperature cycles. In particular, most of the measurements reported in
Section 5.4 have been collected following a standard thermodynamic path:
• the solid sample was first pressurized and then heated until the liquid was
obtained,
• the pressure on the liquid was released up to Pmin keeping constant the temperature,
• heated at Pmin ,
• finally increasing the pressure the solid was recovered.
32
2. Experimental Setup
The absence of sample contamination after each pressure temperature cycle can be
verified by comparing the spectrum of the final solid sample with the first spectrum
collected at the sealing pressure.
2.3
Infrared setup
Infrared spectroscopy is a suitable investigation technique for studying charge delocalization processes, providing information on the low frequency electron dynamics.
Moreover the possibility to probe the sample in a DAC both reflection and transmission configuration makes this technique very powerful to study pressure dependence
of the transport properties of the systems. Transmittance measurements provide
directly information on absorption band, allowing for a simple analysis. The limit
of this technique originates by the saturation of spectra in dealing with high absorbent samples. On the contrary high pressure reflectivity measurements can be
also performed on strongly absorbent samples. A data collection on a wide frequency range allows to extract the optical conductivity via the Kramers-Kroenig
(KK) analysis. Moreover the standard analysis must be corrected to account for the
diamond-sample interface. However no conventional sources have been used in those
experiments because of the high brilliance required due to the small dimensions of
samples. The use of synchrotron radiation is then required and the fact that it is
a broad band source allows to measure in each spectral range only by changing the
beam-splitters.
High pressure infrared measurements presented in this work have been carried
out at the IR beamline SISSI (Source for Imaging and Spectroscopic Studies in
the Infrared) of the synchrotron laboratory ELETTRA in Trieste, Italy [56]. This
infrared source provides high brilliance from the far infrared to the visible frequency
region. As shown in figure 2.9 the radiation is collected from bending magnet and
enters the first vacuum chamber, which hosts the plane extraction mirror M1 and
the ellipsoid mirror M2 which focuses the radiation beyond the shielding wall of the
synchrotron hall in the intermediate focal point F1. The second vacuum chamber
hosts the third M3 (plane) and fourth M4 (ellipsoidal) mirrors, in a symmetric
optical configuration with respect to M1 and M2. This optical design permits the
aberrations by the transfer optics due to the wide emission angle to be minimised.
M4 focuses the infrared radiaton on the diamond window (point F2), which is
the last UHV component of the first branch. The experimental station consists
of a Bruker IFS-66v Michelson interferometer coupled to a Bruker Hyperion-2000
cassegrainian infrared microscope equipped with a 15x objectives, a nitrogen cooled
MCT detector and a Mylar (or a KBr) beam splitter (BS). Within this setup the
300-8000 cm−1 frequency range can be explored.
The screw clamped D’Anvil cell described in section 2.1.1 has been used, in
order to fit the vertical acceptance of the IR microscope. NaCl and KBr cubic
salt were both used as pressure transmitting medium. As for the FIR transmission
measurements, the samples were finely milled and pressed between the diamond
anvils in order to obtain a compact and thin sample slab with well defined thickness
ranging between 2 and 5 µm. Eventually this procedure ensures sharp surfaces, that
are important also for reflectivity measurements, where the quality and the number
of interfaces are critical parameters.
2.3 Infrared setup
33
Figure 2.9. The layout of the SISSI (Source for Imaging and Spectroscopic Studies in the
Infrared) beamline at ELETTRA
Figure 2.10. Scheme of the optical head employed for the ruby fluorescence measurements
at SISSI.
The sample slab was carefully shaped for fitting inside the gasket hole and then
placed on top of a pre-sintered NaCl (KBr) pellet, together with a ruby chip. The
present loading procedure enables the sample slab to maintain a direct contact with
the diamond surface on the side where reflectivity spectra are collected and ensures a
net interface. Moreover this simple multi-layer structure (diamond/sample/pressure
medium/diamond), assures a rather easy theoretical modeling of trasmittance and
reflectivity spectra, thus allowing the calculation of the optical conductivity by
means of a local analysis, instead required the non-local KK analysis (see appendix
B).
The measuring procedure is the following: the DAC is mounted over a home built
holder that ensure to place the DAC always in the same position. This holder is
fixed to the microscope sample stage that allows to finely align the DAC. Adjustable
apertures mounted in the microscope are then fixed so to collect only the signal
from the sample placed inside the DAC. The apertures are kept the same for all the
experimental run. Moreover, mirrors that couple the microscope and interferometer
34
2. Experimental Setup
are aligned so to couple the focus of the IR radiation to the visible focus of the white
light of the microscope, so to be sure that aligning by eye the DAC, exactly the
sample signal is then detected. The focus has to be kept fixed for all the experimental
run.
Ruby fluorescence was measured by means of a spectrometer composed by a
TRIAX monochromator (by Jobin Yvon) equipped with a 1800 lines/mm grating
and a CCD detector whose output signal is analyzed by a dedicated computer.
Resolution was about 3 cm−1 . An optical head represented in figure 2.10 was used
to focus the 514 nm line of a 100 mW Ar ion laser into the DAC and to collect
the backscattered signal without removing the DAC from the sample compartment.
The laser is sent to the objective via optical fiber. The ruby fluorescence collected
by the objective is sent by a notch beam-splitter to the outgoing fiber which is
coupled to the spectrometer.
Measurements are taken first in the mid-IR then the BS is changed and the
near-IR and the visible are measured. In each spectral range IR data are collected
over the sample and also some reference spectra are acquired.
At the end of the each pressure run we measured the light intensities reflected by
Au (ω) and by the external face I D� (ω)
a gold mirror placed between the diamonds IR
R
�
D (ω)/I D (ω) as a correction function, we
of the diamond anvil. By using the ratio IR
R
achieved the reflectivity R(ω):
R(ω) =
�
S (ω)
D (ω)
IR
IR
·
Au (ω) I D (ω)
IR
R
(2.3)
The transmittance T (ω) is obtained using:
T (ω) =
�
D (ω)
ITS (ω)
IR
·
D (ω)
ITDAC (ω) IR
(2.4)
where ITDAC (ω) is the transmitted intensity of the empty DAC without gasket and
with the anvils in tight contact.
Chapter 3
Density functional theory
Density functional Theory (DFT) is a quantum mechanical theory used in physics
to investigate the ground state properties of many-body systems. DFT has proved
to be highly successful in describing structural and electronic properties in a vast
class of materials, ranging from atoms and molecules [5] to simple crystals to complex extended systems (including glasses [78] and liquids [73]) . This method has
the double advantage of being able to treat many problems to a sufficiently high
accuracy, as well as having a relatively low computational cost. For these reasons
DFT has become a common tool in first-principles calculations aimed at describing
- or even predicting - properties of molecular and condensed matter systems.
3.1
The Hohenberg-Kohn Theorem
Let us consider a system of N interacting (spinless) electrons under an external
potential V(r) (usually the Coulomb potential of the nuclei). If the system has
a nondegenerate ground state, it is obvious that there is only one ground-state
charge density n(r) that corresponds to a given V(r). In 1964 Hohenberg and
Kohn [40] demonstrated the opposite, far less obvious result: there is only one
external potential V(r) that yields a given ground-state charge density n(r).
A straightforward consequence of the first Hohenberg and Kohn theorem is that
the ground state energy E is also uniquely determined by the ground-state charge
density. In mathematical terms E is a functional E[n(r)] of n(r). We can write
E[n(r)] =< Ψ|T + U + V |Ψ >=< Ψ|T + U |Ψ > + < Ψ|V |Ψ >=
�
= F [n(r)] + n(r)V (r)dr
(3.1)
where F [n(r)] is a universal functional of the charge density n(r) (and not of
V(r)). For this functional a variational principle holds: the ground-state energy is
minimized by the ground-state charge density. In this way, DFT exactly reduces
the N-body problem to the determination of a 3-dimensional function n(r) which
minimizes a functional E[n(r)].
It is convenient to rewrite the energy functional as follows:
�
E[n(r)] = T0 [n(r)] + Exc [n(r)] + EH + n(r)V (r)dr.
(3.2)
35
36
3. Density functional theory
T0 [n(r)] is the kinetic energy of non-interacting electrons:
�
h̄2 �
T0 [n(r)] = −
2
ψi∗ (r)∇2 ψi (r)dr.
2m
(3.3)
i
The term EH [n(r)] (called the Hartree energy) contains the electrostatic interaction
between clouds of charge:
�
e2
n(r)n(r �)
EH [n(r)] =
drdr �.
(3.4)
2
|r − r �|
(being e the elementary electron charge), Exc is the so called exchange-correlation
contribute to the total energy. This term accounts for the difference in the kinetic energy between the interacting e non-interacting system and for the electron-electron
interaction contribution to the total energy neglected in the Hartree term.
3.2
The Kohn-Sham equations
Using the Hohenberg-Kohn theorem the problem of N interacting (spinless) electrons
under an external potential can be reformulated in a more convenient form. The
system of interacting electrons is mapped into a fictitious system of non-interacting
electrons having the same density. For a system of non-interacting electron the
charge density is
�
n(r) = 2
|ψi (r)|2 ,
(3.5)
i
where the i runs from 1 to N/2 and the orbitals ψi (r) are solution of the Kohn-Sham
equations [50]
�
�
h̄2 2
−
∇ + VKS (r) ψi (r) = �i ψi (r)
(3.6)
2m
(m is the electron mass) obeying the orthonormality constraints:
�
ψi (r)ψj (r)dr = δi,j
(3.7)
It is worth to notice that the Equations 3.6 are the Schrödinger equation of N particles in an eternal one body potential VKS (r). The Hohenberg and Kohn theorem
ensure the existence of such unique potential having n(r) as its ground state chargedensity. Imposing that for any arbitrary variation of the orbital ψi (r) under the
orthonormality constrains of Equation 3.7 the variation of the total energy E must
vanish, one obtain
VKS (r) = VH (r) + Vxc [n(r)] + V (r)
(3.8)
where V (r) is the external potential, VH (r) the Hartree potential
�
n(r �)
2
VH (r) = e
dr �
|r − r �|
(3.9)
and Vxc [n(r)] the exchange-correlation potential
Vxc [n(r)] =
δExc
.
δn(r)
(3.10)
The knowledge of this functional would lead to the exact density and the exact
ground state properties of the system of the N interacting electrons.
3.3 Approximation for the exchange-correlation energy
3.3
37
Approximation for the exchange-correlation energy
The simplest widely used expression for the exchange-correlation term 3.10 is the
local density approximation (LDA), which takes the exchange-correlation energy
density �(nr) at each point in the system to be equal to its known value for a
uniform interacting electron gas of the same density. In this approximation the
exchange-correlation contribute to the total energy writes
�
�
Exc [n(r)] = �xc (n(r))n(r)dr = �hom
(3.11)
xc (n(r))n(r)dr
where the �hom
xc (n(r)) is the energy density of the homogeneous electron gas, which
dependence upon the density is known from Quantum Monte-Carlo techniques. In
this approximation the exchange-correlation potential becomes
�
�
δExc
d�xc (n)
Vxc [n(r)] =
= �xc + n
(3.12)
δn(r)
dn
Even in this simplest approximation, DFT gives quite satisfactory results even in
inhomogeneous system (like molecules) for which an approximation based on the
homogeneous electron gas would not look appropriate. In less homogeneous system
like solids, LDA gives in most case the right structural and vibrational properties. The correct crystal structure is usually found to have the lowest energy; bond
lengths, bulk moduli, phonon frequencies are typically accurate within a few percent. One of the reason of this surprising results is that all the terms, except Exc
term, in Equation 3.2 can be evaluated exactly, and that the terms Ecx that hide
the ignorance about the system is often small. A deeper reason of this success is
that LDA give a good description of the spherical term of the “exchange-correlation
hole”, i.e., of the the charge missing around a given point due to exchange effects
(Pauli antisymmetry) and to Coulomb repulsion.
To improve upon the local-density approximation in density functional theory
calculations, various generalized gradient approximation (GGA) for the exchangecorrelation energy functional can be introduced. Their general form is
�
Exc [n(r)] = �hom
(3.13)
xc (n(r), ∇n(r))n(r)dr,
the difference with respect to LDA is the additional local dependence of the integrand on the gradient of the electron density n(r). The dependence of �hom
xc (n(r), ∇n(r))
from the local gradient of the density is chosen such a way to fit the known exchangecorrelation energy of selected atoms in their ground state. Gradient-corrected functionals yield much better atomic energies and binding energies than LDA, at a
modest additional computational cost. Moreover, they yield a good description of
the equilibrium structure of II-VI semiconductors and of the weak bond in elements
of group VIb.
3.4
The band gap problem
One of the most shortcoming of the DFT is its inability to reproduce with an acceptable accuracy the true band gap in solids which is usually strongly underestimated
38
3. Density functional theory
(up to 50%). The reason for the “band gap problem” lies in the dependence of
the exact energy functional upon the number of electrons and in the inability of
approximate functionals to reproduce it. To better understand this effect it is necessary to generalize the DFT theory to the fractional number of electrons case. The
variational formulation of the DFT
�
�
�
δ
E[n(r)] − µ n(r)dr = 0
(3.14)
δn
can be extended to the fractional particle numbers
�
N = n(r)dr = Z + ω, with Z = integer,
0≤ω≤1
(3.15)
with an opportune definition of the functional E[n(r)], as proposed in Ref. [65].
With this definition the variational principle is well defined and leads to the Euler
equation:
δE[n(r)]
= µ.
(3.16)
δn(r)
∂E(N )
.
(3.17)
∂N
Within this framework the system total energy as function of the number of electron
can be written as
µ(N ) =
EZ (N + ω) = (1 − ω)EZ (N ) + ωEZ (N + 1)
(3.18)
EZ(Z+ω)
and is shown in Figure 3.1. We use the notation EZ (M ) to indicate the total energy
of a system with nuclear charge Z and M electrons. According to Equation 3.18,
I(Z)
EZ(Z)
A(Z)
-1
0
ω
1
2
Figure 3.1. Ground state energy of an atom with nuclear charge Z and Z+ω electrons.
A(Z) and I(Z) are the electron affinity and the ionization potential, respectively (see
text).
the ground state energy of a system (with nuclear charge Z) plotted as function
3.4 The band gap problem
39
of the electron number N consist of straight line segments with possible derivative
discontinuities at integer values of N (see Figure 3.1). This means that the chemical
potential µ can change discontinuously at integer N
�
�
−I(Z) for Z − 1 < N < Z
µ(N ) =
(3.19)
−A(Z) for Z < N < Z + 1
Since the Euler Equation 3.16 is satisfied, the same discontinuity will show up
in the functional derivative
δE[n(r)]
,
(3.20)
δn(r)
if N passes through an integer. Derivatives discontinuities of this kind are of particular importance for the discussion of the band gap of an insulator or semiconductor.
The band gap is rigorously defined as the difference between the lowest conduction band energy and the highest valence band energy. The latter is the energy
required to remove an electron from the insulating Z-particle ground state to infinity; the former is obtained by adding one electron to the insulating Z-particle ground
state, denoting with Z the nuclear charge. Thus, in term of ionization potential
and the electron affinity
I(Z) = EZ (Z − 1) − EZ (Z)
(3.21)
A = EZ (Z) − EZ (Z + 1),
(3.22)
∆ = I − A.
