1
Chapter 15
Relativistic Solid State Theory
N.E. Christensena
a
Institute of Physics and Astronomy,
University of Aarhus
DK–8000 Aarhus C. Denmark
This chapter describes some aspects of how relativistic effects manifest themselves in the quantum theory of solids. Although examples are
mentioned where the energy shifts of energy bands relative to each other
influence the physical properties, we mainly discuss the combined effects
of crystal symmetry and spin-orbit coupling. In crystals without inversion
symmetry this leads to a spin splitting of the bands which in particular
for compound semiconductors produce interesting optical properties which
also may be of technological importance. Also the properties of magnetic
metals are influenced by relativistic effects, again mainly due to the spinorbit coupling. A quantitative description of spin polarization and spinorbit coupling is essential for materials where the spin- and orbital moments
are comparable in magnitude. Often such materials also are those where
simple (local) implementations of the density functional theory are not
sufficiently accurate. Strong electron correlations require other theoretical
methods, self-interaction corrections, ”LDA+U”, for example. Magnetooptical effects (Kerr effect, circular dichroism), magneto-elastic effects and
magnetostriction are fields of great importance, in basic as well as applied
solid state research. An understanding of the relation between magnetic
properties of layered structures and spin dependent transport properties is
essential for the explanation of the Giant Magnetoresistance (GMR) effect,
and thus for the development for novel recording and storage devices.
2
1. Introduction.
Relativistic Solid State Theory will here be limited to a discussion of some
aspects of relativistic effects as they manifest themselves in the quantum
theory of electrons in solids. The relativistic effects which are met in the
theory of electronic states in solids are basically the same as those encountered in the theory of atoms and molecules. In the Pauli picture,
an approximation to the Dirac theory neglecting terms proportional to
1/c4, “inclusion of relativity” is performed by adding three terms to the
Schrödinger Hamiltonian: The mass–velocity term, the Darwin term, and
the spin–orbit (SO) term. For a Coulomb potential, the Darwin term —
the “relativistic s–shift” — is positive for s states and zero for all states
of angular momentum ` > 0. The mass–velocity term is negative, and it
decreases in magnitude with increasing `. These signs are easily understood. The Darwin term results from an enhancement of the wavefunction
at the nuclear site, and as a consequence of the localization the Heisenberg
uncertainty principle this leads to an increase of the electron momentum,
i.e. of the kinetic energy. Only s states have a nonvanishing amplitude at
the radial coordinate r = 0. The mass–velocity shift and the so–called relativistic core contraction are intimately related. In the Bohr model of the
hydrogen atom the orbital radius is inversely proportional to the electron
mass. Thus it appears plausible that the relativistic mass enhancement induces a contraction of the orbits, and the electrons move to regions where
the potential is more attractive, i.e. the energies are lowered. The SO term
is represented by the operator
HSO =
1
σ × ∇V (r) · p,
4m2 c2
(1)
where p is the momentum operator, V (r) the potential, and σ is the Pauli
spin operator. Whereas the former two terms shift energies, the SO operator can induce splitting of states, and these can lead to interesting effects
in the electronic structure of solids. Further, since it couples spin– and angular momenta, the spin–orbit term can influence the magnetic properties.
In general, relativistic effects are less important in the electronic structure theory of solids than in the theory of atoms. The reason is that the
physical and chemical properties of solids are mainly related to the valence
states, the outermost electrons in the constituent atoms, whereas the relativistic effects are largest near the atomic nuclei. The valence electrons
have only small probability amplitudes near the nuclear site, and relativis-
3
tic effects are then to a large extent indirect in the sense that they are due
to influence of shifts of the inner ion–core states through their screening
effects on the outer states. Thus, the relativistic core contraction reduces
the effective nuclear charge felt by the valence electrons. Although these
effects are small in solids, there are nevertheless exceptions from this general picture, and the following sections of this chapter will describe some
examples of important relativistic effects in the electronic theory of solids.
Solids may be structurally disordered or crystalline. Perfect crystals
with completely periodic structures do not exist in Nature. However, most
of the discussion here will be based on such idealized models, and the
electronic structure is described in terms of band structures, dispersion relations between formal one–electron energies, ε, and wavevectors, k; ε(k).
First, in Section 2, we illustrate how the relativistic shifts (mass-velocity
and Darwin) of parts of the band structure with respect to each other may
affect the physical properties, including the crystal structure. The second
subject (Section 3) treated in this chapter concerns the simultaneous influence of the crystal symmetry and the SO–coupling on ε(k), spin splitting
effects, i.e. effects which are without atomic counterparts.
Often the electronic spin mainly plays the indirect role through the Pauli
principle, but in some cases it is directly the source of physical processes.
Magnetism is an obvious example, and the combination of spin- and orbital moments determines the magnetic properties of materials. The theory
must therefore be able to treat spin-polarization and SO-coupling simultaneously. This is therefore the third subject dealt with here. Magnetoelastic
and magnetooptic effects are related to this and are discussed in Section 5.
2. Effects due to relativistic shifts in ε(k).
For solids with heavy atoms, relativistic shifts may affect the bonding
properties, and also optical properties may be influenced. The relativistic
shifts of the 5d bands relative to the s-p bands in gold change the main
interband edge more than 1 eV . Already Pyykkö and Desclaux mentioned
[1] that the fact that gold is yellow is a result of relativistic effects. These
are indirect [2] (see also the introduction, Sect. 1), and the picture was
confirmed by relativistic band structure calculations [3,4]. Also the optical
properties of semiconductors are influenced by relativistic shifts which affect the gap between occupied and empty states, see for example Ref. [5].
Two additional examples may be mentioned where relativistic shifts in the
energy band structure drastically influence the physical properties. First,
4
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0.1
ENERGY (Ry)
0
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5'
5
-0.1
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1
1
5'
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-0.3
-0.4
2'
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3
4'
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-0.5
-0.6
Γ
M
X
Γ
R
X
Figure 1. Self-consistent scalar-relativistic band structure for CsAu. The
Cs-5s and -5p states were included as band states (not shown). In a band
calculation without relativistic shifts included there is no gap at R, and in
that model CsAu would be a (semi-) metal.
consider the bands calculated [6] for CsAu, Fig. 1. The state R 20 (single
group notation) is the highest occupied state which implies that CsAu is
an insulator. In fact it is an ionic insulator, Au being the negative ion.
The gap is caused by the relativistic downshift (mass–velocity “wins”) of
the Au–6s states, and it is by no means a spin–orbit splitting effect. The
second figure, Fig. 2, shows the density of states calculated with SO in-
5
200
C-178-442
Cs Au
Cs 5p
20
3/2
0
-1.1
1/2
TOTAL NOS
TOTAL DOS
5/2
3/2
Au 6s
Au 5d
-0.9
-0.7
-0.5
-0.3
-0.1
0
0.1
ENERGY (RY)
Figure 2. Density-of-states (DOS) and number-of-states (NOS) functions
for CsAu. Spin-orbit coupling included.
cluded. Although the spin–orbit splittings are large in the Au–5d band, the
gap is essentially as in the relativistic calculation where SO was artificially
omitted, Fig. 1.
Lead, Pb, is another example where relativistic energy shifts of valence
states are important. At first it might seem surprising that C, Si, Ge,
and Sn have the diamond structure as the stable (or at least metastable)
structure whereas Pb assumes the face–centered cubic structure with one
atom at each Bravais lattice point. The diamond structure is stabilized by
formation of sp3 hybrid bonds. Although this may seem to be energetically
favorable for Group–IV elements the energy gained by forming these bonds
should be compared to the “cost” of transferring an electron so that the
outer configuration is changed from s2 p2 to sp3. We calculated [7], from
first principles, the ratio δ between the energy gained by formation of sp 3
6
3.0
C-178-422A
C
δ = -4h
Ep-Es
2.0
Si
Sn
1.0
NR Pb
Pb
0
0.5
Ge
0.6
0.7
0.8
BOND ORDER (b)
Figure 3. Ratio δ, between the bond formation energy (-4h) and the s→p
promotion energy plotted against the bond order, b, which is calculated as
the difference between bonding and antibonding character of all occupied
states in the (assumed) diamond structure. Pb crystallizes in reality in the
fcc structure because δ is so small compared to 1 that the sp 3 bonding is
not favored. ”NR Pb” denotes a fully non-relativistic calculation for lead.
bonds and the s → p promotion energy Ep − Es and found the results
summarized in Fig. 3. The quantity b (abscissa in Fig. 3) is the bond
order, B − AB, the calculated difference between bonding and antibonding
7
character. This is a measure of the strength of the covalent bond.[8,9]
For δ > 1 the diamond structure is energetically favorable, for smaller
values the “cost” is too high. Tin is just at the border line, making it
understandable that the α– and β–phases are very close in energy.[10] Pb
has a δ–value which clearly is smaller than 1 (and simultaneously it has
the smallest b value). If we, on the other hand, omit the relativistic effects,
“NR Pb” in Fig. 3, lead becomes represented by a point very close to δ = 1,
in fact close to Sn in the diagram. The reason why lead does not “like” to
form in the diamond structure is that relativity shifts the 6s states so far
below the 6p levels that the s → p promotion energy becomes too large.
This is then a quantitative calculation confirming what Phillips refers [11]
to as relativistic dehybridization.
A more recent example of how relativistic shifts of the bands can influence the crystal structure of a solid was presented by Söhnel et al. [12]
who performed ab initio calculations for gold halides. By comparing relativistic to non–relativistic calculations it was found that the fact that
Au compounds assume chain–like structures and not (like Cu– and Ag
halides) cubic (or hexagonal) structures is indeed a result of relativistic
shifts, mainly of Au–6s and –6p states.
