Journalof Low TemperaturePhysics, Vol. 75, Nos. 1/2, 1989
Calculation of Band Structure and Superconductivity in
the Simple Cubic Phase of Phosphorus
M. Rajagopalan*, M. Alouani, and N. E. Christensen
Max-Planck-Institut fiir Festkiirperforschung, Stuttgart, Federal Republic o f Germany
(Received August 9, 1988)
We report a calculation of the band structure and superconductivity of phosphorus in the simple cubic phase under pressure. The effect of pressure on the
band structure is obtained by means of the linear muffin-tin orbital method.
The superconducting transition temperature (To) is calculated using the AllenDynes formula. It is found that the value of Tc increases continuously with
pressure from 110 kbar up to 210 kbar and then decreases. The change in the
slope of T~ is associated with the appearance of a new piece of Fermi surface.
The calculated values of T¢ are compared with the available experimental data.
1. I N T R O D U C T I O N
Black phosphorus is the most stable form among the many allotropic
modifications of phosphorus under normal condition. It is a layered, narrowgap semiconductor and crystallizes with an orthorhombic structure. Phosphorus shows an interesting sequence of structural transformations from
orthorhombic to rhombohedral and then to a simple cubic phase with
increasing applied pressure. 1-3 Several band structure calculations 4-5 are
reported in the literature for the orthorhombic semiconducting phase. One
calculation 6 exists in which the total energies are calculated for the different
structures as a function of atomic volume. The authors have calculated the
transition pressure between the rhombohedral and simple cubic phase. It
was found to be about 100 kbar, which is in close agreement with the
observed value 3 of about 110 kbar.
On the experimental side, there are a number of studies of the electronic
properties of black phosphorus in the orthorhombic phase. Quantities
measured so far include the electrical anisotropic conductivity, the Hall
coefficient, cyclotron resonance, 7-1° ultraviolet photoemission spectroscopy
(UPS), X - r a y photoemission spectroscopy (XPS), 11-12angle resolved UPS, 13
*Permanent address: Department of Physics, A n n a University, Madras 600025, India.
1
0022-2291/89/0400-0001506.00/0© 1989PlenumPublishingCorporation
2
M. Rajagopalan, M. Alouani, and N. E. Christensen
core-exciton induced resonant photoemission, 14reflectance in the soft X-ray
region, 1°and phonon dispersion by inelastic neutron scattering. 15In addition
to the above mentioned properties, the superconducting transition temperature Tc is also measured for phosphorus under pressure by Wittig et
al. 16 and Kawamura et al. 1~, but the two results disagree. Wittig et al. showed
from their measurement that Tc exhibits two distinct maxima at about
120 kbar and 230 kbar separated by a minimum at about 170 kbar. After
230 kbar Tc begins to decrease. The first maximum occurs at a pressure
where the structural transition takes place from rhombohedral to the simple
cubic phase. Kawamura et al. showed from their measurement that T~
increases continuously with pressure in the simple cubic phase.
The present work is concerned with the calculation of T~ as a function
of pressure in the simple cubic phase. It is motivated by the ambiguity in
the two experimental investigations and intends to give a physical explanation for the variation of T~ with pressure. The parameters needed in the
calculation of T¢ are derived from the theoretical band structures. The band
structure calculations are performed by the self consistent linear muffin-tin
orbital method (LMTO)) 8 Superconducting transition temperatures are
calculated using the Allen-Dynes 19 formula. The organization of the paper
is as follows. In Section 2, the details of band structure calculations are
given. Section 3 deals with the calculation of Tc and the discussion of the
results. The last section contains the conclusions.
2. EQUATION OF STATE AND BAND STRUCTURE OF SIMPLE
CUBIC PHOSPHORUS
The energy bands of phosphorus in the simple cubic phase are calculated by means of the LMTO method within the atomic sphere approximation (ASA). The calculations include the relativistic mass-velocity and
Darwin corrections but neglect the spin-orbit coupling and may be referred
to as scalar relativistic. An "empty" sphere is added in the unit cell to have
a better description of the charge density since the simple cubic structure
is a loosely packed structure. The calculations are thus performed in the
CsCI structure with two atoms in the unit cell, one real atom, and the empty
sphere. To find the equilibrium lattice constant in the simple cubic phase,
the total energies are computed by varying the cell parameter (lattice
constant) from 2.76 A to 2.26 A. The Birch equation of state fitted to the
total energies for the valence electrons is shown in Fig. 1. The total energy
at the equilibrium lattice parameter (2.43 A) obtained in the present work
is -2.65 Ryd/electron which is in good agreement with the value of
-2.642 Ryd/electron obtained by the pseudopotential calculation. 6 The
pressure and the bulk modulus are obtained from the derivatives of the
3
The Simple Cubic Phase of Phosphorus
-13-51
I
I
i
I
2.4
I
2.5
I
2.6
I
52
~
53
e'-
54
•
O
O
i-.,
55
56
2.3
I
2?