(3.23)
∆ = µ(Z − δ) + µ(Z + δ).
(3.24)
the band gap ∆ is defined by
Using the Equation 3.19 the band gap can be written as
By means of the Euler equation 3.16
�
δE[n(r)] ��
µ(N ) =
δn(r) �N
(3.25)
the band gap is rigorously given as
�
�
�
�
δE[n(r)] ��
δE[n(r)] ��
∆=
−
δn(r) �Z+δ
δn(r) �Z−δ n=n
(3.26)
0
denoting with n0 the ground state density of the Z-electron insulator. Using the
representation 3.2 of the total energy functional, one obtains
�
�
�
�
δT0 [n(r)] ��
δT0 [n(r)] ��
∆=
−
+
δn(r) �Z+δ
δn(r) �Z−δ n=n
0
�
�
�
�
(3.27)
δExc [n(r)] ��
δExc [n(r)] ��
−
δn(r) �
δn(r) �
Z+δ
Z−δ
n=n0
For a non interacting system, the gap is readily calculated as
∆ = �Z (Z + 1) − �Z (Z)
(3.28)
40
3. Density functional theory
where the �Z (n) denotes the the n-th single particle level of the Z-particle system.
Moreover in this case Equation 3.27 reduces to
�
�
δT0 [n(r)] ��
δT0 [n(r)] ��
∆=
−
.
(3.29)
δn(r) �Z+δ
δn(r) �Z−δ
If evaluated at the interacting density n0 this expression yields the Kohn-Sham gap
�
�
�
�
δT0 [n(r)] ��
δT0 [n(r)] ��
∆KS =
−
=
δn(r) �Z+δ
δn(r) �Z−δ n=n
(3.30)
0
KS
�KS
Z (Z + 1) − �Z (Z).
Thus in general the band gap can be written as
∆ = ∆KS + ∆XC .
(3.31)
This expression allows to explain the typically band gap underestimation of the
DFT calculations. Indeed the ∆KS is not, by construction, equal to the true gap,
because it is missing the term ∆XC coming from the discontinuity derivatives of the
exchange-correlation functionals. This is absent by construction from any current
approximated functional (be it LDA or gradient-corrected). There is some evidence
that this missing term is responsible for a large part of the band gap problem, at
least in common semiconductors. Despite this systematic error in the band gap
value, its dependence on an external variable (e.g. the pressure) is usually well
reproduced.
3.5
Plane waves and pseudopotentials
In order to apply the DFT theory the Kohn-Sham orbitals in Equation 3.6 are
usually expanded into some suitable basis set. One of the most convenient choice is
the plane-waves (PW) basis set. An element of the basis in the case of crystalline
systems is defined as
< r|k + G >=
1
expi(k+G)·r
V
with
h̄2
|k + G|2
2m
(3.32)
where V is the volume of the system, Ecut the cutoff of the kinetic energy of PW,
k a vector in the Brillouin Zone (BZ) and G is a reciprocal vector of the crystal.
PW have many attractive features: they are simple to use (matrix elements of
the Hamiltonian have a very simple form), orthonormal by construction, unbiased
(there is no freedom in choosing PW’s: the basis is fixed by the crystal structure
and by the cutoff) and it is very simple to check for convergence (by increasing the
cutoff).
Unfortunately this basis is not able to reproduce the localized function of inner
(core) states with a reasonable number of elements. Nevertheless the inner (core)
states do not contribute in a significant manner to the chemical bonding and to solidstate properties. In reality it can been shown that considering the core electron as
frozen leads errors in the energy functional only at second order. The problem is
then reformulated in terms of pseudopotentials. A pseudopotential is a fictitious
3.6 Density functional perturbation theory
41
electron-ion interaction potential, acting on valence electrons only, that mimics the
interaction with the inner electrons-which are supposed to be frozen in the core-as
well as the effective repulsion exerted by the latter on the former due to their mutual
orthogonality.
Modern normconserving pseudopotentials [35] are determined uniquely from the
properties of the isolated atom, while the requirement of norm conservation ensures
an optimal transferability. By the latter expression one indicates the ability of the
pseudopotentials to provide results whose quality is to a large extent independent of
the local chemical environment of the individual atoms. The normconserving pseudopotentials are atomic potentials which are extracted from ab-initio calculations
of the true atom in a such way to mimic the scattering properties of the true atom
and to fulfill the following condiction:
• all-electron and pseudo-wavefunctions must have the same energy;
• all-electron and pseudo-wavefunctions must be the same beyond a given core
radius (rc )
• the pseudo-charge and the true charge contained in the region r < rc must be
the same.
Experience has shown that PP’s are practically equivalent to the frozen core approximation within an all-electron approach: PP and all-electron calculations on
the same systems yield almost indistinguishable results (except for those cases in
which core states are not sufficiently frozen).
3.6
Density functional perturbation theory
A wide variety of physical properties of solids depend on their lattice-dynamical
behavior: infrared, Raman, and neutron-diffraction spectra; specific heats, thermal
expansion, and heat conduction; phenomena related to the electron-phonon interaction such as the resistivity of metals, superconductivity, and the temperature
dependence of optical spectra are just a few of them. As a matter of fact, their
understanding in terms of phonons is considered to be one of the most convincing
pieces of evidence that our current quantum picture of solids is correct. The relationship between the electronic and the lattice dynamical properties of a system is
important not only as a matter of principle, but also because it is only by exploiting these relations that it is possible to calculate the lattice-dynamical properties
of specific systems from ab-initio electronic properties calculation.
The basic approximation which allows one to decouple the vibrational from the
electronic degrees of freedom in a solid is the adiabatic approximation of Born and
Oppenheimer [10]. Within this approximation, the lattice-dynamical properties of
a system are determined by the eigenvalues E and eigenfunctions of the Schrödinger
equation:
� �
�
h̄ ∂ 2
−
+ E(R) Φ(R) = �Φ(R)
(3.33)
2MI ∂R2I
where RI is the coordinate of the Ith nucleus, MI its mass, R ≡ {RI } the set
of all the nuclear coordinates, and E(R) the clamped-ion energy of the system,
42
3. Density functional theory
which is often referred to as the Born-Oppenheimer energy surface. In practice,
E(R) is the ground-state energy of a system of interacting electrons moving in the
field of fixed nuclei, whose Hamiltonian-which acts onto the electronic variables and
depends parametrically upon R-reads
HBO (R) = −
� ZI e2
h̄ � ∂ 2
e2 �
1
+
−
+ EN (R)
2m
2
|r i − r j |
|r i − Rl |
∂r 2i
i
i�=j
(3.34)
il
where ZI is the charge of the Ith nucleus, e is the electron charge, and EN (R) is
the electrostatic interaction between different nuclei:
EN (R) =
e2 � ZI ZJ
.
2
|RI − RJ |
(3.35)
I�=J
The equilibrium geometry of the system is given by the condition that the forces
acting on individual nuclei vanish:
FI ≡ −
as
∂E(R)
= 0.
∂RI
(3.36)
In crystalline solids, the nuclear positions labeled by an index I, can be rewritten
RI = Rl + τ s + uls
(3.37)
where Rl is the position of the l-th unit cell in the Bravais lattice, τs is the equilibrium position of the atom in the unit cell, and usl indicates the deviation from
equilibrium of the nuclear position. In the harmonic approximation the BornOppenheimer energy surface can be expanded as
E(R) = E0 +
1 �
Φlsα,l� tβ ulsα ul� tβ
2
� �
(3.38)
lsα,l tβ
where ulsα denotes the displacement for atoms s in the unit cell l, α and β are
Cartesian components and Φlsα,l� t� β is the force constant matrix, given by the double
derivative ∂ 2 E(R)/∂ulsα ∂ul� s� β evaluated with all atoms at their equilibrium positions. The vibrational frequencies are eigenvalues of the dynamical matrix Dsα,tβ (q)
defined as:
Dsα,tβ (q) = √
�
1
Φlsα,l� tβ exp[iq(Rl� + τt − Rl − τs )]
Ms Mt l
(3.39)
where Rl� + τt represent the equilibrium position of atom t in the primitive cell l’,
and the sum is over the infinite number of primitive cells in the crystal.
The simplest method to calculate the phonon frequencies is the so called “frozen
phonon method” and is based on the definition of the force constant matrix. The
elements of this matrix Φlsα,l� tβ are related to the forces Flsα on a generic atom s
in the cell l when the atom t in the cell l’ is displaced along the Cartesian direction
β by the relation
Flsα = Φlsα,l� tβ ul� tβ .
(3.40)
3.7 Short notes on classical molecular dynamics simulations
43
Thus the elements of the force constant matrix can obtained calculating the forces
on all the atoms when an atom is displaced from its equilibrium position using the
Hellman-Feynman theorem [27, 39]:
Flsα
�
�
�
�
� ∂H �
∂E(R)
�
�
=−
= − Ψ(R) �
Ψ(R)
∂ulsα
∂ulsα �
(3.41)
where Ψ(R) is the electronic ground-state wave functions of the Born-Oppenheimer
Hamiltonian HBO of the system with the atom t in the unit cell l’ displaced from
its equilibrium position. The principal limitation of this approach is the unfavorable scaling of the computational workload with the range of the interatomic force
constants, RIF C . In fact, the calculation of interatomic force constants within the
frozen-phonon approach requires the use of supercells whose linear dimensions must
SCF ∼ R3
be larger than RIF C , thus containing a number of atoms Nat
IF C . As the
computer workload of standard DFT calculations scales as the cube of the number
of atoms in the unit cell, the cost of a complete interatomic-force-constant calcula9
tion will scale as 3Nat RIF
C , where Nat is the number of (inequivalent) atoms in the
elementary unit cell (the factor of 3 accounts for the three generally independent
phonon polarizations).
An alternative approach is to use the perturbative approach to the Density
Functional Theory (PDFT), as described in [6], which avoids the use of supercells.
Indeed the Hellmann-Feynman theorem at second order can be used to calculate
the Hessian of the Born-Oppenheimer energy surface
�
�
�
�
�
�
∂ 2 E(R)
∂2H
�
�
= Ψ(R) �
Ψ(R) +
∂ulsα ∂ul� tβ
∂ulsα ∂ul� sβ �
�
�
�
�
�
� �
�
� ∂H � dΨ(R)
dΨ(R) �� ∂H ��
�
�
Ψ(R) + Ψ(R) �
dul� sβ � ∂ulsα �
∂ulsα � dul� sβ
(3.42)
The calculation of the derivative of the Kohn-Sham orbitals with respect to the
atomic displacements of atom from their equilibrium position can be done with the
standard first order perturbation theory [33]. The second derivatives of the BornOppenheimer energy surface can be done directly in the reciprocal space obtaining
the dynamical matrix defined in Formula 3.39. Moreover the choice of the atomic
displacements can be done in a such a way to obtain a dynamical matrix which is
block diagonal.
A Born-Oppenheimer ab-initio molecular dynamic simulations is a classical dynamic simulation carried out performing a DFT ground state calculation at each
simulation step in order to calculate the force among the atoms using the HellmannFeynman theorem (Equation 3.41). Let us then review some basic concepts of a
classical molecular dynamics simulation.
3.7
Short notes on classical molecular dynamics simulations
Assuming the validity of the Born Oppenheimer approximation (see Section 3.6)
and that ions behave as classical particles, their dynamics can be described by the
44
3. Density functional theory
Euler-Lagrange equations:
d ∂L
∂L
−
= 0,
dt ∂ Ṙi ∂Ri
Pi =
∂L
∂ Ṙi
(3.43)
The equations of motion have to be discretized and numerically solved in a molecular
dynamic simulation. The discrete interval of time is called time step. A sequence
of atomic coordinates and velocities is generated starting from a suitable initial set
of coordinates and velocities.
Let us consider an observable quantity A = A({R}, {P }) function of the set of
the phase space canonical variables {R} and {P }. The observable value of A is
thus given by the ensemble average
�
< A >= ρ(R, P )A(R, P )dRdP
(3.44)
where ρ is the probability of a microscopic state compatible with the constraints used
to describe the macroscopic system. Under the ergodic hypotheses the ensemble
averages can be calculated using
< A >= lim AT
T →∞
where
1
AT =
T
�
T
A(R(t), P (t))dt
(3.45)
(3.46)
0
is the time average of the quantity A. From a practical point of view, the calculation of the thermodynamical averages in classical molecular dynamics simulation is
carried out by means of time average over a sufficient length of time.
3.7.1
Microcanonical and canonical ensemble
Let us consider a system of N atoms in fixed volume V and fixed energy E (microcanonical ensemble). In this case the system’s Lagrangian is
L=
1�
Mi R̈i − V ({R})
2
(3.47)
i
where V ({R}) is the interatomic potential energy. The numerical solution (integration) of the equation of motions 3.43 is generally performed using the Velocity
Verlet algorithm:
δt
[f (t) + f i (t + δt)]
2Mi i
δt2
Ri (t + δt) = Ri (t = +δtV i (t) +
f (t).
2Mi i
V i (t + δt) = V i (t) +
(3.48)
In spite of his simplicity this algorithm is efficient and numerically stable. In particular, it yields trajectories that conserve to a very good degree of accuracy the
energy E.
We are often interested in systems with N particles in a fixed volume V in
thermal equilibrium with a thermal bath at temperature T (canonical ensemble). A
3.7 Short notes on classical molecular dynamics simulations
45
correct sampling of the canonical ensemble can be done by introducing a suitable
thermostat that simulates the contact with a thermal bath at temperature T. A wellknown one is the Nosé-Hoover thermostat. This introduces an extended Lagrangian
adding a fictitious degree of freedom, producing a dynamical friction force having
the effect of heating ions when the kinetic energy is lower than the desired value,
cooling them in the opposite case. The equations of motion become
fi
− ζ̇ Ṙi
Mi
�N
�
1 �
2
ζ̈ =
Mi Ṙi − 3N kb T
Q
R̈i =
(3.49)
i=1
where Q plays the role of “thermal mass”. The extended system samples a microcanonical ensemble with constant energy given by
H=
1�
1
Mi R̈i + V ({R}) + Qζ̇ 2 + 3N kb T.
2
2
(3.50)
i
It can been demonstrated that the system’s canonical variables {R} and {P } sample
a canonical ensemble.
Chapter 4
Measurements and ab-initio calculations on Tellurium at high
pressure
Raman and Infrared measurements under high-pressures are suitable to investigate
the modification of both lattice and electron dynamics properties of a system under
volume compression. In order to investigate the role of the electron-electron and the
electron-phonon interaction in driving the electronic and the structural transitions,
high pressure infrared measurement (0-9 GPa) in the near infrared range (NIR)
and Raman measurements (0-15 GPa) on solid Te have been carried out at room
temperature
In parallel to the experimental effort, high pressure properties of the Te-I and TeII phases have here theoretically investigated by DFT calculation. Once established
that the the experimental results are well reproduced by calculations, theoretical
results have been used to better understand the microscopic origin of observed
pressure dependence, and to extend the analysis to other properties not directly
observable (i.e. full Raman tensors, interatomic force constants and the electron
charge density).