3. Electronic states: SO-coupling and crystal symmetry.
The ideal crystalline solid is an infinite array of identical primitive cells
so that the crystal is invariant under lattice translations
R = n 1 q 1 + n2 q 2 + n3 q 3 ,
(2)
where n1 , n2, and n3 are integers and {q i } are the primitive translations.
In addition there are symmetry operations such as rotations under which
a point is fixed. The full group of symmetry operations is the space group,
whereas the group of covering operations, where the translations in the
space group elements are set equal to zero, is the point group of the crystal.
Considering ordinary symmetry groups (i.e. not the color groups, “magnetic groups”) there are 32 point groups and 230 possible space groups for
crystalline solids. This is different from molecules, where there is no translational symmetry and therefore no limitation of the number of possible
point groups.
The translation symmetry group is cyclic and has one–dimensional irreducible representation. It is simple to show that the solutions, ψ(r), to
8
the one–electron Schrödinger equation (with the crystal potential V (r))
2
p
Hψ(r) =
+ V (r) ψ(r) = εψ(r),
(3)
2m
are of the Bloch form:
ψ(r) = ψk (r) = eik·r uk (r)
(4)
The solutions are thus plane waves modulated by the functions u k (r)
which have the periodicity of the lattice, and the states are labelled by
the wavevector k. With p = −h̄i∇ it is seen that uk (r) satisfies
h 1
i
2
(p + h̄k) + V (r) uk (r) = ε(k)uk (r)
(5)
2m
or
i
h h̄2
2
(∇ + 2ik · ∇) + V (r) uk (r) = λ(k)uk (r)
(6)
−
2m
with λ(k) = ε(k) − h̄2k 2 /2m.
For free electrons V (r) = 0, and uk (r) = constant, λ(k) = 0, i.e.
ε(k) = h̄2 k 2 /2m.
At the center of the Brillouin zone, k = 0, eq. (6) implies that
h
i
h̄2 2
−
∇ + V (r) u0(r) = ε(0)u0(r),
2m
(7)
showing that at k= 0 u(r) has the symmetry of the potential V (r), the
symmetry of the space group.
When spin–orbit coupling is added (eq. (1)), the relation (5) is replaced
by
i
h 1
1
2
(p+h̄k) +V (r)+ 2 2 σ×∇V (r)·(p+h̄k) uk (r) = ε(k)uk (r). (8)
2m
4m c
The term
H0 =
1
h̄
(p +
σ × ∇V (r)) · k
m
4m2c2
(9)
is sometimes treated as a perturbation for small k (or small deviations of
k from a special point ko in the Brillouin zone). This leads to the so–called
“k · p theory” which we shall use later.
9
The Hamiltonian is invariant under lattice translations, if V (r) is invariant, even with inclusion of the SO term. The eigenfunctions will be
of the Bloch form, but they will in general not correspond to pure spin
¯z (the z–axis is taken
states α or β, the spin functions which diagonalize σ̄
as quantization axis). Often one labels the Bloch function by arrows ↑ ↓,
i.e.
ψk↑ (r) = χk↑ (r)α + γk↑(r)β ≡ eik·r uk↑ (r),
(10)
K = −iσy Ko .
(11)
< Kψ | Kφ >=< φ | ψ >
(12)
K 2ψ = −ψ
(13)
¯z | ψk↑ >
where the arrow indicates the dominating spin; ↑ means that < ψ k↑ | σ̄
is positive. In addition to the usual spatial symmetry many physical systems possess time reversal symmetry. The time inversion operator, K,
changes r into r; p into −p, and σ into −σ. Also, it is assumed that
magnetic fields are reversed, and this is only possible if the currents producing these fields are reversed. Therefore we assume that the sources
which produce magnetic fields are included in “the system”. The operator
K is related to the operator Ko for complex conjugation through
Obviously, K(cψ) = c∗ (Kψ), K is an antiunitarian operator. < φ | ψ >=
< Ko ψ, Ko φ > gives with (11):
and
If the system has time reversal symmetry, then the Hamiltonians H and
K commute. If | ψ > is an eigenstate then | Kψ > is also an eigenstate
with the same energy. The functions ψ and Kψ are orthogonal. This is
Kramers’ theorem. For a system with time reversal symmetry all states
are at least doubly degenerate. For a crystal with the wavefunctions (10)
we then have that K | ψk↑ > and K | ψk↓ > belong to the wavevector −k,
and that
ε(k) ↑= ε(−k) ↓; ε(k) ↓= ε(−k) ↑ .
(14)
Each energy occurs (at least) twice, but not necessarily at the same k. If
the crystal potential V (r) has inversion symmetry, then the Hamiltonian,
also including the SO operator, has this symmetry, and
ε(k) ↑= ε(−k) ↑ .
(15)
10
In that case we have:
ε(k) ↑= ε(k) ↓,
(16)
3.1. Spin splitting in semiconductors.
The most important semiconductors are the elemental diamond type
materials (Si and Ge) from group IV in the Periodic System and the III–V
and II–VI compound semiconductors (for example GaAs, GaN, ZnSe etc.).
The cubic forms of the compounds have the zincblende structure which, in
contrast to the diamond structure, lacks the inversion symmetry. Therefore
the band structure of for example GaAs has spin–splittings which are not
found in Ge. (Spin splittings also occur in compounds with the wurtzite
structure, of course, see Ref. [14] and references given therein. For simplicity the presentation here will be limited to the zinc–blende type crystals).
We shall summarize the theory of these effects, partly by describing them
for states near the center of the Brillouin zone by means of k · p perturbation theory, partly by ab initio full zone calculations. The k · p theory
is described by Dresselhaus [15], Kane [16] and an introduction is found in
the textbook by Yu and Cardona [17]. Figure 4 shows the energy bands
along symmetry lines for germanium, and energy levels are labelled according to the standard names of the irreducible double group representations.
The energy scale was chosen so that the highest occupied state (at T = 0)
is at zero. This valence–band top, Γ+
8 , is at the zone center. It is p–like,
+
+
and without SO coupling Γ8 and Γ7 would be generate, Γ250 . The energy
regime near the band gap is shown in Fig. 5 which also serves the purpose
of defining important gaps for later use.
The bands of GaAs are shown in Fig. 6. Also in this case the valence
band maximum is at Γ, the state Γ8 (or Γv8 if we use the extra label “v ”
to identify it as a valence state, to be distinguished from a state Γ c8 in the
conduction band regime). Without SO splitting, single group notation, the
corresponding representation is Γ15, Γv15, and Γc15.
All bands along [110] are split by the SO interaction, and for small k
(i.e. near Γ) we can use the k · p theory to estimate the third–order [18,16]
term:
∆E = γk 3
(17)
The split states for k along [110] belong to either of the two nondegenerate
representations Σ3 and Σ4 , and ∆E in eq. (17) is E(Σ4) − E(Σ3). The actual identification of the sign is discussed in [19,20]. For the Γ6 conduction
11
L4+ + L5+
4
ENERGY (eV) L-- + L-4
5
2
C-178-443
Γ 8--
L6+
Γ 6--
Γ6-L6+
Γ7--
0
-2
Χ5
Γ 8+
Γ 7--
Ge
Γ 8+
Γ 7+
L6--
Γ 7+
-4
-6
Γ 8--
Χ5
L6+
-8
-10
L6--
Γ 6+
-12
L
Λ
Γ
Γ 6+
∆
X U,K
Σ
Γ
WAVEVECTOR k
Figure 4. Relativistic band structure of Ge obtained by a pseudopotential
calculation, Ref. [13]. The valence band maximum (at k=0) is the Γ +
8
state.
band the coefficient γ is γc which has 4 contributions.[19]
γc = A + B + C + D.
(18)
Values of γc were obtained directly by diagonalizing our 16 × 16 k · p
12
C-178-444
Λ4 , Λ5
L--4, L--5
L--6
Λ6
4 Γ 8--
Ε'
1
Λ6
L--6
L+4 + L+5
L+6
E1+ ∆ 1
Λ4 , Λ5
2
Γ 6--
Γ 7-E0' +∆'
0
E1 +
'
E
0
E
Γ8 0
L --
Χ5
∆6
6
Λ6
-2
E2
Γ 7+
eV
k= πa (111)
∆7
k=(000)
∆7
∆6
Χ5
k= 2π
a (100)
Figure 5. Ge: States near the gap.
Hamiltonian [19] as well as by means of by perturbation theory giving the
4 terms of eq. (18):
oi
n
2
1
4h
∆0
0
+
A = PP Q
3
3E0(E0 + ∆0) (E00 − E0 + ∆00) E00 − E0
n
4h
1
∆00
2 oi
0
B = PP Q 0
+
3
(E0 − E0)(E00 + ∆00 − E0 ) E0 + ∆0 E0
4 P 2 Q∆−
C=−
3 Ē02 (Ē00 − E0)
4 P 02 Q∆−
D=−
.
3 Ē0 (Ē00 − E0)2
(19)
The matrix elements of the momentum operator which enters the Hamil-
13
C-178-445
6 L4,5
4
2
ENERGY (eV )
0
Γ8
L6
Γ8
Χ7
Γ7
Γ7
Γ6
Γ6
Χ6
L6
Γ8
Γ8
L4,5
Γ7
Γ7
-2 L6
Χ7
Χ6
-4
-6
-8
Χ7
L6
-10
-12
Χ6
L6
L
Γ6
Λ
Γ
Γ6
∆
X
K
Σ
Γ
WAVEVECTOR k
Figure 6. Energy bands of GaAs calculated by means of a relativistic
pseudopotential method, Ref. [13].
tonian are
P = i < Γv15,x | px | Γ1 >, P 0 = i < Γc15,x | px | Γ1 >
P 000 = i < Γv15,x | px | Γ01 >, Q = i < Γv15,x | py | Γc15,z >,
(20)
and px = −i∂/∂x, E0 the gap E(Γc6) − E(Γv8 ), ∆0 the SO splitting at the
valence band top, E(Γv8 ) − E(Γv7 ), E00 = E(Γc7) − E(Γv8 ), ∆00 = E(Γc8) −
14
E(Γc7), E0000 = E(Γ06) − E(Γv8 ), and the off-diagonal SO splitting is
∆− = 3 < (
33
33
)v | HSO | ( )c >,
22
22
(21)
where ( 32 23 ) represents the angular-momentum-like eigenvector of the Γ 15
eigenstates plus spin.