2.8
Cell parameter (•)
Fig. 1. Calculated total energy as a function of cell parameter of
phosphorus in the simple cubic phase. The arrow indicates the
experimental value of the cell parameter (2.393 •) of the simple
cubic phase (SC) just after the transition from the rhombohedral
phase.
I
I
I
I
200
I
400
I
600
14
A
E
...,a
12-
0
>
10C
8
0
Pressure (kbar)
Fig. 2. Calculated volume versus pressure (P) of phosphorus
in the simple cubic phase. The experimental data of Shirotani
et al. 2 are indicated by (O--O).
4
M. Rajagopalan, M. Aiouani, and N. E. Christensen
fitted total energies, and, in Fig. 2, we compare the theoretical and experimental pressure-volume relation in the simple cubic phase. We find a good
agreement between the calculated and the experimental 2 pressures. The
calculated value of the bulk modulus (B) is 1084 kbar, whereas the experimental value is 955 kbar 3 at V~ Vo=0.75, where Vo is the experimental
volume at ambient pressure in the orthorhombic phase. The values of the
calculated pressure and bulk modulus are given in Table I.
The energy bands are computed for different lattice parameters from
2.393/~ to 2.245 ~. The energy bands near the Fermi level along the
high-symmetry directions are given for three volumes ( V/Vo = 0.715, 0.686,
and 0.68) in Fig. 3. From the figure it can be seen that the band at the
symmetry point R which is mainly s-like is moving up and the d-like band
at M drops down with respect to the Fermi level E~. This drop below the
Fermi level, which changes the topology of the Fermi surface, is achieved
for V/ Vo- 0.686, which corresponds to a pressure greater than 210 kbar.
To examine the appearance of the new branch, we construct the Fermi
surface cross-sections in the unfolded irreducible Brillouin zone (IBZ) as
a function of volume; they are given in Fig. 4 for the same volumes as that
of the band structures in Fig. 3. The size of the hole pocket around R
decreases and an electron pocket appears around the point M. The total
and l partial state densities are calculated by means of the tetrahedron
method 2° using 1220 points in IBZ. This large number of k points was found
to be necessary for the convergence of the density of states at EF. The total
density of states N(EF) and the partial density of states (DOS) evaluated
TABLE I
Cell Parameter, V~ Vo, and Calculated Pressure and Bulk Modulus of Phosphorus in the Simple Cubic Phase
Cell parameter
(A )
V~ Vo
Pressure
(kbar)
Bulk modulus
(kbar)
2.393
2.382
2.374
2.371
2.364
2.355
2.350
2.342
2.331
2.326
2.310
2.288
2.267
2.245
0.725
0.715
0.708
0.705
0.699
0.691
0.686
0.680
0.671
0.666
0.652
0.634
0.616
0.599
110.7
132.8
149.4
156.6
174.1
194.8
208.4
228.6
256.0
270.6
317.4
407.0
488.4
579.4
1372.8
1420.0
1452.4
1463.3
1496.4
1535.7
1546.2
1598.8
1645.5
1669.1
1696.6
1746.2
1847.1
1950.7
The Simple Cubic Phase o f Phosphorus
5
o.6 :iI.........
",L::I
0.3 - " ', "........."~-.
." ..."
.":{
/'%"'..
i
......
EF
[
.'
•
-
." I"."%
-=,,.o.3
• -iiiiiiii %
.:
•
(I)
,r-
'..
- ".,.~...
"
tu -O.6
:
V/Vo:= 0.715
-0.9
/
L
-1.2
....'Y
""...,.....
r
0.6
M
R
•. '...
•
r
=;
;.
•
X
".
•
.;
•
'.
".
'."
."
.
""--
0.3
.
. ".
'.
" .
'~
u..I
0.0
EF
' '.".
V/Vo--0.686 ' .,...