4.1
High pressure Raman measurements and analysis
on solid Te
High pressure Raman measurements have been performed on small Te polycrystalline samples (� 50x10x10 µm3 ) using the apparatus described in Section 2.2. Te
samples have been loaded in the DAC together with a small ruby sphere for “insitu” pressure measurements. Methanol ethanol 4:1 mixture or NaCl have been
used as hydrostatic pressure transmitting media allowing a maximum pressure of
11 GPa and 15 GPa, respectively (see Section 2.1.1). Preliminary measurements as
function of the incident light polarization showed only negligible polarization effects
in the Raman spectra on our Te samples. The high pressure measurements were
carried out in back-scattering configuration collecting light polarized mainly in the
same direction of the incident one owing to the grating efficiency. Room temperature Raman spectra of Te samples at selected pressure are shown in Figure 4.1.
47
48
4. Measurements and ab-initio calculations on Te at high pressure




























 







Figure 4.1. Background subtracted Raman spectra of Te at room temperature at selected
pressures. Panel (a) using ethanol-methanol mixture and panel (b) using NaCl as
pressure transmitting media.
No evidence of chemical reactions between the sample and the two media or of
non-hydrostaticity effects have been observed.
The effect of pressure on Te Raman spectrum appears rather complex, although
on increasing the pressure Raman spectra do not significantly change their peak
pattern and their general spectral shape. The overall pressure behavior and, in
particular, the persistence of the Raman spectrum above 11 GPa is particularly
relevant in addressing the real sequence of structural phases followed by Te on
increasing pressure. It is particularly relevant at this point to recall that the Raman
active modes predicted for the Te-I phase (common to old and new schemes) belong
to the A1 and E symmetries whereas no Raman active modes are expected for both
the β-Po type phase (old scheme) as well as for for the unmodulated Te-III phase
(new scheme). Nevertheless the claimed structural modulation of the Te-III phase
(see Section 1.4) allows the activation of extra Raman active modes (2Ag + Bg ).
Bearing in mind these arguments, the presence of Raman signal above 11 GPa is
thus consistent with the presence of the incommensurate modulation of the Te-III
structural phase [37, 38], and unambiguously non compatible with the old high
pressure scenario [4, 41, 64]. For this reason in the following we definitively assume
the new transition scheme for the discussion of the data.
Measured spectra were fitted by using a standard model curve given by the sum
of the electronic and the phononic contributions [17]:
�
�
N
�
Bνγ
(1 + n(ν)) 2
+
P h(ν, νi , Γi , Ai ) .
ν + γ2
(4.1)
i=1
The parameters B and γ characterize the electronic response, while νi , Γi , and Ai
are the frequency, linewidth, and intensity of the ith phonon peak, respectively.
The quantity 1 + n(ν) accounts for the Bose-Einstein statistics. Damped harmonic
4.1 High pressure Raman measurements and analysis on solid Te
49
oscillators have been employed to describe phonon excitations [17]:
P h(ν, νi , Γi , Ai ) =
Ai Γi νi ν
.
(νi2 − ν 2 )2 − Γ2i ν 2
(4.2)
In order to account for the background in the Raman spectra, a linear baseline has
Figure 4.2. Room temperature Raman spectrum of solid Te from ours samples and best
fit curve (solid line) from equations (4.1) and (4.2). The blue dashed lines represent the
single phonon contributions to the total spectrum.
been included in the fitting curve.
As an example the best fitting curve and the experimental data collected at ambient pressure and room temperature are shown in Figure 4.2. The three phonon
peaks predicted by the group theory for the Te-I phase are clearly observed at
ambient pressure, the total symmetric A1 mode around 115 cm−1 and the doubly
degenerated E(1) and E(2) around 90 cm−1 and 140 cm−1 , respectively. In this
chapter we will use improperly the same labels (E(1),A1 ,E(2)) over the whole explored pressure range, although the symmetry is Ag for the three Raman active
mode of Te-II and 2Ag + 1Bg for those of Te-III incommensurately modulated.
Looking at figure 4.1, an almost regular hardening of the phonon frequencies of
the E(2) mode has been observed over the 0-4 GPa range followed by a softening over
the 4-8 GPa pressure range. On the contrary the A1 mode shows a clear monotonic
softening of the phonon frequencies over the 0-8 GPa pressure range. Increasing
pressure up to 8 GPa, the E(2) mode seems also to soft like the A1 mode. Above
this pressure the frequency of the A1 mode seems to keep constant, whereas that of
the E(2) mode shows a clear hardening.
These findings are quantitatively confirmed by the results obtained applying
the fitting procedure described above. The pressure dependence of the phonon
frequencies is shown in the upper panel of Figure 4.3. On applying pressure up to 4
GPa the A1 mode shows a softening of about 10 cm−1 , while the phonon frequencies
of the E(2) peak weakly increase of about 1-2 cm−1 .
50
4. Measurements and ab-initio calculations on Te at high pressure
Figure 4.3. Pressure dependence of phonon frequencies (upper panel) and of phonon
widths (lower panel) of Te. Open (close) symbols refer to measurements performed
using ethanol-methanol mixture (NaCl) as hydrostatic medium. Dashed vertical lines
mark the transition pressures according to Ref. [36–38]. Green red and black symbols
refer to E(2), A1 and E(1) phonon peak, respectively.
We notice that, using a set of internal coordinates it is possible to write the
dynamical matrix in terms of the strength associated to the displacements along
valence bounds and angles variation between valence bounds. In this way it is possible to see that, the frequency of the infrared active A2 mode (rigid chain rotation),
depends only on the inter-chain force constant, whereas the Raman active A1 mode
frequency mainly depends on the intra-chain strength. The E modes frequencies
mainly depend on the strength among different chains. On these bases the observed
pressure dependence of Raman spectrum of trigonal Te-I is thus consistent with
a progressive weakening of intrachain bonds at expenses of the interchain atomic
interactions.
Within the 4-7 GPa pressure range the frequency of A1 is still linearly decreasing
whereas a significative softening is shown also by the E(2) mode. The slope discontinuity of the E(2) mode at 4 GPa marks the emergence of a new phase whereas
the overall behavior suggests that the weakening of intrachain bonds is still ongoing
over this pressure range. Because of the narrow range in which Te-I and Te-II are
reported to coexist (4-4.5 GPa) [38], our data do not allow for any consideration on
the Te-I/Te-II transition and on the influence of their coexistence on the Te dynamical properties. According to Ref. [37, 38], that reports a Te-II/Te-III coexistence
within the 4.5-8 GPa pressure range, the observed softening of the E(2) mode can
4.1 High pressure Raman measurements and analysis on solid Te
51
be ascribed to the structural instability of the Te-II phase which is progressively
converted into the very similar Te-III phase [37, 38]. We notice that the larger
values of the phonon linewidths can be found within the coexistence region as a consequence of the lattice disorder (see Figure 4.3b). On further increasing pressure
above ∼ 8 GPa all the phonon modes show a rather regular frequency hardening,
indicating that a single and stable phase (Te-III) has finally established. We finally
notice that the spectroscopic signature of the metallization process at ∼ 4 GPa can
be found in the progressive increase of the electronic contribution observed in the
Raman spectra (not shown here).
The presence of modulation implies a one-to-one correspondence between the
eigenvalues of the dynamical matrix at generic k and those at k ± qinc (see Section 1.2.1). The phonon at Γ involved in the extra scattering processes arising
from the modulation have thus the same energy of the phonon at qinc involved in
a inelastic x-ray (neutron) scattering processes. This allows an unusual comparison
between the results of Raman experiments and those of the inelastic x-ray (neutron)
diffraction experiments. Indeed the frequency of the mode Raman mode at higher
energy observed at 8.9 GPa (see Figure 4.3) is the same within the experimental
uncertainties as the frequency of the longitudinal mode at k = qinc observed by
inelastic x-ray diffraction experiment (see Figure 1.12) [55]. This fact is a further
evidence of the presence of the incommensurate modulation in the Te-III crystal
Figure 4.4. The intensity ratio Iratio as function of pressure (see text for details). Dotted
line are guides to the eye. Open and full symbols are referred to pressure measurements
using ethanol-methanol mixture and NaCl as hydrostatic medium, respectively. The
red vertical line marks the insulator to metal transition pressure.
52
4. Measurements and ab-initio calculations on Te at high pressure
phase.
A systematic study of Raman peak integrated areas was carried out and it is
reported in Figure 4.4 only in the pressure range 0-6 GPa since on increasing pressure
above 6 GPa the phonon spectrum is progressively less intense and the comparison
between the areas becomes less reliable. Since a comparison among the absolute
areas among different spectra may be questionable, an analysis of peak areas ratio,
that is of the relative intensities has been performed for each spectrum
Iratio =
I[E(2)]
I[inter]
�
.
I[A1 ] + I[E(1)]
I[intra]
(4.3)
The pressure dependence of Iratio in the pressure range 0-6 GPa is shown in Figure 4.4. Two different regimes below and above the threshold pressure value of 4
GPa, at which the insulator-to-metal transition takes place, are clearly identifiable.
Neglecting the small contribution to Iratio due to the area of the E(1) mode, the
growth of Iratio indicates a strong decrease in the intensity of the A1 peak before the
metallization pressure. Bearing in mind the characters of the A1 mode (intra-chain)
and of the E(2) mode (inter-chain) Iratio in Eq. 4.3 basically provides the intensity
ratio between intra-chain and inter-chain modes. We notice that it is not trivial to
explain the observed pressure behavior and in particular the strong decrease of the
intensity of the intra-chain A1 mode. However we will see in Section 4.3 that the
results of ab-initio calculations allow us to link the evolution of the peak frequency
and intensity on increasing the pressure to the mechanism that drives the MIT.
For the moment we simply note that, for reasons related to the Peierls mechanism, the A1 is the peak most sensitive to the gradual pressure-induced metallization.
Indeed, the trigonal structure can be seen as a distortion of a metallic rhombohedral
one obtained from the Te-I when the nearest neighbor and the next nearest-neighbor
distances become equal, that is the chain radius becomes Rchain = 13 a (being a the
trigonal lattice parameters shown in Figure 1.10). The reduction of the symmetry
that occurs in the trigonal structure (Rchain < 13 a) opens a gap around the Fermi
level. The system compression forces Rchain to approach the value 13 a in Te-I, reducing the stabilizing effect of the Peierls mechanism. Since the A1 vibrational mode
corresponds to a variation of Rchain around the equilibrium value, is therefore clear
that this mode will be stronlgy affected by the removal of the Peierls distortion
which stabilizes the system at ambient pressure.
4.2 High pressure infrared measurements and analysis on solid Te
4.2
53
High pressure infrared measurements and analysis
on solid Te
Figure 4.5. Images of a the sample before it was loaded in the DAC. Blue arrow marks
the c-axis.
In order to understand the role of lattice symmetrization in the metallization
process occurring in solid Te, high pressure infrared reflectivity measurements have
been also performed exploiting the high performances of the infrared beamline SISSI
at Elettra synchrotron in Trieste. Great care has been taken in the sample choice
and in the proper orientation of the crystal loaded in the DAC. At each pressure
we collect reflectivity spectra over the 2000-12000 cm−1 frequency range with polarization of the incoming radiation parallel and perpendicular to the c-axis. A
photo of the crystal before it was loaded into the DAC is shown in Figure 4.5 where
the c-axis direction is shown by the blue arrow. We notice that we were able to
control the correct crystal orientation either by visual inspection or by maximizing
the difference of intensity collected in the parallel and perpendicular configuration.
We indeed know that the reflectivity reaches its maximum values for incident light
polarized along the c-axis and its minimum values for light polarized perpendicular
to this axis (i.e, in the ab plane) [80].
The Infrared setup described in Section 2.3 has been employed for the present
measurements. Possible misalignments and source intensity fluctuations were accounted for by measuring the intensity reflected by the external face of the diamond
D (ω) , at each working pressure. At the end of the pressure run, we meaanvil, IR
sured the light intensities reflected by a gold mirror placed between the diamonds
Au (ω) and by the external face of the diamond anvil I D� (ω). By using the ratio
IR
R
D� (ω)/I D (ω) as a correction function, we achieved the reflectivity:
IR
R
R(ω) =
�
S (ω) I D (ω)
IR
R
.
Au (ω) I D (ω)
IR
R
(4.4)
The reflectivity spectra corrected using 4.4 for radiation polarized parallel and perpendicular to the c-axis are shown in Figure 4.6 at selected pressure. We can ascribe
the progressive increase of the reflectivity for both the directions of light polarization to the metallization process, setting in the system as the pressure is raised
above 4 GPa.
54
4. Measurements and ab-initio calculations on Te at high pressure
Figure 4.6. Collected reflectivity spectra as function of polarization and pressure.
To derive optical constants from reflectivity spectra, a Kramers-Krönig method
is usually used. However, the measured energy range, which is restricted by the cut
off energy of diamond, is too narrow to apply the Kramers-Krönig method. Hence,
an extrapolation for the reflectivity spectrum has to be used at energy lower that
2000 cm−1 and higher than 12000 cm−1 . Since there are no available data at high
pressure, the ambient pressure data from Ref. [80] were substituted as the best
approximation for the high pressure reflectivity above 12000 cm−1 , since the major
changes are expected in the low-energy range.
In order to extrapolate the low-energy behavior of the optical reflectivity at
different pressures, a fit procedure using the Drude Lorentz model has been preformed. This procedure has been applied simultaneously to the experimental reflectivity (measured in the DAC) and to the experimental reflectivity from Ref. [80]
in the frequency range above 12000 cm−1 . In this way we can obtain an estimate
of low-frequency behavior of the reflectivity in the DAC and extend the reflectivity
data at high frequency in order to perform the Kramers-Krönig transformations.
The optical conductivity σ(ω) calculated by the Kramers-Krönig transformation
of the so extrapolated reflectivity at 1.0 GPa is shown in 4.7. In the same figure the
optical conductivity obtained by the best fit parameters of the fit procedure on the
reflectivity spectra is also shown. The sigma calculated with these two methods are
comparable with the optical conductivity at ambient pressure calculated from the
results of Ref. [80]. The shift of the band edge toward lower energy in the calculated
optical conductivity with respect the results of Ref. [80] could be ascribed to the
pressure differences between the two sets of data. Moreover the calculated optical
conductivities with the two different methods are very similar each other, therefore
optical conductivity spectra at different pressures will be calculated using the results
of the fitting procedure just described.
The results at selected pressure is shown in figure 4.8. It is rather clear that
the spectra collected in a perpendicular geometry show a faster band gap closure
4.2 High pressure infrared measurements and analysis on solid Te
14000
σ(ω)Ω−1 cm−1
12000
10000
55
σamb (ω)[80]
σF IT (ω) � c @ 1.0 GPa
σKK (ω) � c @ 1.0 GPa
8000
6000
4000
2000
0
2000
4000
6000
8000
−1
ω(cm
10000
12000
)
Figure 4.7. Optical conductivity spectra obtained both from the Kramers-Krönig transformation (σKK (ω)) as well as from a Drude Lorenz fit of the experimental reflectivity
(σF IT (ω)). The experimental conductivity from Ref. [80] at ambient pressure is also
reported as red squares for comparison.
than those collected with a parallel geometry. On these bases it seems that an
unexpected increase of the electronic anisotropy occurs on increasing the pressure.