The spin-orbit Hamiltonian couples the Γv15 and Γc15 states in a zinc–
blende–type semiconductor since they are p–like and have the same symmetry.[21] This coupling is absent in diamond–type crystals where the state
(Γ250 ) corresponding to Γv15 is even (bonding) and the one (Γ15, antibonding) corresponding to Γc15 is odd. Early estimates of this coupling were
published by Pollak et al. [21] for GaAs, GaP, InP, and AlSb, and by Higginbotham [22] for GaSb, InAs, InSb, ZnSe, and ZnTe. They were obtained
through fits of various spin–orbit splittings with a 30×30 k·p Hamiltonian
and therefore related to experimental data in a nontransparent manner.
Another way of estimating ∆− is based on a tight-binding model where
the Γv15 and Γc15 wavefunctions are taken as bonding and antibonding linear combinations [23] of p–like cation (| pc >) and anion (| pa >) orbitals:
| Γv15 > = α | pa > +β | pc >
| Γc15 > = β | pa > −α | pc > .
(22)
The phases of the wavefunctions are given in Fig. 7, and | pc > and | pa >
are both chosen to have the positive lobes to the right. The diagonal and
off–diagonal matrix elements of the SO Hamiltonian give
∆0 = α 2 ∆a + β 2 ∆c ,
∆00 = β 2∆a + α2 ∆c,
∆− = αβ(∆a − ∆c ),
(23)
where ∆a and ∆c are the atomic spin–orbit splittings, properly renormalized to take into account the compression of the atomic wavefunctions in
the solid. [4] It is also possible to relate ∆− to ∆0 and ∆00 through:
∆− =
η
(∆0 − ∆00),
2
η −1
(24)
where [24,25]
η=
−2Hxx
α
= c
2 ]1/2
β
Ep − Epa + [(Epa − Epc )2 + 4Hxx
(25)
15
ẑ [111]
antibonding
A
C
c
| Γ15
A
C
| Γ1
bonding
A
C
v
| Γ15
A
C
∂
∂ ẑ
Si
Ge
| Γ1
Figure 7. Schematic diagram which demonstrates the phase convention
used for the Γc15 , Γ1 , and Γv15 wave functions. The Si atom is taken to be
at the origin and acting as the anion (A), whereas Ge is regarded as the
cation (C). For the p-like eigenstates Γc,v
15 , the component along k along
the [111] direction is shown.
16
where Epc and Epa are the atomic term values, the diagonal matrix elements
of the tight–binding Hamiltonian, and Hxx is the overlap integral.[25,26]
The expression (23) would give ∆− = 0 for a diamond–type semiconductor
(set ∆a = ∆c ), as it should be, but this is not fulfilled if (24)–(25) are
used. ∆0 and ∆00 are in general not equal. In Ge ∆0 = 0.29 eV, and ∆00
= 0.21 eV. This difference is mainly due to admixture of d–like states into
Γc15 (more than into Γv15), and these d–states are not taken into account in
the tight–binding scheme.
Spin splittings were calculated in yet another way in Refs. [19,20,27,28],
namely by means of relativistic ab initio band structure calculations. We
used the linear muffin–tin–orbital (LMTO) method [30] in a relativistic
formulation [31,32] (see also Section 4.1 ). The value of ∆− was estimated
by comparing LMTO calculations where SO coupling was omitted (“scalar
relativistic” calculations) to the full calculations. In the absence of ∆ −
coupling the Γ15 bands should split into j = 23 and 12 (Γ8 and Γ7 ) components. The shifts with respect to the scalar–relativistic value being in the
ratio 2:1. The ∆− coupling changes this ratio, see Fig. 8. Second–order
perturbation theory gives
3
∆0
∆− 2 1
δ( ) =
−(
) · 0
2
3
3
E0
1
2∆0
2∆− 2 1
δ( ) = −
−(
) · 0.
2
3
3
E0
(26)
For GaAs, GaSb, InP, and InSb ∆0 is 0.340, 0.72, 0.108, and 0.803 eV,
respectively, and the ∆− parameters for these four compounds are given
in Table 1.
Table 1
Values of the off–diagonal SO parameter, which couples the Γ c15 and Γv15
bands, as obtained by tightbindig (LCAO), k · p, and LMTO methods.
(∆− in eV)
GaAs GaSb InP
InSb
LCAO -0.085 -0.28 0.16 -0.20
k·p
-0.07 -0.4
0.4
-0.014
LMTO -0.11 -0.32 0.226 -0.244
Values of ∆− for II–VI compounds are given in Ref. [33], where its influence
17
Γ8v
v
Γ15
δ
3
2
∆-- 2 ´
3 E0
1
3 ∆0
- 23 ∆ 0
-Γ7v
intraband
∆0
δ
1
2
2∆-- 2
E0́
3
interband
∆0
Figure 8. Schematic diagram of the spin-orbit splitting of the Γ v15 bands.
The intraband terms (first-order perturbation theory) and their interband
counterpart (second order) due to interaction with Γ c15 are illustrated.
on the electron g factor is examined. With the parameters for the k · p
Hamiltonian given in Table I of Ref. [19], one finds for InSb the four terms
3
in eq. 19 to be (in eVÅ )
A = 332.1, B = 24.3, C = 117.5, D = 4.1
Table 2 lists the sum of these, i.e. γc , for six compounds in the zincblende
structure. III–V, II–VI as well as I–VII examples [19] are included. Experimental data from Refs. [34] and [35] are also included.
Calculated coefficients for k 3 splittings of other bands, electrons as well
as holes are given in Ref. [19]. Concerning signs, see Ref. 18 in the paper
on GeSi, Ref.[20]
The existence of spin splittings linear in k near Γ of the Γ8 valence
states in zinc–blende–type materials has been known for a long time.[16,18]
They have been measured in magneto–optical[36] and polariton scattering
experiments.[37,38] They result in a slight shift of the position of the top
of the heavy– ( 23 , ± 23 ) and light–hole ( 32 , ± 21 ) Γv8 bands for k along the
< 110 > directions. The splittings do not occur along < 100 >, but
along < 111 > directions the ( 32 , ± 23 ) Γv8 and ( 32 , ± 23 ) Γc8 bands are linearly
split. The coefficients are defined through Ck and Ck0 , respectively (see
18
Table 2
Values of the coefficient of the spin splitting proportional to k 3 of Γ1 –
conduction band for k k [110] as obtained with the LMTO method, the
k · p 16 × 16 Hamiltonian, and k · p perturbation theory (PT). Experimental data are from Refs. [34,35] Units: eVÅ3. For GaAs, GaSb, and
InP experiments only give the magnitude. For InSb also the sign was
determined.[35]
GaAs
LMTO
15.0
k·p
28.4
k · p (PT) 30.0
expt.
25.5
GaSb
109.4
153.9
153.9
186.3
InP
-8.9
-9.3
-11.7
8.5
InSb ZnSe CdTe CuBr
218.7 1.6
11.7 ≈ 0
567.0
477.9
226.8
eq. 27, below). It was shown [28] that the main contribution to C k is the
second–order interaction, bilinear in k · p and in the spin–orbit operator,
between the uppermost d (semi–) core states (like the 3d states in Ga),
and that the contribution of the k–dependent spin–orbit Hamiltonian can
be neglected.[19] The ( 23 , ± 23 ) heavy hole bands (hh) (valence) and heavy
electron (he) bands along [111] have symmetries Λ5 and Λ4 and their linear–
k splitting is
√
(27)
E(Λ5) − E(Λ4) = 2 2Ck k
In (27) Ck is called Ck0 for the he states. The interpolation formula
suggested in Ref. [28] is
Ck = A
∆d,a
∆d,c
+
B
E(Γ8) − Edc
E(Γ8) − Eda
(28)
where “c” refers to cation and “a” to anion, E(Γ8) − Eda,c are the energy
differences between the valence band top and the semi–core d levels, and
the ∆d parameters give the spin–orbit splittings of these core d–states.
The Ck coefficient can of course also be obtained by direct calculations, for
example by means of the relativistic LMTO method as done in Refs.[19,
20,28]. These calculations support the analyses using eq. (28), at least
for the III–V compounds. Figure 9 shows the hh and he splittings along
< 111 > for zinc–blende–type GeSi. For small k the splitting is linear, and
the coefficients obtained by fitting are: Ck = -1.85 and Ck0 = -3.07 meVÅ.
19
4.0
C-178-448
k || <111>
hh
he
--∆E (meV)
3.0
2.0
1.0
0.0
Γ
L
Figure 9. Spin splittings of the heavy hole (hh) and heavy electron (he)
bands in GeSi for k at the Λ symmetry line as calculated with the relativistic LMTO method (Sect. 4.1).
Also, we could obtain Ck from the spin splitting for small k along [110].
For the heavy–hole ( 23 , ± 23 ) (hh) and light–hole ( 23 , ± 21 ) (lh) bands the splittings linear in k are:
√
3 3
Ehh (Σ4) − Ehh(Σ3) =
Ck k
2
√
(29)
3
Elh(Σ4) − Elh(Σ3) =
Ck k
2
For GeSi we get, again by fitting to our ab initio calculations [20], C k =
−1.80 meVÅ, i.e. consistent with the -1.85 meVÅ determined for k along
20
[111].