". ".. i.
"-.
R
"°3r0.6 . -
.
'. _
0.3
"
Fig. 3. Calculated band structure of simple
cubic p h o s p h o r u s along some high symmetry
directions for several value o f V/Vo. (a)
V/Vo=0.715; (b) V/Vo=0.686; (c) V / V o =
0.680 (for (b) and (c), only the bands near the
Fermi level are shown). Vo is the experimental
volume at ambient pressure in the orthorhombic
phase.
''~
0.0
.
".
'.., I.
':-.
.".
WVo=Oe8: ';.
".."...
".. '..
-0.3f.
r
...
"- I •
...
c~
t=
"'~.
":
=
X
",.....
..
•
~,
'.
M
R
M
EF
6
M. Rajagopalan, M. Alouani, and N. E. Christensen
V/Vo= 0.715
F
R
v/v o = 0.686
r"
i2
x
v / v o = 0.68
I-
R
R
Fig. 4. Calculated Fermi surface contours of phosphorus in
the simple cubic phase at V/Vo= 0.715, 0.686, and 0.680 in
the unfolded irreducible Brillouin zone. (The numbers give
the band indices.)
The Simple Cubic Phase of Phosphorus
7
TABLE II
Total and Partial Density of States at the Fermi Level and Number of Electrons of Phosphorus
in the Simple Cubic Phase for Different Values of V~ Vo
DOS
((states/Ry)/atom)
NOS
(electrons/atom)
V~ Vo
N(EF)
((states/Ry)/atom)
s
p
d
s
p
d
0.725
0.715
0.708
0.705
0.699
0.691
0.686
0.680
0.671
0.666
0.652
0.634
0.616
0.599
4.073
4.015
3.976
3.959
3.921
3.883
3.870
3.991
4.037
4.048
4.090
4.111
4.135
4.157
0.520
0.508
0.499
0.495
0.486
0.482
0.480
0.473
0.462
0.450
0.435
0.402
0.383
0.358
2.568
2.520
2.488
2.474
2.443
2.408
2.387
2.376
2.362
2.348
2.321
2.302
2.284
2.252
0.985
0.987
0.989
0.990
0.992
0.994
1.003
1.142
1.213
1.251
1.334
1.407
1.467
1.546
1.864
1.858
1.854
1.852
1.848
1.843
1.840
1.837
1.832
1.829
1.823
1.815
1.806
1.800
2.612
2.612
2.612
2.612
2.613
2.613
2.613
2.612
2.612
2.610
2.608
2.605
2.601
2.596
0.524
0.530
0.534
0.536
0.540
0.544
0.547
0.551
0.557
0.560
0.566
0.580
0.592
0.604
at E F are given a l o n g with the n u m b e r o f electrons ( N O S ) in T a b l e II. The
total a n d t h e p a r t i a l d e n s i t y o f states at Ev are u s e d later in the c a l c u l a t i o n
o f To.
3. C A L C U L A T I O N O F S U P E R C O N D U C T I N G
TRANSITION TEMPERATURE
The s u p e r c o n d u c t i n g t r a n s i t i o n t e m p e r a t u r e is c a l c u l a t e d using the
A l l e n - D y n e s 19 f o r m u l a
T=(l~exp[A-I'04(l+A)
]
-g*(1 +0.62A)J
(1)
w h e r e (w) is the average p h o n o n frequency, A the e l e e t r o n - p h o n o n c o u p l i n g
constant, a n d / z * is the e l e c t r o n - e l e c t r o n i n t e r a c t i o n constant. M c M i l l a n 21
h a d s h o w n t h a t A can b e written as
A=
N ( E F ) ( I 2)
M{w2 }
(2)
w h e r e M is the a t o m i c mass, (w z) an a v e r a g e s q u a r e d p h o n o n f r e q u e n c y ,
a n d ( I z) the s q u a r e o f t h e e l e c t r o n - p h o n o n m a t r i x e l e m e n t a v e r a g e d o v e r
t h e F e r m i surface. G a s p a r i a n d G y o r f f y = h a v e s h o w n t h a t ( I z) can b e
e x p r e s s e d , w i t h i n the rigid muffin-tin a p p r o x i m a t i o n ( R M T A ) , in t e r m s o f
8
M. Rajagopalan, M. Alouani, and N. E. Christensen
the scattering phase shifts. However, in the atomic sphere approximation,
the usual phase shift notation becomes meaningless and (12 ) can be written
as (in atomic Rydberg units) 23
( l + 1)
Nt(Ep)Nt+I(EF)
(12) = 2 ~ (21+ 1)(21+3) M'21~-a N ( E F ) N ( E ~ )
(3)
where Mr,t+1 is the electron-phonon matrix element which can be expressed
in terms of the logarithmic derivates
d In ~bt[
D~- d l n r Ir=s
(4)
evaluated at the sphere boundary23:
M~,t+l = -6,6t÷l[(Dt(Ee) - l)(Dt+~(EF) + l + 2) + (Ep - V ( S ) ) S 2]
(5)
where V(S) is the one electron potential and tkt the sphere boundary
amplitude of the l partial wave function evaluated at EF. Nl in Eq. (3) is
the partial density of states function for the angular momentum quantum
number/.