This behavior is well summarized by the calculated spectral weight defined as
SW (P ) =
�
6000
σ(ω, P )dω.
(4.5)
2000
Figure 4.8. Optical conductivity spectra obtained from reflectivity data using the fitting
procedure (see text). The the x-axis range was set equal to the range explored in the
experimental investigation of reflectivity.
56
4. Measurements and ab-initio calculations on Te at high pressure
The SW(P) calculated for the same σF IT (ω) of Figure 4.8 is reported in Figure 4.9.
From the inspection of this Figure it is noticed that:
• the anisotropy of the response to the incident radiation grows even in the
Te-III phase (at 8.7 GPa)
• the Te-III phase shows a clear metallic character despite is reported to be
a CDW states. Indeed we know (see Section 1.1.2) that a transition into a
CDW state is usually associated with e metal to insulator transition, since the
lattice distortion opens gaps near the Fermi level.
Figure 4.9. Optical conductivity spectra obtained from reflectivity data using KramersKrönig relations.
As far as the increase of the anisotropy is concerned, this seems to be related
to the persistence of covalent like bonds with strong directional properties even in
the high Te-III pressure phase despite the higher symmetry of the crystal structure.
As we will see in the Section 4.4 even the (bcm) structure, i.e., the unmodulated
structure underlying that of Te-III structure, presents an high anisotropic response
to the incident light.
As far as the metallic character of the Te-III phase is concerned, this behavior
suggests that the dynamical instability that lead to the Te-II/Te-III transition could
not be related to an electron instability, i.e., to a divergence of the Lindhard response
function as it happens in the unidimensional linear chain. As matter of fact the
Peierls instability is not the only mechanism that can cause the formation of an
incommensurate modulation in the one dimensional linear chain. It is known that
the introduction of the next-nearest-neighbor interaction and of a nearest-neighbor
4.2 High pressure infrared measurements and analysis on solid Te
57
anharmonic coupling among atoms of a linear chain can lead to the formation of an
incommensurate phase [43].
58
4.3
4. Measurements and ab-initio calculations on Te at high pressure
Ab-initio calculations on Trigonal Tellurium
In Section 4.1 we saw how the knowledge of the eigenvector of the A1 phonon mode
and of the pressure behavior of its frequency allowed precise considerations on the
evolution of the bonding properties in the Te-I phase. In the case of a A1 and A2
modes of trigonal Te-I the eigenvectors are fixed by symmetry reasons. However,
this is a special case, since they are generally determined by the force constant
matrix of the system. Providing that DFT gives an accurate description of the
system ground state properties, it can then be used to determine the movements of
atoms involved in the normal modes.
In the Te-I case, as well as to determine the eigenvectors of the E modes, ab-initio
calculation can be particularly useful in establishing the pressure behavior of the
infrared active A2 mode. Indeed it can provide a clear indication on the change of
the interaction between different chains since its frequency depends only to the intrachain strength as seen in Section 4.1. Moreover the calculation of Raman tensors
will allow us to give a possible explanation of the marked intensity decrease of the
A1 mode. Finally, once verified that the DFT reproduces satisfactorily the system
ground state properties, the Te-I electron charge density behavior on increasing
pressure will be investigated.
Ab-initio calculations were performed within Density-Functional Theory, using a Plane-Wave basis set and pseudopotentials as implemented in the Quantum
Espresso package [32]. Convergence tests showed that the energy and forces on
atoms for Te-I are well converged using a plane waves cutoff of 82 Ryd and a 7x7x7
Monkhorst-Pack of 343=73 k-points (see fig 4.10). To obtain the lattice parameters
Figure 4.10. Convergence of total energy and forces versus plane waves cutoff and k-point
mesh
and the wyckoff positions for each target pressure, the unit cell were relaxed applying the Car-Parrinello technique under the influence of the Hellmann-Feynman
forces until no force component exceeded 0.004 eV/Å. The determination of the
external pressure is usually affected from a systematic error known as Pulay stress
due to the fact that the stress tensor is defined at constant plane waves number,
whereas it is calculated at constant energy cutoff. This introduces a negative ficti-
4.3 Ab-initio calculations on Trigonal Tellurium
59
Table 4.1. Pressure estimated by Car-Parrinello variable cell dynamics and by a Murnaghan fit
Pscf
∆Pscf
Pfit
∆Pfit
∆P = Pfit -Pscf
0.48
0.90
1.55
25.85
49.84
0.07
0.08
0.02
0.23
0.21
-0.06
0.10
1.30
25.10
50.00
0.09
0.09
0.14
2.40
6.00
-0.54
-0.80
-0.25
-0.75
0.16
tious contribution to the stress tensor, so that the calculated value is always greater
than the true one. The entity of this error can be estimated by fitting the energy
E as function of the cell volume V by the Murnaghan function:
�
�
�
B0 V (V0 /V )B0
B0 V0
E = E0 +
+1 − �
,
(4.6)
�
�
B0
B0 − 1
B0 − 1
where E0 and V0 are the energy and the volume at equilibrium, B0 and B’0 are
the bulk modulus and the first derivative of the bulk modulus at P=0, respectively.
The pressure corrected from the Pulay stress can be obtained from the formula:
�� � �
�
B0
V 0 B0
P (V ) = �
−1 ,
(4.7)
B0
V
using the best fit parameters. As we can see in table 4.1, the error in the pressure
calculated by the Hoemberg-Kohn theorem is less than 1 kbar and, therefore negligible over the explored pressure range. The cells relaxed to the target pressures of
0, 2.6 and 5 GPa have been used for structural, electronical, dynamical, and optical
properties calculation of Te-I. The relaxed lattice parameters are in good agreement
with experimental ones while the internal parameter (the chain’s radius) is slightly
overestimated by about a 10% at ambient pressure and its pressure dependence is
slightly different from the experimental one (see Figure 4.11). This means that the
relaxed structure is more orthorhombic like (i.e., less Peierls distorted) than the
real one. Despite this, the evolution of the lattice parameters (see Figure 4.11) and
the bulk modulus value Bcal = 22 GPa, obtained from fitting to the third order
Murnaghan equation of state 4.6, are in good agreement with the experimental
ones given by x-rays measurements [47] (Bexp = 19.4 GPa), while its first pressure
derivative is slightly overestimated by the calculation (B’cal = 7.3, B’exp = 5.1).
In the experimental configuration (see chapter 4.1) the differential Raman cross
section for the Stokes component of the ith eigenmode far from resonance is given
by
�
�
dσf
(2πνs )4 �� ∂ α̃ ��2 h(nbi + 1)
=
êS
êL
,
dΩ
c4 � ∂Qf �
8π 2 νi
�
�
�
�−1
hνi
nbi = exp
−1
.
(4.8)
KT
In Eq. 4.8, νS is the frequency of the scattered light, êS and êL are the unit vectors
of the electric-field direction (polarization) for the scattered and the incident light,
60
4. Measurements and ab-initio calculations on Te at high pressure
α̃ is the polarizability tensor, and nbi is the Bose-Einstein statistical factor. Since
the crystal of our polycrystalline sample was oriented randomly, this expression has
to be appropriately space averaged. The result of the averaging procedure depends
on the relative orientations of the direction and polarization of the incident and
scattered beams. In our experiments, a plane-polarized incident laser beam was
used. Furthermore, we used a back scattering geometry and collect light mainly
polarized in the same direction of the incident beam. Under these circumstances,
by space averaging the (4.8), one yields a Raman cross section
dσf
(2πνs )4 h(nbi + 1)
=
I ,
dΩ
c4
8π 2 νi �
(4.9)
where
1
2
I� = Σ0 + Σ2
3
15
0
2
denoting with Σ and Σ the tensor invariants
Σ0 =
1
|α̂xx + α̂yy + α̂zz |2
3
1
Σ2 = {|α̂xy + α̂yx |2 + |α̂xz + α̂yz |2 + |α̂zx + α̂xz |2 },
2
1
+ {|α̂xx − α̂yy |2 + |α̂yy − α̂zz |2 + |α̂zz − α̂xx |2 }.
3
0
The quantity Σ represents the isotropic part of the tensor, while the symmetric part
of the anisotropy is given by Σ2 . We should notice that if the light were collected
with the polarization direction perpendicular to the incident one the term I� must
be replaced by
1
I⊥ = Σ2 ,
10
and therefore a measurement of I⊥ yields a relative value of the anisotropy. The
ratio of the intensities perpendicular and parallel to the incident polarization is
Figure 4.11. Calculated lattice parameters (a) and chain radius (b) as a function of pressure (open symbols). Experimental data from Ref. [47] are also shown (sully symbols).
4.3 Ab-initio calculations on Trigonal Tellurium
61
Table 4.2. Calculated and experimental phonon frequencies. The last column labeled as
Iratio represents the calculated values for the intensity ratios (see text)
exp
cal
P
(GPa)
E(1)
(cm−1 )
A1
(cm−1 )
E(2)
(cm−1 )
A2
(cm−1 )
Iratio
0.0
2.5
3.50
0.0
2.6
5.0
93.00
93.86
94.38
80.8
80.2
78.0
115.0
112.4
108.6
104.8
96.0
90.0
140.0
140.8
143.2
125.8
124.0
124.6
85.6
89.3
89.0
0.22
0.25
0.32
0.30
0.33
described by the depolarization ratio
ρ=
I⊥
3Σ2
=
,
I�
10Σ0 + 4Σ2
(4.10)
which is 3/4 for a non-totally symmetric vibration (like the E mode).
The optical mode frequency at Γ were calculated using the first-order perturbation
theory of the DFT, while the nonresonant Raman coefficients, obtained using secondorder response as in [53], was properly space averaged according to (4.9). Calculated
phonon frequencies and the intensity ratio Iratio
Iratio =
I[E(2)]
I[E(1)] + I[A1 ]
(4.11)
are compared with the experimental values in Table 4.2. We notice that the calculated frequencies show the correct pressure dependence although their absolute
values are underestimated by a factor ∼ 0.9. The latter value is in agreement with
the typical accuracy of DFT calculation in the corrected gradient approximation
in these complex materials (see Ref. [76] and reference therein). The calculate
intensity ratio is slightly underestimate but also in this case it shows the same experimental pressure dependence. Using the calculate frequencies and intensities and
the experimental phonon peak linewidth we have are able to simulate the Raman
spectrum at ambient pressure shown in Figure 4.12. The agreement between the
theoretical and the experimental spectra is very good (compare the spectra shown
in Figure 4.2 and Figure 4.12).
Such a overall agreement between experimental data and calculated lattice dynamic properties make us confident about the reliability of our theoretical analysis.
This allows us to carefully study how the real atomic displacements of the atoms
are modified by the lattice compression for each phonon mode. At ambient pressure
the displacements of the two E modes eigenvectors show a significant admixture of
the axial helix compression and the asymmetric dilation normal to the helix axis.
Under compression the two modes decouple and the E(1) modes become more helix
compression like while E(2) modes more asymmetric dilation like. As described
in section 4.1, the A1 mode corresponds to a intra-chain dilation and compression
normal to the chain axis. Thus this mode involves both bond-length and bond
angle distortion. On the contrary, the infrared active A2 mode is a rigid chain rotation, thus its frequency depends only to the inter-chain strength. The increase
62
4. Measurements and ab-initio calculations on Te at high pressure
Figure 4.12. Raman spectrum calculated unsing DFPT
of the frequency of the A2 mode with respect to the decrease of the A1 mode frequency indicates an enhancement of the inter-chain interaction at the expenses of
the intra-chain bonding.
As to the pressure dependence of the intensity, our calculation shows that the
depolarization ratio for the E(1) and E(2) modes keep constant whereas it strongly
increase for the A1 mode. Bearing in mind that the Raman experiment and the
theoretical calculation have been carried out in a parallel polarization configuration,
the observed pressure dependence of the intensity ratio Iratio can thus be ascribed
to the different pressure behavior of the depolarization ratios for the E(1), E(2)
and A modes. Their variation on increasing the pressure points out an increase
of the anisotropy of the symmetrical part of the Raman tensor (see eq. (4.10)).
More precisely, with increasing pressure, the motion of the total symmetric A1
mode results in a change of dielectric tensor in the direction of the chain, more
pronounced than along the line joining the chains, i.e.
�
�
�
�
∂ α̃
∂ α̃
b=
�a=
.1
(4.12)
∂(QA1 ) 3,3
∂(QA1 ) 1,1
This means that the system becomes more and more polarizable along the c direction
(axis chain) on increasing pressure in comparison to a direction in ab plane. This
effect is related to the charge transfer process between different chains activated by
lattice compression. The increasing of charge delocalization in the direction joining
the chains suppresses Raman response for incident light polarized in the ab plane.
In order to demonstrate the charge transfer, the calculated valence electron
density was shown in Figure 4.13 for the plane containing three nearest neighbor
1
a 0 0
1
The Raman tensor of the A1 mode has the form
= @ 0 a 0 A with b > a in
0 0 b
the Te case. Given this expression for the Raman tensor the depolarization ratio in Equation 4.10
(a − b)2
3
writes ρ =
where Σ =
. Thus a grow of the depolarization ratio ρ means an
−1
5Σ + 4
(a + a + b)2
increase of Σ in this phonon mode.
„
∂ α̃
∂(QA1 )
«
0
4.3 Ab-initio calculations on Trigonal Tellurium
63
and one next nearest neighbor (see picture in Figure 4.13). The norm-conserving
pseudo-potential used in the calculations, ensure us that the pseudo charge density
beyond the cutoff radius (∼ 8 Å) is the same density of an all-electron calculation
(see Section 3.5, i.e. the density constructed starting from the Kohn-Sham orbitals
(see Formula 3.5). Moreover, since our calculations well reproduce the system’s
ground state properties, the Hoemberg-Kohn theorem (see Sectrion 3.1) ensure us
that the density constructed starting from the Kohn-Sham orbitals can not differ
much from the real one. It is worth to notice that the coupling between adjacent
chain arising from the charge transfer process could be closely related to the onset
of the incommensurately lattice modulation in the high pressure phase.
Figure 4.13. Charge density plot for trigonal Te at 0 GPa and 5 GPa.
As seen in the chapter 4.2, optical conductivity spectra show an enhancement of
dichroic properties on increasing pressure. In order to evaluate the system optical
response starting from Konh-Sham eigenfunctions and eigenstates, the many-body
approach recently developed in the YAMBO code was employed. The calculated
optical conductivity was obtained starting from imaginary part of the dielectric
function calculated in the Random phase approximation without local field effects,
since tests have been shown that their effects on Te optical properties are very
small. The theoretical optical conductivities at various pressures are reported in
� c (dashed lines) and E⊥�
� c (solid lines). A comparison on panels
Figure 4.14 for E��
a) and b) shows that zero-pressure calculations are in fairly good agreement with
the experiments. The main discrepancy lies in a overall shift of theoretical results
to lower energies due to the known gap underestimation tendency of the DFT.