GeSi consists of two Group–IV elements, but nevertheless we see that
the (artificial) zinc–blende–type compound will exhibit spin splittings. A
charge transfer occurs, and the value of the ionicity becomes non–zero.
However, the small splittings calculated for GeSi indicate that the value of
the ionicity [8,11] is small in magnitude, and it is even difficult a priori to
know its sign, i.e. whether Ge will be the “anion” or the “cation”. The
fact that we call the compound “GeSi”, and not “SiGe” indicates that we
[20] believe that Ge plays the role of the cation. Van Vechten estimated
the electronegativity C to be 0.25 with Ge acting as the cation.[39,40] Estimates of the ionicity cited in Ref. [20] agree that it is small in magnitude
but there is no unambigous determination of the sign from experiments.
We therefore made additional calculations of the polarity, α p . It is possible
to extract [8] from the self–consistent LMTO calculations all the parameters entering the tight–binding scheme used by Harrison.[26] This then
allows us to calculate αp using these ab initio parameters:
αp = (Epc
Epc − Epa
−
Epa )2
+
2 1/2
Hxx
,
(30)
where Epc and Epa are p–band centres for the “cation” and the “anion”, and
Hxx is the overlap integral. In Ref. [26] the E’s are atomic term energies
and a simple formula is given to estimate Hxx . We use eq. (30), but with
values of the parameters which are deduced from the actual self–consistent,
relativistic band calculation. In this way we find αp = 0.07, provided that
we take “c” in eq. 30 to mean Ge and “a” Si. The positive value for α p is
then consistent with Ge being the cation.
Like other zinc–blende type compounds GeSi is an infrared active material. The optical phonon at the Γ–point is associated with an electric
dipole moment which is related to the “ transverse effective charge” e ∗T . In
terms of αp and its volume derivative (see eq. (49) of Ref. [8]), e∗T is given
by
e∗T = 4{αp +
dαp
}
d ln V
(31)
The calculated value of dαp /d ln V is -0.047, and we get e∗T = +0.09, and
the + sign implies that the negative charge resides on Si, and again Ge
plays the role of the cation.
21
Now, return to the formula eq. (28). The constants A and B are positive.
Silicon has no d-like core states, and since we now concluded that Si is the
anion, then the last term must vanish, and the first term (“cation”) is
to be evaluated for Ge parameters. First, it is noticed that in that case
eq. (28) yields a negative value for Ck , and that agrees with the value
already quoted (-1.85 meVÅ). In Refs. [19,28] it was suggested to use
A = 220 meVÅ for group–IV materials. With this value, and the other
Ge data from photoemission experiments [41] one finds Ck = −4 meVÅ.
This is off by more than a factor of two from our ab initio calculations,
probably indicating that the value of A should be reduced for group–IV
materials.[20]
We examined in detail the relativistic band structure of GeSi because
Ge and Si are the constituents in thin (Ge)n /(Si)m superlattices, artificial
layered structures grown with high accuracy using molecular–beam epitaxy.[42,43] One reason for examining these systems is that, by choosing
properly the substrate, it may be possible to form a new semiconductor,
which easily can be incorporated in silicon based technology, but which, in
contrast to the pure constituent materials (Si and Ge), have a direct optical
gap.[44–46] This is important in optoelectronic applications. GeSi is the
shortest period (Ge)n /(Si)m superlattice. It lacks the inversion symmetry,
and this is also the case for all other (Ge)m /(Si)n superlattices with both
m and n odd.
Recently, Foreman demonstrated that k–linear terms in semiconductor
heterojunctions [47] are enhanced near the interface, and that the associated mixing of Γ8 hh and lh states is increased so that it influences the
quantum–well Pockels effect.[48] The reduced symmetry in quantum well
structures (from Td in the bulk to C2v ) increases the optical anisotropy.
The work in Ref. [47] shows that the k–linear splitting contributes significantly to this anisotropy because the Ck coefficients are often an order of
magnitude larger in the heterojunctions than in the bulk materials.
The spin splitting of the energy bands of semiconductors without inversion symmetry has other interesting and presumably also technological
consequences. Photoemitted electrons from GaAs are spin polarized when
the light used for the irradiation is circularly polarized.[49,50] (such effects
may also be observed for metals, see for example [51]). The so–called “three
step model” for photomission considers the three processes, optical excition
from a valence–band state to a state in the conduction band, transport of
the excited electron to the surface, and — finally — emission through the
22
surface into the vacuum. In order to achieve an efficient emission of the
electrons into vacuum the electron affinity at the surface of GaAs is lowered by deposition of Cs and O2 . This creates a depletion layer and thus
a downwards band bending at the surface. By means of an elegant experimental technique Riechert et al. [52] examined how the spin polarization
of the excited photoelectrons changed during their transport through this
layer. In the analyses of their data they used that the spin–splitting of the
conduction band may be expressed via the Hamiltonian [53]
HΩ =
h̄
σ · Ω,
2
(32)
with
h̄Ω = 2γκ,
(33)
where γ is material dependent, and the vector κ is
κ = {kx (ky2 − kz2), ky (kz2 − kx2 ), kz (kx2 − ky2 )}.
(34)
(This is the k 3 term valid for small k as discussed earlier). The Hamiltonian
is equivalent to a magnetic field B i ( i for “internal”) parallel to κ. B i
depends on k, but for small widths of the band bending regime electrons
with high kinetic energy can pass this regime ballistically with k normal to
the surface, and consequently all these escaping electrons will feel the same
direction of B i . Their polarization vector, P , will then be rotated away
from the initial direction defined during the excitation. The rotation will
be maximum for k in [110] directions and zero for k along [100] or [111].
The precession angle, θp, could be measured [52], and it could be related to
the width of the depletion regime and the band bending energy. Numerical
estimates were not (eq. 4 in Ref. [52]) based directly on eq. (32), but used
the ab initio results obtained by the relativistic LMTO calculations.[27,54]
These splittings are shown in Fig. 10.
The work by Alvarado et al. [27] reports the observation of a spontaneous
spin polarization of photoemitted electrons from GaAs (110) with linear
polarization of the exciting light. As mentioned earlier (eq. 10) a “spin
up” state in Fig. 10 has ↑ as well as ↓ components. For very small k
only almost pure spin states will be found, but nevertheless photoemitted
electrons which were excited to energies lying so close to the conduction
band minimum will not exhibit a net polarization. Contributions from ↑
23
C-178-449
∆Ec x5
Ecrit
E (eV)
1.5
"spin up"
band
"spin down"
band
1.0
k || <111>
0.5
0.0
Γ
Γ6
k
k
K
Figure 10. Lowest conduction band along Γ-K for GaAs as calculated
by means of the relativistic linear muffin-tin-orbital method. The spin
splitting was exaggerated for clarity by a factor of 5. The dashed line
shows the band without spin splitting. The part K-X of the [110] line is
not included in the figure. The splitting is zero at the X-point, (1,1,0)2π/a
(equivalent to (2,0,0)2π/a).
and ↓ states will be equal. For higher energies, as illustrated in Fig. 10,
a certain energy will correspond to two different k-vectors, k ↑ and k↓ , in
the “up” and the “down” band. The hybridization to other states will be
24
different for ψ↑ (k↑) and ψ↓ (k↓ ) and a net polarization can occur. This was
indeed observed and a reasonable agreement between theoretical estimates
and experiment was found.[27]
This observation is interesting in itself, but one might get the idea that
it could be possible to reverse the process, i.e. from the vacuum side inject electrons which are polarized (from a magnetic material) into suitable
spin–split bands. These electrons with their spin could then be transported
through the semiconductor. Such an injection has in fact been shown to be
effective. Alvarado and Renaud [55,56] used a scanning tunneling microscope (STM) with a ferromagnetic tip and LaBella et al. [57] have extended
this in their study of surface structure and spin–dependent STM tunneling.
A theory for spin polarized tunneling through barriers of dilute magnetic
semiconductors was presented by Chang and Peeters.[58] Hammar et al.
[59] use a spin–injection method which also is based on the spin splitting
in the semiconductor bands. They send a spin polarized current from a
ferromagnet into the spin split states of a high–mobility two–dimensional
electron gas at the interface between the semiconductor and the ferromagnetic material. A theory of electrical spin injection was presented by
Rashba.[60] The efficiency of the injection of spins into a semiconductor
is of major importance in connection with new spin–based electronics, the
rapidly growing field of basic and applied solid state physics now referred
to as “spintronics”, see Ref. [61] and references given therein.
4. Electronic states: SO-coupling and spin polarization.
Earlier it was mentioned that the relativistic theory of electronic states
in solids in many respects is identical to that of atoms. Since this is well
described elsewhere, this section will only deal with some features of specific implementations of the theory in actual calculation methods used for
solids, and the importance of relativistic effects — apart from those already discussed — will be illustrated by examples. Although Section 3
did refer to results of LMTO calculations, we did not describe how these
included relativity. This section will deal with these items in the form of
an overview, and the basic band structure calculations described relate to
the density–functional theory [62,63]. Since magnetism is one of the most
important solid state physics fields we shall discuss the simultaneous inclusion of spin–polarization and relativistic effects, in particular the spin–orbit
coupling. In that context it appears that several of the materials where
such effects are particularly large and interesting are those where electron
25
correlations are strong, and where the standard (local) implementations of
the density-functional theory become inaccurate.