The parameters that are entering into the calculation of M~,~+~are taken
from the band structure results. The contributions to the matrix elements
Msp and Mpd from the empty site is found to be small when compared to
those from the phosphorus site. The average of the square of the phonon
frequency (w 2) to be used in Eq. (2) is calculated using the Debye temperature 0o. Following the work of Papaconstantopoulos eta/., 24 (co2) is
set equal to 0.5 0 2. Moruzzi et al. 25 has derived a simple relation to calculate
the Debye temperature from the electronic structure which is
/ Soa~ '/2
Oo = 41.03 ~"~--)
.
.
.
.
(6)
where B is the bulk modulus evaluated at the equilibrium Wigner-Seitz
sphere radius So and M is the atomic mass. They have used the above
relation to calculate 0o for 14 nonmagnetic metals and found good agreement with experimental values. This relation was used in the present work
to calculate 0o which is found to be 413 K.* The electron-electron interaction
constant /x* is obtained from the empirical relation given by Bennemann
and Garland 26
~* -
0.26N(EF)
1+ N ( E F )
(7)
*Since no experimental data are available for the variation of the Debye temperature with
pressure, relation (6), which is valid only at the equilibrium lattice constant, is used to
calculate the Debye temperature. The variation of the phonon spectrum under pressure is
not studied in the present work.
The Simple Cubic Phase of Phosphorus
I
I
I
9
I I"
As"
10
o
8
x,"
o
6
~
i I"
-,// :o
-
•
oO, ;
O
k---
t,
•x
4
•
I
A
I
I
50
I
I
I
100 150 200 250
Pressure (kbar)
Fig. 5. Calculated superconducting transition temperature Tc versus theoretical pressure (P) of phosphorus in the simple cubic phase. The experimental data
of K a w a m u r a et aL 17 are indicated by ( A - - A ) and those
of Wittig et al) 6 are indicated by ( A A O 0 x[S]).
TABLE III
Electron-Phonon Matrix Element (IZ), Electron-Phonon
Coupling Constant h, and Tc for different volumes o f V~ Vo
V~ Vo
(12 )
((eV/A) 2)
;t
T~(K)
0.725
0.715
0.708
0.705
0.699
0.691
0.686
0.680
0.671
0.666
0.652
0.634
0.616
0.599
22.659
23.627
24.312
24.603
25.293
26.107
26.457
25.245
24.934
24.604
24.260
23.751
23.329
23.012
0.598
0.614
0.626
0.631
0.642
0.656
0.663
0.653
0.652
0.645
0.640
0.632
0.624
0.619
6.86
7.45
7.86
8.03
8.44
8.93
9.21
8.82
8.81
8.55
8.35
8.08
7.82
7.63
10
M. Rajagopalan, M. Aiouani, and N. E. Christensen
The superconducting transition temperature is calculated as a function of
pressure and is given in Fig. 5. Table I I I contains the values of Tc along
with the other parameters that enter into the expressions to calculate To.
It is seen both from Fig. 5 and Table I I I that the value of Tc increases
with increase in pressure up to about 210 kbar and then begins to decrease.