However, the pressure induced changes in optical conductivities are insensitive to
this shift and are in good agreement with experiment, since the increase of the
dichroic properties is well reproduced. It is worth to notice that the calculate
optical conductivity at 5.0 GPa is much more similar to the experimental one at 8.7
Gpa than the experimental one at 5.0 GPa, that is, the effects of the compression
are overestimated in the calculations. This is consistent with the underestimation
of the first derivative of the bulk modulus and with the pressure behavior of the
internal parameters (see Figure 4.11) which indicate that the calculated system is
much more compressible than the real one.
To understand this pressure behavior we turn to examine the electronic density
64
4. Measurements and ab-initio calculations on Te at high pressure
Figure 4.14. Calculate and experimental optical conductivity as function of ω and pressure
of states shown in Figure 4.15. We can observe the gap closure around the Fermi
energy, and a general broadening of all the bands related to the increase in overlap
between wave function on different chains. This change of the wave functions also
affects the dipole matrix elements, which determinate the increase of the anisotropy
of the optical response.
EF
DOS (arb. units)
0 GPa
5 GPa
-15
-10
-5
0
5
Energy (eV)
Figure 4.15. DOS of the Te-I relaxed cells at 0 GPa and 5 GPa
4.4 Ab-initio calculations on Te-II high pressure phase.
4.4
65
Ab-initio calculations on Te-II high pressure phase.
As seen in Section 1.4, there is a jump in volume at the Te-I/Te-II transition around
5%. However Raman data show no jump in the frequency evolution of the vibrational modes at 4GPa, i.e, the transition pressure (see Figure 4.3). This behavior
is rather unusual considering that the atoms are arranged in a very different way
in the two structures. As shown in Figure 4.16 atoms forms a helix chain along the
c-axis in the Te-I structure, while they are arranged in a zig-zig-zag pattern to form
an almost planar chain in Te-II phase. A detailed study of electronic properties of
Figure 4.16. The atom position in the unit cel in Te-I and Te-II phase.
Te-II phase was also carried out together with the stability analysis of the Te-I and
Te-II phases.
The triclinic Te-II cell was relaxed applying same technique used in the Te-I case.
In order to study the stability of the high pressure Te-II phase, the total energy E
as function of the unit cell volume V was fitted using the Murnaghan equation of
state (Formula 4.6) for both the Te-I and Te-II relaxed cells. The obtained best fit
parameters was used to calculate the external pressure on the cells (see Formula 4.7)
and to obtain the enthalpy as function of the external pressure for the two phases.
The difference between the enthalpy so obtained for the Te-II and Te-I phase
∆H(P ) = HT e-II (P ) − HT e-I (P )
(4.13)
is shown in Figure 4.17. Above 5.5 GPa ∆H becomes negative, suggesting the
presence of the Te-I/Te-II phase transition. As far as the phonon frequencies at
the estimated pressure transition (∼ 5.5 GPa) their values show a rather sharp
discontinuity, since the modes of the Te-II phase are between 20 cm−1 and 50 cm−1
lower than those of the Te-I phase. This is in contrast with the continuous evolution
of the phonon frequencies observed in our Raman experiments on Te samples.
A possible explanation for this result is that the structure of the Te-II relaxed
cells deviate from the experimental one more than the Te-I relaxed cells structure.
Indeed in Te-I most of the crystallographic constants are fixed by the hexagonal
geometry. Nevertheless the DFT underestimates the anisotropy of the system leading to a slight overestimation of the ratio of chain radius to interchain distance. In
4. Measurements and ab-initio calculations on Te at high pressure
66
3
2
∆H (mRyd)
1
0
-1
-2
-3
-4
4
4.5
5
5.5
6
Pressure (GPa)
6.5
7
Figure 4.17. The difference ∆H = HT e-II − HT e-I between the enthalpy values of the
Te-II and Te-I cells relaxed at different pressures using the variable cell shape method.
the Te-II case, where all the crystallographic constants and the atom positions are
left free to vary during the cell relaxation, this trend is dramatically emphasized so
that, relaxing the structure, the experimental zig-zig-zag puckering is entirely lost.
The zig-zig-zag puckering of the Te-II phase is not related to any crystal symmetry,
thus it isn’t possible to impose any constrain during the cell relaxation in order to
obtain the experimental puckering structure in the relaxed cells.
In the Te-II case the dynamical properties were thus recalculated optimizing
the atomic positions while keeping fixed the lattice parameters to the experimental
values obtained for the Te-II phase at 4.5 GPa [38]. For sake of comparison the
Raman frequencies have been recalculated by optimizing only the atomic positions
keeping constant the lattice parameters to the experimental values observed for the
Te-I phase at 4.0 GPa. The results for the two phases are reported in 4.3. We
Table 4.3. Calculated Raman phonon frequencies for Te-I and Te-II optimizing the atom
position and keeping constant the lattice parameters to the experimental values.
Te-I at 4 GPa
Mode ν (cm−1 )
E(1)
78
A1
92
E(2)
122
Te-II at 4.5 GPa
Mode ν (cm−1 )
Ag (1)
38
Ag (2)
94
Ag (3)
125
notice that only the low-frequency phonon shows a remarkable discontinuity at the
transition whereas the differences between the values calculated in the two phases
for the high-frequency modes are within the typical DFT calculations accuracy. In
agreement with the experimental data our calculations thus predict a continuous
evolution of the frequencies across the Te-I Te-II transition for the high-energy
modes.
As to the estimate of the frequency of the low-energy mode further discussion is
4.4 Ab-initio calculations on Te-II high pressure phase.
67
required. In order to explain the discrepancy between experimental and calculated
frequencies for the low energy phonon in the phase TeII, it can be helpful to examine the phonon dispersion relation of TeIII at 8.5 GPa for the momentum transfers
parallel to the modulation vector. The solid symbols in Figure 4.18 represent the
Figure 4.18. Phono dispertion relation of Te-III at 8.5 GPa for momentum transfers
�q = (0ξ0) parallel to the modulation vector. The solid symbols represent the results
for the LA phonon dispersion branch, and small open symbols mark additional weak
features observed in the IXS spectra. The solid line is a guide to the eye; the dashed line
shows a sine-type dispersion relation and the dash-dotted line ∆ shows the difference
between the sine-type dispersion and the observed LA dispersion. (b) Calculated phonon
dispersion curves of bcm structure, i.e, the unmodulated structure underlying that of
Te-III one (solid symbols). Phonons with imaginary frequencies are displayed with
negative energies. The experimental results for the LA phonon branch in Te-III at 8.5
GPa are shown by small open symbols.
results for the longitudinal acoustic (LA) phonon dispersion branch observed in inelastic x-ray experiments spectra, while the solid symbols are the calculated phonon
dispersion curves of the (unmodulated) bcm Te [55]. The dispersion of the LA mode
is in perfect agreement with the experimental data, while the low-frequency transverse acoustic (TA) mode shows a strong Konh anomaly, so that its frequencies
are negative in the neighborhood of qinc . Then the low-frequency TA mode of the
(unmodulated) bcm is more affected by the presence of the dynamic instability, because the vibration of atoms occurs along directions close to the frozen-in lattice
distortion. The analysis of the TeIII structure can give important and relevant hints
for discussing also the TeII phase. Indeed these two crystal structures are closely
related. The Te-III to Te-II transition involves a tripling of the Te-III unit cell along
b, equivalent to the wave vector locking in at the commensurate value of (0 1/3 0);
a slight distortion of both the α and γ cell angles of Te-III away from 90◦ ; and a
slight rearrangement of the atoms in the ac plane.
Bearing in mind that Te-II structure is very close to the (unmodulated) bcm
structure (i.e. the unmodulated structure underlying that of modulated Te-III) with
the b-axis tripled in length, the phonon mode behavior in Te-II can be related to
those calculated at wavevector k = 13 in the (unmodulated) bcm one. Since the
low-frequency mode in the (unmodulated) bcm is strongly affected by the presence
of the instability, it seems reasonable to assume that the frequency of the low en-
68
4. Measurements and ab-initio calculations on Te at high pressure
ergy Te-II mode is underestimated too. Indeed their eigenvectors, as those of the
low-frequency TA mode in the neighborhood of k = qinc , are close to the direction
of the incommensurate modulation. Given that there are phonon modes with negative energies throughout a neighborhood of qinc in the (unmodulated) bcm phase
[37], the underestimate of the Te-II low energy mode seems thus to be related to
DOS (arb. units)
bcm
TeII
-20
-15
-10
-5
0
5
10
15
20
25
Energy (eV)
Figure 4.19. DOS of the (unmodulated) bcm structure and of the Te-II structure. The
Fermi Energy for both Te-II as well as the bcm structure was set to 0 eV.
an excessive sensitivity of the DFT theory to predict the incommensurate phase, or
in other words it seems to be related to an overestimate of the width of the saddle
point of the energy landscape in the direction of the modulation. This coexistence,
together with the incommensurate nature of Te-III, makes hard to perform reliable
optical properties calculations for this system. As seen above the Te-II and the
(unmodulated) bcm structure are closely related. Hence, the electronic density of
states (DOS) calculated for the Te-II and the unmodulated Te-III structure are
very similar (see Figure 4.19). However, the calculated optical properties differ remarkably, thus the distortion from the (unmodulated) bcm play a key role in the
determination of the system interaction with light (Figure 4.20).
4.4 Ab-initio calculations on Te-II high pressure phase.
(bcm) ⊥ c
(bcm) � c
Te-II ⊥ c
Te-II � c
18000
σ(ω) Ω−1 cm−1
69
14000
10000
6000
2000
3
5
7
9
11x103
ω(cm−1 )
Figure 4.20. Optical conductivity of the (unmodulated) bcm structure and of the Te-II
structure
Chapter 5
Insulator to metal transition in
liquid Iodine
5.1
Insulator to metal transition in molten elements
It is known that a number of elemental molecular systems under high pressure undergo a transition to a metallic state or, generally speaking, to a high electrical
conductivity state when the distances between bonded and non bonded atoms become comparable. It is worth to notice that these conductivity transitions occur
in both the solid and the liquid phase although with a different phenomenology.
Whereas theoretical and experimental investigations on the transitions occurring in
the solid phase have clarified a number of relevant points, those occurring in the
liquid phase are much less investigated owing mainly to the experimental difficulties in carrying out investigation where high pressure and high temperature must
simultaneously be applied to the sample under study. Despite the several similarities between the transitions shown by the solid and the melt system, remarkable
differences exist between the paths from the insulating to the high conductive state
followed by the system under applied pressure.
In particular it is peculiar and quite unexpected that for wide class of elemental
fluids (Hg, Cs, Rb, and the molecular H2 , I2 ) the conductivity transition appears
abrupt and sharp with a variation of the conductivity of some order of magnitude
over a rather narrow pressure-temperature range. Moreover the metallization in
melts often occurs at lower pressure than in corresponding crystals, and shows a
common phenomenology in the thermodynamic region of the metallization. For
instance, if the conductivity is plotted versus the scaled density n∗ = nr3 (n is the
atomic number density and r the effective atomic radius), an almost universal behavior is observed. The underlying idea is that both the high temperature involved
(2000-4000 K for Cs, Rb and H2 800-900 K for I2 ) and the disorder, peculiar to the
fluid phase, contribute to smear out the differences among the various solids, mainly
due to the crystal symmetry and to the intensity of quantum effects, leading to a
sort of universal fluid in the transition region [67].
Among these systems I is particularly interesting for several reasons. Analyzing
the results of the research carried out on I in the last decades we find out that the
pressure induced transitions in the solid phase have been extensively investigated
and that a clear picture emerges from the theoretical and the experimental liter71
72
5. Insulator to metal transition in liquid Iodine
ature. Moreover, since the transitions in the liquid phase occurs over a pressure
and temperature range accessible to many experimental techniques (400<T<1000
K and 0<P<6 GPa), several experimental information are already available. In
particular, it is well known that both solid and liquid I show the same sequence of
transitions on applying pressure (i.e. from molecular semiconductor to molecular
metal to atomic metal [11, 25, 49, 71, 72]). I is finally interesting in itself since it is
considered as the classical counterpart of the mainly quantistic hydrogen for which
an enormous effort has been devoted and it is still actually devoted to study its
pressure induced metallization transitions in the liquid and solid phases.
5.2
Pressure induced transitions in solid Iodine
In order to gain a deeper understanding on the pressure induced MIT in fluid I2 it
is useful to discuss at first the effects of pressure on solid I2 . At ambient pressure
solid I2 has a layered structure where molecules within a layer are ordered along parallel zigzag patterns. The crystal structure (Phase I) is face centered orthorhombic
(D18
2h -Cmca). Besides the electron pair repulsion and dispersive attraction, multipolar and charge-transfer effects manifest themselves in the dense packing of the
molecular solid. The distance between two adjacent zigzag patterns is 3.50 Å, and
the corresponding distance between the layers is 4.27 Å. The interatomic distance
in molecular I2 is somewhat longer in the solid state (2.715 Å) than in the isolated
molecule (2.666 Å) thus showing that the intermolecular bond is formed at the expense of the intramolecular bond [75]. Moreover even at atmospheric pressure there
is a charge transfer (CT), or a weak covalent bonding between adjacent molecules in
solid I [79]. Solid molecular I can be considered as a two-dimensional semiconductor
with a much higher conductivity in the layer plane than perpendicular to it.
1200
LIQUID
1100
mol.
met.
1000
atom.
met.
T (K)
900
mol. semic.
800
700
600
500
400
300
Phase
I+V
SOLID
Phase I
(mol. semic.)
0
2
4
6
Phase I
(mol. met.)
15
20
Phase
II+V
(atom.
met.)
25
30
P (GPa)
Figure 5.1. I phase diagram. Vertical dashed lines mark the sequence of transitions at
room temperature in the solid phase.
As well as Te, solid I phase diagram has been recently revised [49] after the
5.2 Pressure induced transitions in solid Iodine
73
discovery of a new incommensurate high pressure phase (phase V) above 23 GPa at
room temperature. In terms of the superspace-group representation, this structure
is written as Fmm2(a00)0s0, with the modulation wave vector propagating along the
a axis (q = (0.257, 0, 0) at 25.5 GPa)[49]. On increasing the pressure above above
30 GPa (see Figure 5.1) I is an atomic solid with atoms placed in a body centered
orthorhombic lattice (Phase II, space group D25
2h -Immm)[74]. Both Phase II and
Phase V have a metallic behavior. Moreover the occurrence of the incommensurately
modulated Phase V is accompanied by coexistence of different phases over a wide
pressure range. Indeed Phase I coexists with Phase V over a pressure range of 23-24
GPa, while Phase V coexists with Phase II over a larger pressure range, 25-30 GPa.
It is worth to notice that the occurrence of the Phase II involves the dissociation of
the molecules in the solid I. The room temperature high pressure phase diagram of
solid I is depicted in Figure 5.1.