4.1. Relativistic band structures.
The first accurate band structure calculations with inclusion of relativistic effects were published in the mid–sixties. Loucks published [64–67]
his relativistic generalization of Slaters Augmented Plane Wave (APW)
method. [68] Neither the first APW, nor its relativistic version (RAPW),
were linearized, and calculations used ad hoc potentials based on Slaters’s
Xα scheme, [69] and were thus not strictly consistent with the density–
functional theory. Nevertheless (or, maybe therefore!) good descriptions of
the bands, Fermi surfaces etc. of heavy–element solids like W and Au were
obtained.[3,65,70,71] With this background it was a rather simple matter
to include [4,31,32,72] relativistic effects in the linear methods [30] when
they (LMTO, LAPW) appeared in 1975.
Conceptually, the formation of the energy bands is most easily described
within the ASA, the atomic–spheres approximation.[4,30,73] A pure band
of `-character (` is the angular momentum), “pure” in the sense that hybridization is omitted, extends over the energy range E B (`) to EA (`) where
the logarithmic derivative
D` (E) =
∂ ln φ` (E, r)
|r=S
∂ ln r
(35)
of the solution φ` to the radial Schrödinger equation in the atomic sphere
(radius S) is negative. The center C` of the pure `–band is the energy at
which D` (E) = −` − 1. The width of the `–band is W` = EA (`) − EB (`).
Relativistic generalizations of the logarithmic derivatives (eq. 35) are
Dκ (E) =
Sgκ0
cfκ (E, S)
= −κ − 1 + S
gκ
gκ (E, S),
(36)
where fκ and gκ are the “small” and “large” solutions of the radial Dirac
equation, and
(
−` − 1 j = ` + 12
(37)
κ=
`
j = ` − 21
There are cases where the inclusion of the relativistic shifts of the bands
is essential, but where inclusion of spin–orbit coupling is not needed. In
26
such cases an averaging scheme [30] can be applied by solving
D` (E) = Eκ=`−1(E−) = Dκ=`(E+)
(2` + 1)E = (` + 1)E− + `E+
(38)
2
(2` + 1)φ2` = (l + 1)g−`−1
+ `g`2
(39)
1
(2` + 1)ξ`(E) = E− − E+,
2
(40)
D` (E) = Dκ=−`−1(E−) = Dκ=`(E+)
(41)
Other methods of formulation a scalar relativistic scheme based on a
differential equation where SO coupling is eliminated have been developed.[74,75]
In the free atom a spin–orbit parameter, ξ` , is associated with each one–
electron state. In the solid `–states broaden into bands of non–zero widths,
and the spin–orbit parameter varies with energy over the band. We define
ξ`(E) through
where
The central SO parameter is
ξ`0 =
2
(C− − C+ )
2` + 1
(42)
with
Dκ=−`−1(C−) = Dκ=`(C+) = −` − 1
(43)
Figure 11 illustrates how the characteristic energies are defined. The
band widths are (approximately)
`
W− ∼
= W` + (ξ`A − ξ`B )
2
(` + 1)
W+ ∼
(ξ`A − ξ`B )
= W` −
2
(44)
and the energy derivative of ξ` is
ξ˙` ' (ξ`A − ξ`B )/W`
(45)
27
W
EA+
EB
j= +
j=
-- 12
1
2
2
D
+1
2 ξA
ξB
EB--
EA
C+
2
ξA
C C--
0
EA--
E
+1
ξB
2
- -1
- -1
D+
D
D--
-∞
-∞
-∞
Figure 11. Dirac logarithmic derivatives for κ=−` − 1 (D−) and κ=`
(D+), together with D` , the corresponding ”scalar relativistic” logarithmic
derivative. The figure illustrates the effects of SO coupling and how the
spin-orbit parameter ξ varies across the ` band. Bonding states (B) are at
the band bottom, antibonding (A) at the top. Note that W− > W` > W+ ,
and that ξ`A > ξ`0 > ξ`B .
Thus, the spin–orbit parameter increases with energy across the band. To
first order:
1
W− ' W` (1 + `ξ˙` )
2
`+1 ˙
W+ ' W`[1 −
ξ` ].
2
(46)
28
Knowing ξ`0, the spin–orbit parameter at the band center (C`) we can,
approximately, find ξ` (E) from
ξ` (E) ' ξ`0 + (E − C` )ξ̇`,
(47)
where ξ˙` may be obtained from
2` + 1 ˙
W−
ξ` ,
'1+
W+
2
(48)
with the bandwidth ratio calculated from
2
g−
(C−, S)
W−
µ+
=
' 2
,
W+
µ−
g+ (C+, S)
(49)
where the µ’s are mass parameters.[30,73,4]
Since the wave functions φ are normalized to the atomic sphere we expect
the wave function at the band centre (C`) to increase its amplitude, also
near the nucleus, if the sphere radius S is reduced. Consequently,
dξ`0
< 0.
dV
(50)
This does not necessarily mean that the SO splittings in the band structure decrease with volume. For example, while the SO splitting at the
top of the 5p–band in fcc Xe has a negative volume coefficient,[4] the
Γ025 → Γ7 , Γ8 splitting in the Au 5d band has the opposite volume dependence. This is a consequence [76] of the volume dependent SO–induced
hybridization between the Eg and T2g states.
The relativistic LMTO and LAPW methods were used to calculate [77–
80] the Fermi surface of UPt3. This is a heavy fermion compound, and its
physical properties are strongly influenced the presence of the narrow U-f
bands at the Fermi level. The shape of the Fermi surface is then sensitive
to relativistic effects, in particular the SO-coupling. The results of the
calculations [78] were surprising since they showed that the topology of
the Fermi surface was well described by these band structures although
they were obtained within the LDA. A similar precision was not found for
the effective cyclotron masses which were off by up to a factor of 30 when
compared to experiments. The crystal potential enters in the LMTO via
the potential parameters [30,73] for each ` (or each j in the relativistic
version [4]), including the mass parameters µ (eq.(49)). A convenient way
29
30
States/eV Cell
25
C-178-453
UPt 3
U 5 f 5/2 and f 7/2 PDOS
20
15
10
5
0
-4
-3
-2
-1
5/2
7/2
0
1
2
E (eV)
Figure 12. The U-f5/2 and -f7/2 density of states in UPt3 calculated within
the LDA. The Fermi level is at E=0. (Ref. [77]).
of modifying the LDA band structure to obtain a better description of the
heavy fermion systems, including the larger masses of the more localized
states (the narrow f bands) consists in renormalizing the mass parameters
for selected channels (j=7/2 and 5/2 for f states) and leave those for other
states unchanged. The formal theoretical background for this renormalized
band theory was given by Fulde et al. [81,82] and by Strange and Newns
[83]. For applications, see Refs. [84–89]. Reference [89] describes the
results of relativistic band structure calculations for CeRu 2 Si2, and also
in this case it was found that the topology of the Fermi surface is well
described by the LDA, although the T -linear specific heat coefficient is very
large, γ ≈ 350 mJ/molK2. This, and the similar observation made for UPt3
were explained [85,86] by showing that the Fermi surface topologies derived
from renormalized bands and an LDA calculation for this kind of systems
30
agree if 1): The LDA f -band width is small compared to the SO splitting
of the f states; and 2): the crystal electric field (CEF) splitting is small
compared to the width of the renormalized f bands. The first condition
implies that the f7/2 states lie so high in energy, that their contribution
to the scattering phase shift at the Fermi level is vanishingly small, in
the LDA calculation as well as in the renormalized band structure model.
If condition 2) is fulfilled, then the CEF can be neglegted, and all phase
shifts corresponding to the spin-orbit ground state multiplet can be chosen
identical. Further, all non-f phase shifts are assumed to be identical in
the renormalized band model and the LDA calculation. The boundary
condition imposed by the particle number then implies that the remaining,
single parameter of each of the two schemes, the f5/2 phase shift at the
LDA
Fermi level must be the same, ηfRen
5/2(EF ) = ηf 5/2 (EF ). The heavy mass
does not enter, all phase shifts (and then also the logarithmic derivatives)
at the Fermi level are the same, and the renormalized bands and the LDA
bands yield the same Fermi surface topology in that case. Figure 12 shows
that the SO splitting of the U-f states in UPt3 indeed is large, and that
the 7/2 states essentially is above the Fermi level, and Fig. 13 shows cuts
in the Fermi surface with illustration of the f -5/2 character.
Magnetism is a central field in condensed matter research, basic as well
as applied. Several physical effects such as, for example, the magnetooptic
Kerr effect, are caused by the simultaneous occurrence of spin polarization
and spin–orbit coupling. It is therefore necessary to include spin polarization in the (fully) relativistic band structure formalisms. Feder et al.
[90] and Strange et al. [91] developed reltivistic Korringa–Kohn–Rostoker
(KKR) methods which fulfil this requirement. Jones et al. [92] have made a
systematic study of the cohesive properties, including equilibrium volumes,
bulk moduli, of the light actinides using relativistic full-potential LMTO,
LAPW and linear combination af Gaussian-type orbital methods, with and
without SO coupling within the LDA as well as GGA (generalized gradient
approach) schemes. Ebert included [93] spin polarization in his relativistic
LMTO implementation, and a similar scheme was used by Solovyev et al.
[94] in their study of the volume dependence of the electronic structure
and magnetic moment of plutonium. This work presented in fact the first
self–consistent relativistic spin–polarized calculation for a solid.
The density-functional theory (DFT) with spin polarization, even in the
local approximations (LDA and LSDA with spin polarization) has been
applied to describe many physical properties with good precision. The
31
C-178-454
H
~100%
L
A
mj = 1
2
H
A
1
1
2
2
3
5
4 3
3
Γ
3
mj = 3
2
M
K
K
4
mj = 5
2
5
Γ
UPt 3
U-f 5/2 characters
Figure 13. UPt3 Fermi surface calculated within the LDA using the Diracrelativistic LMTO method. The width of the hatched stripes is proportional to the U-f5/2 component. The U-f7/2 content is very low all-over
the Fermi surface. Dotted stripes show the |mj |=1/2 contribution, righthatched the |mj |=3/2, and left-hatched the the |mj |=5/2 projections. Note
that band 1 and 2 have regions with very low f -character. On these parts of
the Fermi surface there is a strong hybridization with other states, mainly
Pt-p and -d. (Ref. [80]).