There is a continuous s - t o - d electron transfer under pressure which is
responsible for the increase of Tc with pressure up to 210 kbar. The experimental curve of Wittig e t al. 16 shows to distinct maxima at about 120 kbar
and 230 kbar, separated by a minimum at about 170 kbar. They attributed
the first m a x i m u m to p h o n o n - m o d e softening in connection with the structural transition from rhombohedral to simple cubic phase. 16 They also
suggested that there are covalent bonds in the simple cubic phosphorus just
above the structural transition. With increasing pressure the covalent bond
character diminishes and phosphorus becomes a simple metal. However,
the decrease in the value of T~ in the experimental work of Wittig e t al. in
the pressure range from 120 kbar to 170 kbar may be due to the presence
of mixed phases. The experimental curve a K a w a m u r a e t al. 17 shows that
Tc increases continuously with pressure in the simple cubic phase.
The decrease of T¢ above 210 kbar is due to the fact that there is sudden
increase in the total density of states at the Fermi level. This can be
understood as follows: Fig. 6 shows the decomposition of the matrix element
25
--" 2O
5~
I--I
v
~
o - - __i~._.4~.. o_ . . . .
•
100
i
( :r2pd )
o- - o - - - i
i
150
200
P r e s s u r e (k bclr)
....
-o--.o
n
250
Fig. 6. The Fermi surface average of the electron-phonon interaction (I 2)
and the individual contributions to (12) from the s, p and p, d scattering
processes as a function of pressure.
The Simple Cubic Phase of Phosphorus
11
(12) into (I,,p)
2 and (Ip,2 d). Obviously, (I 2~.p) is responsible for the change in
the value of To. From our calculations we find that M]pN~(Ev)Np(EF) is
continuously increasing under pressure. For pressures exceeding 210 kbar,
a new piece of Fermi surface appears around M. These states are predominantly of d-character and cause the d partial density of states and the
total density of states at the Fermi level to increase. Since N,(Ev) and
Np(EF) are varying smoothly with pressure, it therefore follows from Eq.
(3) that (I~.p)
2 decreases for pressures above 210 kbar.
If the variation of an external parameter (such as here, with pressure)
causes changes in the Fermi surface topology, the variation is likely to affect
the otherwise monotonous variation of T~. This was pointed out by Lifshitz, 27
who has demonstrated that there will be a change in the density of states
and consequently a change in the associated thermodynamic properties if
a closed surface transforms into an open surface by the formation of a new
"neck". It was also shown by Makarov et aL28 for thallium that there is a
change in the density of states due to a change in the topology of the Fermi
surface, which led to a nonlinear contribution to To. A similar behavior
was also observed by Chu et al. 29 for rhenium. Thallium and rhenium are
the two elements which have been found to show an anomalous pressure
dependence of To. Merriam 3° has shown that if the Fermi surface is very
close to the Brillouin zone boundary and not touching the boundary, normal
scattering processes play an important role in determining the electronphonon coupling constant. I f the Fermi surface touches the Brillouin zone
boundary, the U m k l a p p scattering process will also be present in addition
to the normal scattering processes. Since (•2) is averaged over the Fermi
surface, the strength of the coupling constant is determined by the nature
of the scattering process. Depending upon which process dominates, the
value of the coupling constant may increase or decrease.
4. C O N C L U S I O N S
We have calculated the band structure of phosphorus in the simple
cubic phase as a function of pressure by means of the LMTO method. The
total energies are computed as a function of volume from which we obtained
the equilibrium lattice constant, pressure, and bulk modulus. From the band
structure calculation, we find that the s-like states around R move towards
higher energies, and the d-like state at M drop below the Fermi level, i.e.,
there is a continuous s-to-d electron transfer under pressure. The changes
in the Fermi surface topology under pressure are also studied. We find that
the size of the hole pocket around R is decreasing and a new piece of Fermi
surface appears around M at about 210 kbar. The superconducting transition temperature is calculated as a function of pressure using the Allen-
12
M. Rajagopalan, M. Alouani, and N. E. Christensen
D y n e s formula. We find that the value o f Tc increases with pressure u p to
210 kbar a n d then start to decrease. The initial increase o f Tc is attributed
to the c o n t i n u o u s s - t o - d electron transfer u n d e r pressure. Since a n e w piece
of Fermi surface appears a r o u n d M, which is d-like in character, the density
of states at the Fermi level begins to increase after 210 kbar. This causes
the value o f Tc to decrease.
ACKNOWLEDGMENTS
We are greatful to Dr. K. Syassen for a critical r e a d i n g of the m a n u s c r i p t .
O n e of us (M. R.) w o u l d like to express sincere gratitute to the m e m b e r s
of M a x - P l a n c k - I n s t i t u t for the w a r m hospitality he received d u r i n g his stay
in the Institute.
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