The nearest interatomic distances in Phase V are continuously distributed over
the 2.86-3.11 Å range. The shortest of these interatomic distances falls between
the bond length of I in the molecular crystal (2.75 Å) and the nearest interatomic
distance in the fully dissociated monoatomic crystal (2.89 Å). The nearest distance
Figure 5.2. Distribution of the near interatomic distances for phases I (19.1 GPa), V
(24.6 GPa) and II (30.4 GPa). The arrangement of atoms in each phase is shown on
the right, which depicts atoms on the initial molecular plane (the b-c plane of Phase
I). The solid and broken lines indicate respectively the unit cell and the relationship
between the different unit cells. The four nearest-neighbour distances are continuously
distributed in phase V. The diagram for phase V is hence obtained by calculating the
interatomic distances for 2500 unit cells along the a axis, and taking the histogram with
a step of 0.004Å. The vertical scale of the diagram is normalized so that the integrated
frequency becomes equal to four, as in phases I and II.
(2.86 Å) of Phase V is definitely larger than the bond length of I2 molecule (2.75
74
5. Insulator to metal transition in liquid Iodine
Å) in Phase I, clearly showing that no diatomic molecules exist in this new phase.
It is, however, difficult to decide whether it has molecular or monoatomic character. Indeed, as a consequence of modulation, linear chains of atoms exist in some
places in the crystal, but not in other places (see Figure 5.3). Moreover the distribution of interatomic distances is exactly intermediate between those for phases
I and II (see Figure 5.2) implying that the incommensurate intermediate phase is
a transient state during molecular dissociation. This behavior is very similar to
that observed and previously discussed for Te since the incommensurate modulated
phase is an intermediate state between the Te-II close-packed structure and the high
pressure rhombohedral one, where the packed structure completely disappear (see
Section 1.4).
Figure 5.3. Crystal structure of I for phases V at room temperature and 24.6 GPa. The
figure shows the projection of atoms onto the planes indicated. Filled and open circles
indicate respectively the atoms lying in the plane and in the adjacent plane halfway
below. The unit cell is indicated by a small rectangle. The interatomic distances are
distributed in the range 2.86-3.11 Å. In order to distinguish the small difference in the
distances, atoms at a distance 2.86-2.92 Å are connected by thick lines, and those at
2.92-3.05Å by thin lines. Atoms at a distance 3.05-3.11 Å are not connected for the sake
of clarity. Phase V has no molecular units, however three or four atom chains appear,
which form domains as indicated by parallelograms. Below the crystal structure is shown
the modulation wave, whose amplitude is exaggerated so as to clarify the modulation.
The positions of atoms progressively shift on the wave, showing the incommensurate
nature.
At room temperature, on increasing pressure, solid I becomes metallic at 16
GPa without changing the crystal structure stable at ambient pressure [49, 71, 72].
The metallization of solid I, as well as that occurring in other Halogens such as
Br2 and Cl2 , results from indirect band overlap. On increasing the pressure on
these molecular systems a progressive and continuous band gap closure occours. In
Figure 5.4 the band structure just above the metallization pressure is reported for
solid Br2[59].
As demonstrated by DFT charge density calculation [85], band overlap transfers
the electrons from the intramolecular to the intermolecular region in solid I. This
mechanism affects heavily the pressure behavior of the Raman spectrum. Indeed
a detailed study of the intensities of the Raman peaks [18] have shown that the
5.2 Pressure induced transitions in solid Iodine
75
Figure 5.4. The band structure of molecular solid bromine at 35 GPa, within the LDA.
intensities of the intermolecular (external) librational modes steeply increase with
respect to the intensities of the intramolecular (internal) stretching ones. As in the
Te case, this behavior is the spectroscopic signature of the onset and of the increased
role of the CT interactions as the pressure is increased.
Raman spectra on liquid I could give some insight into the role of the CT interaction into driving the MIT in liquid I. Indeed MIT in liquid I has some similarities
to the solid case that suggest a key role of the CT interactions. On the other
hand, it is reasonable to expect that the disorder present in the liquid phase at
high temperature can allow mechanisms completely absent in the solid phase. It
is therefore useful to review the electronic and structural properties of the liquid
phase to highlight similarities and differences with the solid phase.
76
5.3
5. Insulator to metal transition in liquid Iodine
Pressure induced transitions in liquid Iodine
The structure of liquid I has also received considerable attention. X-ray diffraction studies have indicated a short-range orientational order suggesting that the
structure of a layer in the solid state is somehow preserved in molten I. The bond
length of I2 in liquid (2.70 Å) is intermediate between the solid and gas phase values. At atmospheric pressure and 400 K, the I melt is a semiconductor with the
energy gap of about 1.34 eV found from optical absorption measurements [67]. On
increasing the pressure (and the temperature) liquid I shows the same sequence of
transitions as the solid phase (see Figure 5.1). Liquid I becomes a metal retaining
its molecular character at much lower pressure (3-4 GPa) that in the solid case,
albeit at much higher temperature (850 K). On further increasing the pressure and
the temperature, above 4.5 GPa and 1000 extended x-ray absorption fine structure
(EXAFS) experiments [15] indicate a complete molecular dissociation. In this phase
diagram region liquid I is an atomic metal. The microscopic mechanism leading to
a lower metallization density (i.e. pressure) in the fluid phase is still not clear. The
transition scheme from both the liquid and solid phase is summarized in Figure 5.1
Figure 5.5. Electrical conductivity σ for fluid I2 and H2 versus reduced density n∗ (see
text). Dashed and full lines are σth calculated for a partially ionized plasma model of H2
at T=15000 K and T=10000 K respectively [69], open and full circles are experimental
conductivities σ for H2 (T=3000 K Ref. [84]) and I2 (T=900 K Ref. [83]). In the inset
the pressure dependence of the experimental conductivity σ for I2 is shown (Ref. [83]).
As mentioned above the MIT in liquid I can be described on a graph of the
conductivity versus the scaled density n∗ . In this way it is possible to see that the
metallization transition in I2 occurs at nearly the same reduced density of H2 and
in the same region of the calculated plasma transition for a partially ionized plasma
model of hydrogen [69]. The steeper increase displayed by the experimental conductivities with respect to those calculated could be ascribed to the large temperature
differences between the two sets of data. Indeed it is clearly evident from the Figure
that the slope of σth at the transition increases as the temperature is decreased. In
the inset of Fig.5.5 the conductivity jump for liquid I is displayed on an expanded
5.3 Pressure induced transitions in liquid Iodine
77
scale, showing that the increase occurs in two separate steps, as it does also for
hydrogen. These fndings suggest that, in the transition region, fluid I2 follows a
path similar to that calculated, at much higher temperatures and pressures, for the
partially ionized plasma model for H2 [69]. Following theoretical calculations for
the H2 plasma the first step of the transition is a large, density-driven, increase in
molecular ‘ionization’, induced by the strong intermolecular interactions brought
about by the disorder of the fluid phase, which produce a rapid increase of the
electrical conductivity. A further increase in density quickly drives the system to
molecular dissociation, as suggested by the EXAFS high-pressure data[67].
5.3.1
Electronic properties of liquid Iodine under pressure
The structure and electronic states of the I2 molecule have received considerable
attention for about a century starting from the observation of the variation of the
color of the I2 solution in different solvents [7]. Recently it is increasingly being
used in novel geometries in nanoscopic systems. An important breakthrough in
TiO2 based photocatalyst was recently achieved by introducing I molecules during
fabrication of a nanovoid-structured TiO2 material [2]. The encapsulated, thermally
stable (I2)n adducts on the cavity walls act as light harvesting antennas to create
excitons transferred to the catalytic reaction zone, and thereby significantly enhance
the TiO2 photoactivity.
Indeed, I is a chemical element with many peculiar properties. Although at ambient pressure it consists in all states of the matter of well defined diatomic molecules
with closed electronic shells, it is capable to interact strongly with other molecules,
including I2 itself. The most widely accepted explanation is the association of an
electron-donor electron-acceptor to produce the so-called charge-transfer complex.
It has been shown that in this theoretical framework it is possible to give a satisfactory interpretation of the observed changes in the UV and visible spectra of I2 with
the molecular environment. The strong correlation between the ionizing potential
of the donor and the blue-shift of the visible absorption band of I2 [60] indicates
that the environment appreciably changes the characteristics of the molecule.
Absorption spectra measurements along the melting line have revealed a marked
effect of the intermolecular interaction in the determination of the optical properties
also in liquid I at high pressure. The spectrum shows an exponential tail in the lowfrequency region, a broad and almost structureless absorption band in the visible
and near UV region and a peak around 18000 cm−1 [67]. This last feature is known
to arise from an intramolecular absorption transition; indeed this peak is almost
identically present also in the gas phase spectra [86] shown in the inset of Figure 5.6
The comparison between gas and liquid spectra reveals also that, apart from the
intermolecular peak, the absorption band originates from forbidden molecular transitions and that it is activated by intermolecular interactions. The low frequency
limit of the spectra presents exponential tails (generally referred to as Urbach tails
[46, 81]), depending strongly on the temperature according to the empirical law
�
�
−(Ef − hν)γ
α(ν) = α0 exp
,
KT
(5.1)
where Ef , α0 , and γ are temperature-independent parameters. The region just
78
5. Insulator to metal transition in liquid Iodine
Figure 5.6. Absorption spectrum of liquid I2 at T=410 K in logarithmic scale. The solid
line is the exponential fit of the Urbach tail and the dashed line is the power law fit
over the band edge region. The inset shows the molar absorption coefficient of liquid
and gaseous [86] I2 in linear scale
above the exponential behavior is well described by the expression
α(ν) =
C(ν − νg )2
ν
(5.2)
where C is related to the absorption intensity and νg is the frequency of the optical
energy gap. At high pressure, where the Urbach tails are relevant, the absorption
spectra do not show a well defined band origin. Thus, the temperature-pressure
dependence of the optical gap was studied making use of the empirical energy gap
Eg defined as the energy value for which the absorption coefficient reaches the fixed
value of αg = 190 cm−1 . The Eg value is mainly pressure (i.e., density) independent,
while a reduction of the band gap occurs when the temperature is increased. The
temperature dependence of the band gap is the same of the solid phase, however
by a linear extrapolation of the data in the liquid case, the optical gap appears
to vanish at temperature values well beyond the observed metallization transition.
This means that in the fluid phase the metallization can not be achieved by a
progressive linear closure of the band gap as it does in the solid case (see Figure 5.7).
As far as the C factor is concerned, its value is almost temperature independent,
while it grows on increasing pressure. As previously pointed out, the band intensity
is related to the intermolecular interactions, thus the huge increase of the absorption
shows that relevant changes of the molecular electronic states occur with increasing density. Moreover, the intensity of the absorption band is about one order of
magnitude larger in the liquid system under pressure than in the solid one, thus
indicating that the intermolecular interactions are more effective in perturbing the
molecular electronic states in the disordered phase, although the density is lower.
5.3 Pressure induced transitions in liquid Iodine
79
Figure 5.7. Values of the energy gap as a function of pressure. Solid and dashed line
are the extrapolated quadratic and linear fit to the data, respectively. The triangle
indicates the pressure at which the beginning of the metallization has been observed
[11, 83]
5.3.2
Structural properties of liquid Iodine under high pressure
The strong dependence of the electronic properties on the environment remarkably
affects the structural properties of the I2 molecules too. Liquid I can be seen as a
CT complex, where some molecules act as donors, while others like acceptors. The
CT decreases the screening effect in the donor molecules, inducing a shortening of
the bond length, while it has an opposite effect in the acceptor molecules. EXASF
analisys had proved a very detailed informations on I molecular parameters in different phases and solvents revealing a marked dependence of both the bond length
and bond length variance on the solvent [13].
In Table 5.3 the bond length of I2 in different solvents, in the liquid, solid and
gas phases obtained from EXAFS experiment [13] are reported together with the
bond length variances σ 2 . It is worth to notice that the variance bond length σ 2
can be placed in relation with the molecular vibrational frequency using the onedimensional harmonic oscillator model
�
�
h̄
βhω
σ=
coth
(5.3)
2νω
2
1
where ν is the reduced mass, β = KT
and ω is the vibrational frequency. The
calculated frequencies using the experimental results from EXAFS measurements
for I in different phases and in different solutions are in good agreement with the
experimental ones as shown in Table 5.1.
The errors reported in Table 5.1 are derived from the statistical errors on R and
σ calculated refining the empirical parameters used in the fit function together with
R and σ [13]. This allows to compare the values in the Table 5.1 with the experimental ones. Nevertheless, the relative comparison among the different samples
in the ambient pressure experiment 5.1 can be performed with reduced statistical
80
5. Insulator to metal transition in liquid Iodine
uncertainty fixing the values of the empirical parameters to those obtained in a
selected sample of the series (i.e., the gas phase). Thus, the relative shifts of the
calculated frequencies become more accurate, and the difference between frequencies in different solvents becomes greater than the statistical uncertainty on the
frequencies shifts.
The clear bond length augmentation and the shift to lower frequency of the
phonon mode, on increasing the ionization potential of the solvent, can be related
to a shift of the minimum of the I-I effective potential to larger distances and
to a broadening of its width, respectively. This ionic potential modification can
be seen as a mark of the CT complex formation between I and solvent molecules,
considering the strict relation between solvent ionization potential, bond length and
atomic vibration shift.
Figure 5.8. Phase diagram of I according to Ref. [11, 83]. The boundary lines between
the molecular semiconducting liquid phase and the molecular metallic liquid (L’) and
monoatomic metallic liquid phases (L”) [11, 83] are also indicated with dashed lines.
The thermodynamic history of the three samples measured in Ref. [15] is indicated
schematically by the arrows.
A series of EXAFS measurements [15] carried out on liquid I at high pressure
and temperature along the melting curve (see Figure 5.8), showed an abnormal
Table 5.1. Results of EXAFS experiment from Ref. [13]
Phase/Solvent
Gas
Liquid
Solid
Benzene
Toluene
2
R(Å)
σ 2 (10−3 Å )
2.681(2)
2.696(8)
2.720(8)
2.687(4)
2.688(2)
4.0(3)
5.0(8)
3.7(8)
3.2(4)
3.0(2)
νEXAF S
207
172
176
191
199
±
±
±
±
±
12
8
11
14
15
νexp
214
194
184
205
203.6
5.3 Pressure induced transitions in liquid Iodine
81
expansion of the bond length. At ambient pressure the bond length in the liquid
phase is known to be shorter than in the solid phase (2.70 Å [13] and 2.715 Å [82],
respectively). In the 0-2 GPa pressure range the bond length in the solid is nearly
Figure 5.9. Pressure dependence of interatomic distances of I in room temperature solid
(open circles) and in liquid along the coexistence curve (full circles). The dashed lines
show the pressure dependence of the first neighbor (molecular) bond distances R1 and
the second (intermolecular) bond distances R2 as measured on solid I2 by XRD (from
Ref. [19]). The solid line is a guide for the eyes.
constant while in the liquid phase it increases appreciably and eventually the order
is reversed at pressure higher than 2 GPa. Moreover, moving along the melting
curve, the bond length variance changes from 5.4 10−3 to 8 10−3 Å [15] while the
vibrational frequencies calculated using 5.3 seem to show a softening (see Table 5.2).