DFT [95] was generalized to include relativistic effects by MacDonald and
Vosko [96] and by Ramana and Rajagopal.[97] By omitting small diamagnetic effects one–electron equations were obtained that contained a scalar
effective potential as well as an effective magnetic field caused by the polarization. In cases where the paramagnetic currents cannot be neglected,
it is necessary to resort to a current–density formalism.[98,99]
Details of the relativistic methods are found in the references given
above, and only the essential features will be described here, exemplified by
32
the LMTO–ASA scheme. Rydberg–atomic units are used (m = 12 , h̄ = 1,).
With the simplifications suggested in Ref. [96] the Hamiltonian can be
written as
H = cα · p + (β − 1)
c2
+ V (r) + βσ · B(r)
2
(51)
inside the spheres, where the effective scalar potential (V (r)) is spherically
symmetric, and B(r) is the previously mentioned effective magnetic field.
The radial solutions φνΛ (E, r) to the one–electron equation inside the
spheres are of the form
(
µ
` gκ (E, r)χκ (r̂)
(52)
φΛ (E, r) = i
ifκ(E, r)χµ−κ(r̂)
χµκ (r̂) =
1/2
X
1
C(` j; µ − ms , ms )Y`µ−ms (r̂)χms
2
1
(53)
ms = 2
Here Λ stands for (κ, µ), the relavistic quantum numbers (eq. 37), g and
f as defined earlier. As in the non–relativistic case [30,73] φ and φ̇ (the
energy derivative) satisfy the relations
(H − E)φΛ(E, r) = 0
(H − E)φ̇Λ(E, r) = φΛ (E, r)
< φΛ (E, r) | φΛ (E, r) >= 1
(54)
< φΛ (E, r) | φ̇Λ (E, r) >= 0
where < | > means that the matrix element is obtained by integrating over
the atomic sphere.
Letting “ν” label Eν , a chosen energy [30,73], the logarithmic derivative,
Dνκ , is already defined (eq. 36), and for the energy derivatives a similar
function is introduced:
Dν̇κ = S
cf˙νκ (S)
−κ−1
ġνκ (S)
(55)
Within the linear approximation [30] the basis set is made up of muffin–tin
orbitals consisting of envelope functions (decaying), nΛ (r−R) on each site,
33
R, and which are augmented inside the spheres by functions Φ Λ (D, r − R),
where
ΦΛ (D, r) = φνΛ (r) + ωκ(D)φ̇νΛ (r)
(56)
and
ωκ (D) =
−gνκ (S) D − Dνκ
·
ġνκ (S) D − Dν̇κ
(57)
The muffin–tin orbital χΛ (r − R) with its origin at a given site R contains
ΦΛ(−` − 1, r − R) and from all other spheres at R 0 (6= R) ΦΛ (`, r − R0 ).
As a final step from individual spheres to the periodic crystal we make the
basis functions as a Bloch sum of χΛ :
X
k
χΛ (r) =
eikR χΛ (r − R)
(58)
R
which can be rewritten as
2 1/2h Φ(−` − 1, r) X
i
φΛ0 (`0, r)
k
k
−
S 0 .
χΛ (r) =
0 + 1)g 0 (`0 , S) Λ Λ
S
gκ (−` − 1, S)
2(2`
κ
0
(59)
Λ
The structure constants SΛk0 Λ are related to the non–relativistic structure
constants [30,73,4] S`0m0 ,`m by the transformation:
1
SΛk0 Λ =
2
X
1
1
C(l0 j 0 ; µ0 − ms , ms )Sl0µ0 −ms ,lµ−ms C(l j; µ − ms , ms ) (60)
2
2
1
ms =− 2
With the basis set (59) the secular equations
X
(HΛk0Λ − E jk OΛk 0 Λ )Ajk
Λ =0
(61)
Λ
are obtained, and they yield the eigenvalues, E jk and the eigenvectors AΛ .
From the solutions to the one–electron equation for all k in the (irreducible
part of the) Brillouin zone one calculates the charge and spin densities, and
from these new potentials V R (r) and B R (r) are determined according to
the prescription of the LSDA.[95] Iterations towards self-consistency are
then made in the usual way.
34
The total moment in sphere R is obtained as
X
MJR =
µnR`µ ,
(62)
`,µ
where nR`µ is the number of `µ electrons in the sphere centered at the cite
R and with the quantum numbers ` and µ. The spin moment in sphere R
is found by integrating the the spin density, mR (r), over the sphere:
Z SR
R
dr4πr2 mR (r),
(63)
MS =
0
and the orbital moment is
MLR = MJR − MSR
(64)
For δ-Pu (fcc structure) the self–consistent relativistic LMTO–ASA calculation of Ref. [94] gave MJ = 2.1 µB , MS = 4.5 µB and ML = -2.4 µB
((100) as quantization axis). As compared to a scalar–relativistic LSDA
calculation [100] (where ML = 0) which gave a moment of 5µB , the relativistic calculation gives a considerably smaller total moment. A similar
calculation, but without the shape approximations of the ASA, performed
by Bouchet et al. [101] gave similar values, MS = 4.23 µB , ML =-1.94 µB ,
and MJ =2.29 µB . The same authors also applied the ”LSDA+U” scheme,
see below, and found MJ =1.55 µB if U (see below) was set to 0.23 Ry, and
MJ =1.55 µB with U =0.33 Ry. Measurements of the magnetic moments of
pure δ-plutonium (above 600 K) have not been published. Apparently, the
δ phase is non-magnetic [102], but this does not exclude that local (but not
ordered at the high T ) moments exist. Méot-Reymond and Fournier [103]
stabilized the fcc structure of Pu by adding 6 at. % Ce and 6 at. % Ga and
found effective magnetic moments of 1.7 µB and 1.2 µB , respectively, for
the two kinds of samples. But still, it is not clear whether this allows us
to draw conclusions about the moments of pure δ-Pu.
4.2. Beyond LSDA.
Plutonium has physical properties which cannot be accounted for within
the local density-functional approximations. It undergoes a very large lattice expansion (α → δ) at 600 K, but the L(S)DA cannot predict this. The
reason is that this scheme cannot describe correlations sufficiently accurately. The f states are itinerant in the light actinides but localized in the
35
heavier (Am and further on). Plutonium is just at the point in the series
where the transition from delocalized to localized f states occurs. The
recent work by Savrasov et al. [104] suggests that the f –electrons in fact
are neither really localized nor delocalized, but the “f –electron is slightly
on the localized side of the interaction–driven localization–delocalization
transition”.[104] These conclusions were drawn from calculations using a
dynamical mean–field approach [105] built “on top” of a DFT formalism.
It appears to be a rather general trend that the magnitudes of the orbital
moments are too small in the relativistic LSDA calculations.[106–109] The
LSDA is derived from the interaction properties of a homogeneous electron
gas. This has no spin–orbit interaction, and therefore Hund’s second rule
is not built in. It results in atoms from orbital exchange interactions. It
has been suggested by Brooks et al. [107,109] to compensate for the lack of
these interactions in the LSDA by adding a term to the energy functional
which is of the form
1
(65)
∆EOP = − E 3L2Z ,
2
where E 3 is a Racah parameter, a linear combination of Slater Coulomb
integrals. This “orbital polarization” (OP) leads to different energies of
the | m > states when there is an orbital moment.
The orbital polarization scheme has been applied to several systems (see
references above) where it improved the agreement between theory and
experiment. A recent application to americium was reported by Söderlind
et al.,[110] who examined structural changes of Am under pressure. The
results were consistent with a high–pressure phase with delocalized 5f electrons and a low–pressure phase with localized and non–bonding 5f states,
a Mott transition.
A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self–interaction corrections (SIC) [111–114] in electron structure methods for solids, Svane et al.
[115–120]. In the Hartree-Fock (HF) theory the electron–electron interactions are usually divided into two contributions, the “Coulomb term” and
the “exchange term” although they both are Coulomb interactions. The
separation though, is convenient because simplifications of self–consistent–
field calculations can be obtained by including in both terms the interaction
of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms
exactly cancel each other. However, when the exchange term is treated
36
in the L(S)DA formalism, the cancellation of the self–interaction terms is
no longer perfect. Corrections for this were included in atomic physics for
example as described in Refs. [112–114], but SIC is much more difficult to
include in selfconsistent electronic structure calculations for solids. [115]
The energy functional in the SIC–LSDA is similar to that of the LSDA
apart from a correction term
∆
SIC
=−
occ
X
δαSIC ,
(66)
α
where
LSD
δαSIC = U [nα ] + Exc
[nα↑, nα↓ ].
(67)
Here α labels the orbitals, and the sum in eq. (66) runs over the occupied
LSD
states. U [nα ] is the Hartree term for a single state α and Exc
[nα↑, nα↓ ]
is the exchange–correlation energy in state α. The latter is calculated in
the LSDA, and consequently the cancellation of self-interactions is still not
exact.