It is worth to notice that the high temperatures used to keep the I in the liquid
phase makes the harmonic approximation used to derive Formula 5.3 less reliable.
Table 5.2. Results of EXAFS experiment from Ref. [15], and intra-molecular bond stretching mode frequency calculate using 5.3. Errors as in Table 5.1
2
Pressure (GPa)
Temperature (K)
R(Å)
σ 2 (10−3 Å )
νEXAF S
1.1
1.8
2.1
630
710
850
2.710(5)
2.723(5)
2.736(8)
5.4(8)
6.(1)
8.(1)
208 ± 10
209 ± 10
198 ± 10
The pressure dependence of the bond length, which is much stronger in the
high density liquid than in the solid, indicates that intermolecular interactions are
more effective in weakening the molecular bond in the disordered phase. Direct
observation of the softening of the intramolecular vibration mode would thus allow
to demonstrate that even in the high pressure liquid, like in the liquid at ambient
82
5. Insulator to metal transition in liquid Iodine
pressure in different solvents, there is a broadening of the I-I potential shape related
to a CT process. Moreover, the extent of softening could offer an understanding
of the role of CT interaction into driving the MIT in the molecular liquid phase at
densities much more lower than in the solid case.
5.4 High pressure and temperature Raman experiment
5.4
83
High pressure and temperature Raman experiment
The vibrational properties, together with detailed molecular structure information
provided by EXAFS for I in different solutions, allowed to relate the simultaneous
presence of both bond length expansion and shift to low frequency to the CT interaction. The molecular vibrational mode is particularly sensitive to the extent of CT
interaction. Namely, a large increase of this interaction should reflect in a marked
softening of this vibrational mode as suggested by the values of the bond length
variance (see Table 5.2).
Raman spectra were measured using the experimental apparatus described in
Section 2.2. Spectra are collected over the 80-600 cm−1 frequency range being the
low frequency limit due to the notch filter cutoff. Since liquid I2 is particularly
reactive at high temperature, Mo gaskets (250 µm thick foil) have been chosen [14].
Under working conditions the gasket thickness ranged from 30 to 50 µm with a
typical 300 µm diameter hole. The I sample (99% purity from Merk) was always
manipulated in a glove box under inert atmosphere. Finely powdered I was put in
a small cave (∼100 µm diameter) carved in the center of a LiF pellet previously
synthesized within the gasket hole. This loading procedure avoids the direct contact
of I2 with the gasket wall preventing chemical contamination of the sample and long
term corrosion of the M o.
The I sample was pressurized following a standard thermodynamic path as described in Section 2.2.1. This procedure allow us to asses the absence of sample
contamination after each pressure temperature cycle by comparing the spectrum of
the final solid sample with the first spectrum collected at the sealing pressure (see
Section 2.2.1 for further details). Efforts to cross the metallization line have failed
due to sample contaminations (as already reported in Ref. [15]). This procedure
has allowed us to obtain points essentially along two defined thermodynamic paths:
an isobaric path at P=0.9 GPa from 665 to 819 K and an isothermal path at about
T= 715 K ranging from 0.9 to 1.8 GPa. We want to point out that the occurrence
of the melting is clearly marked by the disappearance of the factor group splitting
of the vibrational stretching mode (see Figure 5.10) and that all the melting points
we determined lie on the known melting [83].
Raman spectra collected in the liquid phase along the 715 K isotherm are reported in Figure 5.11. The spectra have been collected without polarization selection and corrected only for the transmission of the notch filter to recover the correct
low frequency lineshape at least down to 75 cm−1 . All spectra show a pronounced
low frequency Rayleigh wing and a well separated strongly asymmetric vibrational
band. The asymmetry of the vibrational peak suggests the existence of “hot bands”
originating from transitions starting from thermally excited vibrational states. In
the absence of a polarization analysis isotropic and anisotropic scattering components cannot be separated and thus a detailed analysis has not been attempted.
We will focus on the temperature and pressure evolution of the peak position of
the vibrational band which allows a direct comparison with data in the literature,
see e.g. [57]. Peak frequencies for the spectra at all investigated thermodynamic
states are reported in Table 5.3. We note that while the effect of temperature at
constant pressure is quite modest, there is a marked pressure driven softening of the
vibrational peak frequency. Indeed while at ambient pressure and T=663 ◦ C this
is centered around 180 cm−1 (in agrement with previous results [57]), its position
84
5. Insulator to metal transition in liquid Iodine
Ag
Solid
Liquid
Intensity
B3g
140
150
160
170
180
190
200
-1
210
220
230
240
Raman Shift (cm )
Figure 5.10. Factor group splitting of the vibrational stretching mode in solid I.
decreases down to 171 cm−1 at P = 1.8 GP a and T = 728 K.
Table 5.3. Results of Raman experiment on liquid I at high pressure and temperature
T (K)
P (GPa)
ρ (g/cm3 )
ν̃peak (cm−1 )
692(5)
667(5)
665(5)
728(5)
712(5)
717(5)
718(5)
819(5)
1.6(1)
1.1(1)
0.9(1)
1.2(1)
1.3(1)
1.8(1)
0.9(1)
0.9(1)
4.38
4.25
4.20
4.28
4.30
4.44
4.20
4.20
174(1)
184(1)
185(1)
180(1)
179(1)
171(1)
184(1)
183(1)
In Figure 5.12 we report the density dependence of the vibrational Raman peaks
in the high-pressure liquid phase obtained from the present experiment together
with previous determinations along the liquid-vapor coexistence curve [57] and in
the solid phase [63] (in this case an average of the two Raman active vibrational
frequencies in the solid phase is shown). The continuous decrease of the vibrational
frequency in the fluid phase, which actually begins in the low density gas, is ascribed
to an increase of intermolecular CT interaction. Although a monotone trend is
observed in the liquid phase the softening of the molecular vibration in the highdensity liquid region (present data) is much steeper so that, for density values still
well below those of the solid phase, it brings the vibrational frequency in the liquid
well below the minimum value of 183 cm−1 observed in the solid around the density
of 5.8 g/cm3 . This behavior strongly suggest the presence of a pressure-activated
(i.e. density-activated) threshold mechanism in the disordered phase.
5.4 High pressure and temperature Raman experiment
85
Figure 5.11. Pressure dependence of selected Raman spectra of liquid I along the T=715
isotherm. The vertical dashed line marks the Raman peak frequency at the lowest
pressure.
The estimate of the sample density requires some comments. Since, to our
knowledge, no experimental data are available in the literature on the high-pressure
liquid we have interpolated a theoretical estimate which predicts a density increase
of about 20% going from the melting point (ρM = 3.96 g/cm3 ) to the insulator to
metal transition at P = 3 GP a along the melting curve (these are the densities
reported in Table 5.3 and shown in Figure 5.12). However, since large uncertainties
could affect this estimate, we wish to point out that a lower and upper bound to
the density of our sample can be set between ρM and the density of the solid at
the melting curve at the corresponding sample pressure. The latter is obtained
from the room temperature-high pressure X-ray diffraction data [75] through the
thermal cubic expansion coefficient. Upper and lower density limits reported by
dashed lines in Figure 5.12 clearly show that even if the interpolated densities could
be questionable the basic argumentation on the fast density-induced softening of
the molecular vibration remain unchanged.
The comparison of the density-dependence between the present vibrational frequencies and the I2 bond length obtained in a recent EXAFS experiment [15] could
be quite relevant to understand the origin of pressure-activated microscopic mechanism. In Figure 5.13 the pressure dependence of both the vibrational frequency and
the bond length is shown as functions of the pressure in the solid and liquid I. These
quantities exhibit an analogous behavior so that the pressure effect is of the same
sign in both phases but clearly more pronounced in the liquid one. This comparison
clearly shows that in the fluid phase the CT interaction drives the observed anomalous bond length expansion and that a remarkable weakening of the molecular unit
takes place quite more rapidly than in the solid phase. These findings are also at
the origin of the early molecular dissociation in the liquid phase (P=4 GPa in the
86
5. Insulator to metal transition in liquid Iodine
Figure 5.12. Density dependence of the peak position in the first-order Raman spectra in
the liquid phase along the liquid-vapor coexistence curve [57] (full triangles), in the solid
phase [63] (full line) and in present experiment high-density liquid phase (full circles).
The area between the two dashed lines represents the maximum fluctuation interval for
the density values in the reported measurements. The full triangles and the full line
represent previous measurements along the liquid- vapour coexistence curve [57] and in
the solid phase [63], respectively. The two open triangles are points on the coexistence
curve obtained with the present experimental set-up from an I sample in a heated sealed
quartz cuvette; the consistency with previous data is evident.
liquid vs. P=21 GPa in the solid) [83].
As a whole, the experimental results show that, even in the fluid phase, the
density is the critical thermodynamic parameter in driving the metallization. Moreover, as for the intramolecular structural properties, the density dependence of both
electronic and structural properties of the molecule is much stronger in the dense
liquid phase than in the solid one. On the other hand, this finding is well consistent
with the weak temperature dependence of the measured metallization and dissociation critical lines [83]. The role of the temperature is essentially that of keeping the
system disordered, thus allowing for local configurations with lower binding energy
and larger overlaps of the electronic clouds which are forbidden in the solid phase
[16]. In this context a classical molecular dynamic simulation study [20] has shown
that at high densities and small distances, the steric hindrance favors more compact
geometries (e.g. X-shaped dimers) with respect to more open pair configurations
(e.g. L- or T-shaped dimers), regardless of their lower binding energy. These close
packed geometries thus activate strong CT interactions which lead to a real charge
delocalization.
The activation of short range intermolecular interactions is made easier in the
liquid than in the solid phase where the molecular rotations are mainly frozen by the
crystal lattice. Also in the case of the solid phase, the metallization density is the
leading parameter and the intermolecular CT is the driving microscopic mechanism
5.4 High pressure and temperature Raman experiment
87
Figure 5.13. (a) Measured vibrational frequencies on liquid I (blue circles) as function of
the pressure along the melting line together with the vibrational frequencies on solid I
at room temperature (red circles) from Ref. [63] as function of the pressure. (b) Bond
lengths in both in liquid (blue circles) and solid (red circles) I as functions of the pressure
from Ref. [15].
Figure 5.14. Leading geometry of the I2 dimer for each configuration during the classic
dynamics simulations from Ref. [20].
[18]. The metallization occurs when the pressure increases the density up to about
7 g/cm3 and reduces the minimum distance between non-bonded atoms, rmet , in a
crystalline plane up to about 3.2 Å. On the contrary, in the high density fluid the
transition occurs at a density value which can be estimated around 4.5 g/cm3 [54].
88
5. Insulator to metal transition in liquid Iodine
These early metallization could be understood within a very simple scheme. We
can assume that: a) the molecules in the fluid phase are free rotator, that is, they
occupy a spherical effective volume, b) the dimension of the effective volume depends
on the measured expansion of the bond length, that is, it depends on pressure, c)
rmet is a threshold value for activating a real transfer of charge between molecule.
Under these very crude assumptions the probability P(rmin ≤ rmet ) that a couple of
molecule have a minimum distance lower than rmet can be easily obtained. For the
liquid at the density of the transition P(rmin ≤ rmet ) is around 20 % and therefore
well above the percolation threshold. In this framework the picture that emerges
more consistently is that the number of strongly linked molecules rapidly raises
as function of the increasing density and eventually, after crossing the percolation
threshold, we have the onset of a metallic conductivity through percolative path.
5.5 Ab-initio molecular dynamics simulations on liquid iodine
5.5
89
Ab-initio molecular dynamics simulations on liquid
iodine
Ab-initio Born-Oppenheimer Molecular Dynamics (see Section 3.6 and Section 3.7)
was performed on liquid I to understand the microscopic mechanism at the origin of
the observed metallic behavior at high temperature and pressure in the molecular
liquid phase. In particular, ab-initio molecular dynamics offers the possibility to
connect the structural and the dynamical properties to the electronic ones. However
this approach can be applied to systems with a relatively small number of electrons,
since the computational cost grows quite rapidly with the number of electrons in
the system. For this reason the choice of simulation cell size is a crucial issue. The
size of the simulation box must be large enough to prevent periodic artifacts and to
allow the forces and electronic properties calculation to be converged. However, a
too large box would prevent the simulation of a sufficient long time. Another crucial
issue is the choice of the time step used to integrate the equation of motions (see
Equations 3.43). It must be chosen as long as possible but at the same time it must
allow a correct integration of the equations of motions (i.e., a satisfactory energy
conservation during the simulation) and an appropriate sampling of the bond length
oscillation of the I dimer. The frequency of this oscillation falls around 200 cm−1 ,
corresponding to an oscillation period of 80 fs. A time step of 2.9 fs allows thus to
sample 25 positions along the trajectory for each oscillation which are enough to
properly integrate the equations of motions. Tests conducted with this time step
on an isolated molecule and a cutoff for plane waves of 70 Ryd showed a percentage
change of total energy in 100 ps less than 5 10−6 %.
8e-06
thr = 2 10−6 Ryd
thr = 10−6 Ryd
thr = 10−7 Ryd
Eenergy (Ryd)
6e-06
4e-06
2e-06
0
-2e-06
-4e-06
-6e-06
0
50
100
150
200
250
300
time (fs)
Figure 5.15. Total energy for a I2 dimer during a 0.2 ps simulation as function of threshold
in wave function optimization. A remarkable drift can be observed in the case of
simulation with the 2 10−6 threshold.
The initial positions of the 64 atoms in the simulation box was obtained by
a classical molecular dynamics simulation of 32 I rigid dimers at the density of
the liquid at ambient pressure (3.9 g/cm3 )[3] and at a temperature T=730 K. An
optimization of the wave function at these initial positions gave an external pressure
of 1.6 GPa and a gap of 0.27 eV. Hence, even if the calculated external pressure is
90
5. Insulator to metal transition in liquid Iodine
F3×3×3
FΓ
|F3×3×3 − FΓ |
Force (atomic units)
0.02
0.015
0.01
0.005
0
0
10
20
30
40
50
60
70
Atom index
Figure 5.16. Magnitude of the forces on the 64 atoms in the simulation box calculated
using the the Γ point (FΓ ) and using a uniform 3×3×3 grid of points in the reciprocal
space (F3×3×3 ). The blue line represents the module of the differences between FΓ and
F3×3×3 .
higher than room pressure, the system has an insulating character, as expected for
liquid I under 3 GPa [11].
To study the convergence of the forces and of the electronic properties of the
box with the 64 I atoms, forces and the Kohn-Sham eigenvalues calculations were
performed with and without a k points mesh. In this way it was possible to study
the effect of the simulation box size on the calculated properties and to asses the
accuracy of the results without the k points mesh. These tests showed that using
only the Γ point introduces an error of a few percent in the forces calculation (see
Figure 5.16) and only a slight changes in the density of electronic states (see Figure
5.17), while does not influence the value of the gap.
Thus the simulation was carried out with 2.9 fs time step employing a NoséHoover thermostat set to 730 K using only the Γ point for the wave function optimization at each simulation step. The radial distribution function (rdf) obtained
from a 10 ps ab-initio simulation is shown in the Figure 5.18 and labeled with the
symbol gDF T (r). The experimental rdf (gEXP (r)) and the rdf obtained form the
classical rigid dimers dynamics simulation (gCM D (r) ) used to obtain the initial
positions of the ab-initio simulation are also shown in Figure 5.18. The gEXP (r)
shown was obtained by neutron diffraction study [3] for liquid I at ambient pressure
and 407 K, while gCM D (r) and gDF T (r) are computed for liquid I at the same density of the neutron diffraction study (3.9 g/cm3 )[3] but at a temperature of 730 K.
This high temperature value set in the thermostat during the ab-initio simulations
ensures that the system remains liquid. Indeed, the pressure obtained from the
ab-initio simulation box at this density averaging over different configurations is 1.5
± 0.5 GPa. According to the phase diagram [11] at this pressure, the I exists in the
liquid state for temperatures above 650 K.
Looking at the experimental radial distribution function gEXP (r) the structure
around 4 Å seems to have a double-peaked profile (see arrows labeled as A in
Figure 5.18). Moreover a weak structure, absent in the gCM D (r), appears around 5
5.5 Ab-initio molecular dynamics simulations on liquid iodine
91
Å (see arrow labeled as B). Differently from the classical simulation which basically
shows only two peaks at ∼ 4 Å and ∼ 6 Å, the gDF T (r), also with the present low
statistic, seems to reproduce these structures. We notice that besides a number of
differences between the classical and the ab-initio approach, the most relevant one
is that dimer dissociation it is not allowed in the former. On the contrary dimers
dissociate during the ab-initio simulation. In particular, dissociation occurs when
the dimers are arranged in a L-shaped configuration with a third I atom along the
dimer axis.
In order to understand the origin of the structural features appearing in the
gDF T (r) and not in the gCM D (r), I atoms have been classified into two classes
according to their history during the simulation:
• IM OL , atoms that, throughout the simulation, remain in the dimer formed at
the initial time;
• IDIS , atoms that at some time during the simulation leave the dimer formed
at the initial time.
Then the rdf was calculated for atoms belonging to different classes:
• gIM OL ,IM OL (r): rdf of IM OL type atoms;
• gIDIS ,IDIS (r): rdf of IDIS type atoms;
• gIDIS ,IM OL (r): that is the probability of finding an atoms of type IM OL at
distance r from an atom of type IDIS .
The results are shown in Figure 5.20 and Figure 5.21. We can note that there
is no structure around 5 Å in the gIM OL ,IM OL (r), whereas it is present both in the
gIDIS ,IDIS (r) see Figure 5.20 and in the gIDIS ,IM OL (r) (Figure 5.21). Furthermore,
Γ
3×3×3
200
DOS
150
100
50
0
-6
-5
-4
-3
-2
-1
0
1
2
Energy (eV)
Figure 5.17. Calculated density of electronic states (DOS) of the simulation box at initial
time using the Γ point and a 3×3×3 grid of points in the reciprocal space. In order to
obtain a quite smooth DOS, the bin dimension was chosen to be 0.22 eV. This results
in a “apparent” narrowing of the gap near the chemical potential which is set to 0 eV
in the both calculation.
92
5. Insulator to metal transition in liquid Iodine
2.5
gEXP (r)
gDF T (r)
gCM D (r)
2
g(r)
1.5
1
0.5
0
A
2
3
4
B
5
6
7
8
r(Å)
Figure 5.18. Experimental radial distribution function gEXP (r) for I atoms (black line)
from neutron diffraction study [3], gDF T (r) from ab-initio simulation (blue line) and
gCM D (r) from the classical rigid dimer simulation (green line) used to obtain the initial
positions of the ab-initio simulation.
Figure 5.19. Selected frames from the ab-initio simulation. The formation of a trimer
starting from the two dimers placed in a L-shape configuration on the top of the figure
is clearly observable.
the presence of the intramolecular peak in the gIDIS ,IDIS (r) indicates that an atom
that dissociates tends to form a new dimer. Hence the liquid remains molecular
despite the presence of dissociation. The above analysis means that the features in
the experimental rdf not reproduced by the classical simulation seem to be ascribed
to a dynamic dissociative-recombination process.
The comparison between the atomic density map and the gDF T (r) shown in
Figure 5.22 allow us to correlate the main features of the gDF T (r) to the I atom configuration during the intermediate stage of the dynamic dissociative-recombination
process. Indeed, the intermediate stage of this process is the formation of linear
chains of three I atoms.
The increase in the number density map in Figure 5.22 at distance of 3 Å from
g(r)
5.5 Ab-initio molecular dynamics simulations on liquid iodine
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
93
Experimental
gd (r)
gnd (r)
A
2
2.5
3
3.5
4
4.5
B
5
5.5
6
r(Å)
Figure 5.20. The gIM OL ,IM OL (r) and gIDIS ,IDIS (r) (see text for the meaning of the notation) obtained from the an initio molecular dynamic simulation. The intramolecular
peak is clearly observed both in gIM OL ,IM OL (r) and in gIDIS ,IDIS (r). Arrows labeled as
‘A’ and ‘B’ indicate the features absent in the classical rdf. Black line is the experimental
radial distribution function.
1.6
1.4
g(r)
1.2
1
0.8
0.6
gEXP (r)
gIDIS ,IDIS (r)
gIM OL ,IM OL (r)
gIDIS ,IM OL (r)
0.4
0.2
0
2
2.5
3
3.5
4
4.5
5
5.5
6
r(Å)
Figure 5.21. The gIM OL ,IM OL (r) and gIDIS ,IDIS (r) together with the gIDIS ,IM OL (r) (see
text for the notation). Black line is the experimental radial distribution function. The
intramolecular peak is absent in the gIDIS ,IM OL (r). Se text for further details.
the Ia atom (see the black arrow) is due to the presence of a third I atom which
is added to the dimer to form a trimer. This explains the weak structure in the
gIDIS ,IM OL (r) at distance slightly greater than the bond length, i.e. 2.71 Å (see
Figure 5.21). Moreover the increasing of the density in this regions of the map is
related to the peak around 5.8 Å too, since the circumference centered at the Ib
atoms with a 5.8 Å radius passes through this region.
Similarly we can deduce that the peak around 5.2 Å is related to the presence
of a fourth I atom at distance of 5.2 Å from the Ib atom along the dimer axis (see
blue dashed circumference centered in the Ib atom). The presence of this atom
94
5. Insulator to metal transition in liquid Iodine
1.6
1.4
1.2
g(r)
1
0.8
0.6
0.4
0.2
0
gDF T (r)
2
3
4
5
6
7
r(Å)
Figure 5.22. Number density map of a dimer that forms a linear trimer during the
simulation. The gDF T (r) is reported for sake of comparison. Dashed line circumference
in the density map represents the distance from the I atoms at which the main features
of the gDF T (r) occurs. The radii of the circumference are reported in the gDF T (r) graph
as vertical lines with the same line style.
allows the formation of two new I dimers. Indeed, the considerable increase in the
density of particles in the opposite side of the Ia atoms makes much more likely the
dissociation of the Ia -Ib dimer.
According to the classical molecular dynamics simulation reported in Ref. [20],
5.5 Ab-initio molecular dynamics simulations on liquid iodine
95
on increasing the density (i.e. the pressure) more compact geometries should be
favored (e.g. X-shaped dimers) with respect to more open pair configurations (e.g.
L- or T-shaped dimers). There is no evidence of these compact geometries in our abinitio simulation. It should be thus very interesting to perform ab-initio molecular
dynamics simulation at higher density in order to asses the presence of these compact
geometries and to see the effects of these compact geometries in the dissociativerecombination process.
Considering the results of our ab-initio simulation and the presence of the features in the experimental gEXP (r) (labeled as ‘A’ and ‘B’ in Figure 5.18) which
are not observed in the gCM D (r) resulting from the classical molecular dynamics
simulation used to obtain the initial positions of the ab-initio simulation, we can conclude that in the molecular insulating liquid phase there is a dynamic dissociativerecombination process. This process could be the threshold mechanism that leads
to a metallic molecular liquid phase at pressure much lower than in the solid case.
The ab-initio molecular dynamic simulations carried out at longer times and at
other points of the phase diagram which are currently in progress, could be particularly useful to understand the microscopic mechanisms leading to the metallic
behavior of the high pressure molecular liquid phase and to relate the dynamic
dissociative-recombination process to the gap closure.
Chapter 6
Conclusions
Recent advances in high-pressure experimental and theoretical methods have led to
a new level of understanding of phase diagrams and structures of materials under
pressure. In particular pressure induced insulator to metal transitions and new unconventional high density structural phases have been deeply investigated. It worth
to notice that, in particular, the small distortions involved in the formation of a
incommensurately modulated phases often observed as a precursor of the metallization transition make it particularly difficult to distinguish between modulated and
unmodulated phases using only diffraction techniques. Mainly for this reason the
investigation of high pressure phases have to be carried out by properly combining different experimental and theoretical techniques. In the recent years indeed a
number of phase diagrams, including those for simple elemental molecular systems,
have been deeply revised also by recognising the onset of new incommensurately
modulated crystal phases under pressure such as in the case of Te and I2 .
In the present research work we have focused the attention on the high pressure
phases of solid Te and liquid I2 where the incommensurately modulated solid phases
can be seen as a transient state during the pressure induced molecular dissociation
which occurs in both liquid and solid phases. High pressure Infrared and Raman
spectroscopy measurements have been carried out to study both the systems. The
experimental results obtained have been combined with those derived from specific
theoretical DFT calculations to gain a deeper understanding of the nature and
of the physical rules governing the high pressure phases. Beside investigating the
insulator to metal transitions in the two systems a particular attention has been
devoted to the incommensurately modulated structures and their relevance on the
pressure induced molecular-atomic transition.
6.1
Solid Tellurium
Room temperature Raman and infrared measurements over the 0-15 GPa pressure
range have been carried out on crystalline Te. Complete DFT calculations have been
performed over almost the same pressure range. These theoretical results provided
reliable reconstructions of the optical spectra and allowed us to investigate the
nature and the stability of the Te high pressure phases. The most important results
obtained can be schematised in the following four points:
• The simple observation of a structured Raman signal above 11 GPa confirms
97
98
6. Conclusions
the recently claimed new high pressure scenario for Te.
• The pressure dependence shown by the Raman spectra on approaching the
MIT (i.e. over the 0-4 GPa range) indicates a remarkable intra- / inter-chain
charge transfer which is identified with the leading microscopic mechanism
driving the system toward the metallic phase above 4 GPa. Our ab-initio
calculations allows us to confirm this experimental finding and to directly visualize the progressive charge transfer from inside to outside the spiral chains.
• The analysis of the Raman spectra collected above 8 GPa combined with recent inelastic x-ray scattering measurements have been coherently interpreted
as the signature of the incommensurately modulation of the Te-III phase.
• Polarized Infrared reflectivity measurements have shown an unexpected strong
enhancement of the optical anisotropy response of Te on increasing the pressure. Preliminary ab-initio calculations show this anomalous pressure dependence due to the persistence of strong anisotropic bond even in the Te high
pressure phase rather than to specific crystal structure.
The last point deserve a further theoretical effort, already started, which help
us in understanding this peculiar pressure behavior.
As a whole our results on high pressure solid Te thus provide a new deeper understanding of the effects of lattice compression on the physical-chemical properties
of solid Te. In particular it well evident that our combined theoretical/experimental
approach is mandatory for investigating these unconventional phases and that the
onset of incommensurate lattice modulation can have a deep impact on the developments of metallic and purely atomic states at high pressures. Our investigation
allow us the to identify the charge transfer as the precursor mechanism of the MIT
in solid Te. Moreover this mechanism could be related to the onset of the incommensurate lattice modulation which, weakening the chain structure of the ambient
pressure structure, favors the onset of an isotropic metallic lattice.
6.2
Liquid Iodine
High-pressure, high-temperature Raman experiments were carried out on liquid
Iodine over the 1.0-2.0 GPa and 665-920 K pressure and temperature ranges. Abinitio molecular dynamics simulation of liquid Iodine at the ambient pressure density
and at the temperature of T=730 K have been carried out. The most important
results we have obtained on liquid Iodine are:
• Owing to the remarkable experimental difficulties, the realization of Raman
measurements at high pressure and high temperature on a very high chemical
reactive system such as liquid iodine is a results in itself.
• The analysis of the Raman spectra collected in the liquid approaching the
MIT shows a strong anomalous softening of the intramolecular vibrational
mode. This finding, together with the remarkable bond length expansion
previously observed by EXAFS experiments, is the signature of a strong charge
transfer from inside the molecules to the region in between the molecules. This
microscopic mechanism although active also in the solid phase is much more
6.2 Liquid Iodine
99
pronounced in high pressure liquid and can be seen as the leading microscopic
mechanism driving the system toward the metallic phase.
• Our ab-initio molecular dynamics simulations show a dynamical exchange of
iodine atoms between adjacent molecules. The charge transfer interaction
are indeed so strong in liquid iodine as to lead to a dynamic dissociationrecombination process. The analysis of our theoretical results show that this
mechanism clearly results in some characteristic features in the radial distribution function which are well observable in the experimental radial distribution
function.
The discovery of this dynamic dissociation-recombination process is in a full
agreement with the weakening of the molecular unit showed by our Raman experiments. The charge transfer interaction thus produces strong charge density fluctuations in liquid iodine which resemble the onset of the incommensurately modulated
phase in solid phase. This analogy is even stronger if we consider the that the intermediate step of the dissociation-recombination process is the formation of linear
trimers in liquid iodine as well as the incommensurate modulation in the high pressure crystal phase involves the formation of trimers in some place in the crystal and
isolated iodine atoms in others. In solid iodine we thus have spatial charge density
fluctuations whereas in the liquid phase these charge fluctuations are both space
and time dependent. These strong fluctuations could be related to the formation of
dynamical percolative paths which could be at the origin of the early metallization
transition in the liquid. Ab-initio molecular dynamic simulations carried out at
longer times and in other regions of the phase diagram are currently in progress
to demonstrate the existence of a percolative threshold for the onset of a metallic
phase. The study of a possible link between the dissociation-recombination mechanism rate and the values of the gap could be decisive in the understanding of the
MIT in liquid iodine.
Generally speaking the evolution of the dynamical and of the bonding properties is related to the Born-Oppenheimer surface shape, i.e. to the electronic ground
state properties. The evolution of the dynamical properties are thus closely linked
to the evolution of the dynamical properties. The approach used in the present thesis allows us to obtain a detailed study of this connection. Moreover it is clear how
this approach is useful in the investigation of the microscopic precursor mechanism
leading the insulator to metal transition. Moreover the importance of incommensurate lattice modulations in the high pressure solid phases and its relation with
the pressure induced MITs in both the systems investigated strongly suggest a
multi-techniques approach. Indeed the combined use of experimental spectroscopic
investigations and theoretical ab-initio calculations has thus revealed a powerful
tool in the investigation of the effects of the lattice compression in molecular (or
molecular-like) systems.
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