The SIC–LSDA treats localized and delocalized states on equal footing,
and by comparing total energies corresponding to different distributions of
electrons in a shell on localized resp. delocalized states the energetically
most favorable configuration can be selected.[119]
When applied to transition–metal monoxides [115] the SIC–LSDA gave
results for energy gaps as well as for total moments, which were substantially improved over the LSDA results. Recently, several f –electron systems have been considered, see Refs. [117–120]. The latter work on Pu
monochalcogenides shows that in these systems there is a coexistence of localized and delocalized Pu-5f states. By considering different Pu valencies,
determined by choosing different numbers of localized 5f states, a ground
state configuration could be found. The total angular momenta were calculated and the effective magnetic moments were derived and compared to
experiments. The magnetic properties of these Pu compounds are not well
[120] described in the SIC–LSDA, and this was ascribed to spin fluctuations
and other effects which are not included in a mean–field theory.
Another scheme known as “LDA+U” has been developed [121–125] to
add aspects of the Hubbard model [126,127] to self–consistent band structure calculations. It introduces additional interactions which depend on the
occupation of the individual orbitals, and in that way an extra symmetry
37
breaking is introduced. The extra terms added to the L(S)DA functional
thus depend on the occupancies nσm of the (m, σ) orbitals:
ELDA+U = ELDA − Edc +
1 X
Umm0 , nσmn−σ
m0
2
m,m0 ,σ
1 X
(Umm0 − Jmm0 )nσm nσm0 (68)
+
2
0
m6=m ,σ
Here Umm0 and Jmm0 are the elements of the effective Coulomb and exchange
matrices for the strongly correlated electron states (such as the f –electrons
in the actinides), and Edc corrects for double counting of the correlation
effects. It is the average of the last two terms in eq. (68). The effective
Umm0 and Jmm0 are related to Slater integrals F (k) . The average Coulomb
term Uav = F (0) can be derived from a constrained LDA scheme.[128,129]
The extra terms in ELDA+U have some similarity to the “orbital polarization” [107] term eq. (65) but a major difference is that this only includes
the symmetry breaking which restores Hund’s second rule, but it does not
include the (large) shifts which in LDA + U enters via F (0) . The orbital
polarization can produce a large orbital moment, but it cannot as LDA+U
induce large splitting of the correlated states (f states, for example). This
difference between the two approaches was examined quantitatively by
Liechtenstein et al. [125] who calculated magneto–optical effects in CeSb.
They calculated (see also Section 5) the diagonal and off–diagonal elements
of the optical conductivity σij (ω), where
π X nm
Pi (k)Pjmn (k)fnk (1 − fmk )δ(ω − ωnm (k)) (69)
Im[σij (ω)] =
3ωΩ
k,n,m
The LSDA+U results for the diagonal terms agree well with experiments
[130] and with calculations [132] using LDA plus orbital polarization. The
off–diagonal part, on the other hand, disagrees with the calculation [132]
with OP, but agrees well with experiments.[131] The spin and orbital moments calculated in Ref. [125] are −0.92µB and 2.86µB giving a total
moment, 1.94µB , in good agreement with the experimental value for the
antiferromagnetic ground state, (2.10±0.04)µB .[133,134] Svane et al. [135]
examined the phase changes of CeSb and other rare–earth pnictides and
chalcogenides using the SIC–LSD scheme. This is less suited for spectral calculations which can be compared to the LDA+U results. Cerium
38
monopnictides exhibit several interesting physical properties, and some of
these are difficult to explain. Their magnetic properties are unusual. The
antimonide is especially interesting since it has a complicated magnetic
phase diagram [133,134], crystal–field splitting [136,137] and large magnetic anisotropy. It is the compound with the largest known Kerr angle
[131,130], see also Section 5.
Unusual magnetic properties are also found in UGe2. This compound
crystallizes in the orthorhombic Cmmm base–centered structure, like ZrGa 2.
The magnetic structure is collinear and the moments, 1.49µ B are ferromagnetically ordered. Experiments have shown that the magnetocrystalline
anisotropy is very strong in UGe2. The a–axis (the shortest) is in the easy
direction. The most unusual behavior, however, is observed when pressure
in the range 1–1.6 GPa is applied. In that range, and for T < 1 K UGe2 is
found [138] to be a superconductor. This means that the superconducting
and the ferromagnetic phases coexist. The moment in UGe 2 is comparable
to that of Ni — “strong” ferromagnetism. This would suggest that magnetism is the source of the pairing which then must be a triplet pairing.
Shick and Pickett [139] examined the electronic structure and the magnetic properties (including spin–orbit coupling) of UGe2 by means of the
LDA+U scheme. The parameters U and J were chosen so that the ground
state magnetic moment MJ = MS + ML was reproduced. The total energies E(a), E(b), and E(c) for moments along the a, b, and c axes were
calculated and the magnetocrystalline anisotropy energy was derived. The
results, E(b) - E(a) = 0.55 mRy/f.u., and E(c) - E(a) = 0.67 mRy/f.u.,
are large and reflect a strong spin–orbit coupling. For usual ferromagnets
the magnetic anisotropy energies are much smaller, of the order of µRy.
This makes it very difficult to calculate these effects. The following section
deals with these in more details.
5. Magnetooptical and magnetoelastic effects.
Magneto–crystalline anisotropy (MCA) can be substantial in some f –
electron systems as mentioned in the last part of Section 4. For the “usual”
(3d) magnetic metals Fe, Co, and Ni they are smaller. The magnetic
anisotropy energies (MAE) are extremely small, a few µeV/atom in Fe,
see Ref. [140,141] and references therein. These effects as well as magnetooptical effects like the magneto–optical Kerr effect (MOKE), soft x–ray
magnetic circular dichroism (MCD) are also useful tools in the characterization of magnetic thin films and layered structures. [142] Transport
39
properties of such layered magnetic structures, where spin dependent scattering processes are important, exhibit new striking effects such as the
Giant Magnetoresistance (GMR) [143,144]. Special magnetic properties
like the oscillatory exchange coupling are found in multilayers consisting
of alternating layers of magnetic and non–magnetic metals.[145] The simultaneous presence of spin–polarization and spin–orbit coupling plays an
important role in several of these effects.
5.1. Magnetic dichroism.
X–ray scattering experiments are important in determining structures
of solid phases as well as monitoring the electronic structures, for example
by photoelectron spectroscopy. In addition to this, magnetic scattering experiments can give information about the magnetic structure of solids, and
they then supplement neutron scattering techniques. The magnetic scattering monitors relativistic effects in the sense that in a non–relativistic
limit the x–rays would couple only to the charge of the electron, whereas
in the relativistic theory the Compton amplitude depends also on the spin
of the electron. Platzman and Tzoar [146] showed how this could be used
to determine the spin–dependent moment distribution in ferromagnetic
materials and to examine magnetic structures. The increasing number of
synchrotron radiation facilities has improved the possibilities of performing
high–resolution studies of magnetic scattering in antiferromagnets and of
interfering charge and magnetic scattering in ferromagnets, bulk and thin
films, see for example Ref. [147]. Also, the properties of the synchrotron
light allow systematic studies of the polarization dependence of the magnetic scattering [148]. Shütz et al. [149] studied near–edge photoabsorption
of circularly polarized x–rays in iron above the K edge. The circularly polarized x–rays create (partially) polarized photoelectrons excited into the
continuum of p states, which are spin–orbit split. The spin–dependent
part of the absorption depends on the induced polarization of the photoelectrons and the difference of the spin density ∆ρ = ρ ↑↑ − ρ↑↓ of final
states with spins parallel (↑↑) and antiparallel (↑↓) to the spins of the 3d
electrons. This observation of the (soft) x–ray magnetic dichroism (MCD)
was important because the technique became a tool for investigations of
magnetism in many different materials, transition metals as well as rare
earths. The principal mechanism of the dichroism in core–level photoelectron spectroscopy (PES) is easy to understand in the “three–step model”
combined with the one–electron band structure picture. The three–step
model for PES assumes that, i) an electron is optically excited from an
40
p core states
Energy
Continuum
initial state |i > to a final state |f > in the band structure, then, ii) follows transport towards the surface without inelastic scatering (“primary
spectrum”) and finally, iii) the transmission through the surface into the
vacuum region. We consider here the first step and assume that |i > is a
core state, taken to be a p state below (could be a 2p state in a 3d transition metal). As a final state we choose |f > to be an s–like continuum
state. The core state is spin–orbit split into j = 3/2 and j = 1/2 states,
see Fig. 14.
1/2
±1/2
Difference Signal
3/2
-3/2
-1/2
1/2
3/2
1/2
1/2
-1/2
j
mj
∆m= -1
∆m= +1
left
right
Figure 14. Allowed dipole transitions from a core p state to an empty s
state (the uppermost two levels being almost degenerate in this sketch),
when the exciting radiation is left, resp. right circularly polarized light.
The right-hand side of the sketch illustrates a difference photoemission
dichroism spectrum.
41
In a magnetic material, here a 3d ferromagnetic metal (like Fe), the
polarization creates a magnetic field which causes a splitting of the j =
3/2 and 1/2 levels into sublevels, mj = −3/2, −1/2, 1/2, 3/2 and mj =
1/2, −1/2, as illustrated in Fig. 14. According to the dipole selection rules
transitions are allowed with right/left circularly polarized light between
states with ∆m = +1/ − 1. Such transitions are shown as arrows in Fig.
14. The right–hand side of the figure is a sketch of a (broadened) difference
signal.[150]
A sum rule derived by Thole et al. [151], and also by Altarelli,[152] states
that (approximately) the integral of the circular dichroism signal from two
spin–orbit partners of a core edge is related to the orbital moment:
R
(σ − σ−)dE
− < Lz >
R +
(70)
= Idic =
[`{2(2` + 1) − n}]
(σ+ + σ−)dE
where Idic is the dichroism intensity, σ+ and σ− absorption cross sections
for left and right polarized light, and ` the angular–momentum quantum
number of an ion with an incomplete outer shell with `n configuration. The
denominator of the right–hand side of eq. (70) then contains the number
of holes, 2(2` + 1) − n, in the `–shell. The result is in particular interesting
because eq. (70) implies that MCD, yielding a direct measure of < L z >,
provides a method for measuring the orbital contribution to the magnetic
moment. In this way MCD experiments also represent a supplement to
neutron scattering methods. The sum rule, however, is approximate when
applied to real solids, because it was derived for electric dipole transitions in
a model of a single ion. Therefore the validity for applications to materials
with strong multishell hybridization needs to be verified. Wu and Freeman
[153] calculated MCD spectra for 3d transition metals, in bulk as well as
surface regions, in order to examine the sum rule as well as the importance
of the magnetic dipole term which enters in a second sum rule derived by
Carra et al. [154] Considering transitions from the 2p states, L 3 and L2 ,
in 3d transition metals, the sum rules, eq. (70) (with `= 2) and the one
related to magnetic dipole terms are [153]
R
σm dE
< Lz >
Im
= RL3
=
(71)
It
2N
σ
dE
h
t
L3
R
(σm,L3 − 2σm,L2 )dE
Is
< Sz > +7 < Tz >
R
=
=
(72)
It
3N
σ
dE
h
t
L3 +L2
42
Here σm ≡ σ+ − σ−, σt ≡ σ+ + σ− , Nh is now the number of holes obtained
by integrating the density of unoccupied states, and Tz is the z–component
of the magnetic dipole operator
1
T = [S − 3r̂(r̂ · S)]
2
(73)
In Ref. [153] it was found that the angular–momentum rule eq. (71) for
Fe, Co, and Ni holds within ' 10% but that the spin rule, eq. (72), in
particular for Ni has substantial errors. With
< Se >≡
< Sz > +7 < Tz >
3
(74)
the quantity
I s . < Se >
R1 ≡
− 1,
It
Nh
(75)
is 0.36 for bulk Ni and 0.52 for a Ni surface layer, implying up to 52%
error. If the hybridization between d– and (s, p)-states was switched off,
the largest error was reduced to less than 10%. Further, it was shown that
the errors mainly originate in the denominators of eqs. (71) and (72), and
that the applicability of the sum rules is improved by combining them into
Im
< Lz >
=
.
Is
2 < Se >
(76)
5.2. Magnetic anisotropy, magnetostriction.
The 3d transition metals have very small magnetic anisotropy energies
(MAE), and their calculation is a challenge to ab initio relativistic electronic structure calculations. The MAE may be calculated by means of
the so-called Force Theorem,[155–158] (FT):
X
X
∆E M AE = E(→) − E(↑) =
εi −
εi ,
(77)
occ→
occ↑
i.e. differences in one–electron energy sums alone, the first sum in (77)
meaning the sum evaluated for a band calculation where the moments are
in one direction (→), the second sum for a calculation where the moment
are in another (↑) direction. The FT, eq. (77) assumes that one calculation
43
is self–consistent, whereas the other is not, but uses essentially the same
potentials (see Refs. [155,156] for further details). Using this method
Daalderop et al. found MAE values for Fe, Co, and Ni of the right order
of magnitude [141], but the wrong easy axes were predicted for Co and Ni.
In multilayer structures, (Co/Pd, Co/Cu, and Cu/Ag), on the other hand,
they obtained [159] good agreement with experiment, mainly because the
anisotropy energies for these systems are about a factor 10 larger than for
bulk cobalt. A later work [160] by the same authors report calculations
for Co and a series of compounds, FePd, CoPd, YCo5 , FePt, and CoPt.
The calculations were performed with and without inclusion of “orbital
polarization” [107,109], eq. (65). In order to increase the accuracy of the
MAE calculations by means of the FT, Wang et al. [161] used a tracking
procedure in which the occ → and occ ↑ states in eq. (77) are determined
according to their projections back to the occupied set of states without
the perturbation from the spin–orbit operator.
In order to reduce further the uncertainties due to spin–orbit interactions
between nearly degenerate states close to the Fermi level Wang el al. [162]
proposed a torque method. They illustrated the method by considering a
uniaxial system where the energy can be well approximated by
E(θ) = E0 + K2sin2θ + K4 sin4 θ,
(78)
where θ is the angle between the direction of the magnetization and the
normal axis. A torque is defined as the derivative of E(θ):
T (θ) =
dE
= K2 sin(2θ) + 2K4sin(2θ)sin2 θ.
dθ
(79)
From this it follows that
T (45◦) = K2 + K4 = E(90◦) − E(0◦) = ∆E M AE
(80)
The Hellman–Feynman theorem gives [162]
∆E M AE = T (45◦) =
X
occ0
< ψk |
∂HSO
|ψk >θ=45◦
∂θ
(81)
Therefore only one k–space integration is needed and one Fermi surface is
involved. This improves the numerical accuracy.
With sufficiently accurate methods for calculating the magnetic anisotropy
energies it should also be possible to treat magnetostriction phenomena in
44
transition metals by ab initio calculation methods. When the magnetization in a crystal is rotated a strain (magnetostrictive) is induced. As a
particular example, consider a magnetization in a direction specified by
the direction cosine αz , then the change ∆l in the lattice constant in the
(001) direction is given by
and
1
3
∆l
= λ100 αz2 −
l
2
3
λ100 =
3 ∆lz − ∆lx,y
2
l
(82)
(83)
Near the equilibrium value lo of the length the total energy is well described
by a quadratic form:
E = al2 + bl + c
(84)
The MAE varies essentially linearly with l,[163,164]:
∆E M AE = E(x, y) − E(z) = k1l + k2
(85)
and then
λ001 = −2k1/3b
(86)
Artificial, layered structures of magnetic materials are becoming technologically important since they can be “tailored” to have properties which
cannot be found in natural bulk–like materials. The magnetic moments
at the interfaces differ from those in the bulk on a detailed understanding of the anisotropy effects require models which combine magnetic and
elastic properties. Recently Schick et al. [164] performed relativistic spin–
polarized calculations in order to examine the magnetoelastic coupling and
the magnetic anisotropy strain dependence in Co/Cu(001) layered structures. In addition to bulk magnetoelastic coupling coefficients also surface
magnetoelastic coupling must be included.
5.3. The Kerr effect.
Kerr observed [165] 125 years ago that linearly polarized light by reflection from a magnetic solid has its polarization plane rotated with respect
to that of the incoming light. This magnetooptic Kerr effect (MOKE) is
45
thus related to the Faraday effect as well as to the circular dichroism already discussed. Letting the z–axis be normal to the surface, the complex
Kerr angle can be expressed as
φK (ω) = θK (ω) + iεK (ω)
σxy (ω)
4πi
=−
(1 +
σxx (ω))−1/2
σxx (ω)
ω
(87)
where the interband contribution to the conductivity tensor is
Παif Πβfi
ie2 X X 1 h
σαβ (ω) =
h̄m2 Ω
ωif (k) ω − ωif (k) + iδ
k
i,f
+
(Παif Πβfi)?
ω + ωif (k) + iδ
(88)
i
fik (1 − ff k )
Here Ω is the unit cell volume, the k–sum runs over the Brillouin zone,
the Fermi factors fik and (1 − ff k ) ensure that only occupied initial states
and empty final states are included, h̄ωif = εf (k) − εi (k), and the matrix
elements Παif are [166]
h̄
e
Παif (k) =< f |(p + A +
σ × ∇V )α |i >,
c
4mc2
(89)
where A is the vector potential, p the canonical momentum operator
and the last term in (89) is due to spin–orbit interaction. The spin–orbit
coupling affects the conductivity through its influence on the wavefunctions
of states |i > and |f > and directly via its presence in the transition
matrix element eq. (89). Also the magnetic field, represented by the
vector potential in the present formulation, influences the Kerr signal, and
the relative importance of these two contributions [167] vary from one
material to another.
Model studies where the weight of the SO term was varied were performed by Oppeneer et al. [168] As examples we show in Fig. 15 calculated
[169] and experimental Kerr spectra, θK (ω) and εK (ω). (The experimental
results for FePt are from Ref. [170])
Equation (89) only contains contributions from interband transitions,
and only direct transitions are taken into account; the final state |f >
and the initial state |i > relate to different band indices, but they have
the same k vector. The small momentum transferred from the photon
46
C-178-452
COMPLEX POLAR KERR EFFECT (degrees)
0.4
θK
εK
0.0
-0.4
-0.8
exp.
(001)
(110)
-1.2
0.2
θK
εK
0.0
-0.2
-0.4
-0.6
(001)
(110)
-0.8
-1.0
0
2
4
0
2
4
PHOTON ENERGY (eV)
6
Figure 15. Calculated interband contribution to polar Kerr spectra of CoPt
(Ref. [169]) (lower panel) for the (001) and (110) orientations of the magnetic moment. (A 0.03 Ry broadening was included). The experimental
data (Ref. [170]) are for (001) oriented FePt.
47
is neglected, εi and εf have the same argument k, as also indicated in
h̄ωif (k). However, the optical conductivity of metals usually have nonnegligible intraband contributions, in particular for small photon energies.
This is also the case for Kerr spectra as demonstrated in the work by Uba
et al. [171] for platinum.
6. Conclusion.
Relativistic effects are considered to be “small” in solid state physics
where mainly properties of the outermost electrons in constituent atoms
are important. These are, however, not independent of the core states and
relativistic effects may therefore also “propagate” even from the innermost,
deep lying, states. This happens through screening and orthogonality requirements (in one–particle pictures). The fact that the relativistic effects
are “small” does not imply that condensed matter theory in general can
neglect them. In this overview we have only briefly discussed the effects of
energy shifts in the electronic states. The major part of the presentation
has concerned the influence on the physics from the spin–orbit coupling.
We concentrated on two main aspects, i) the spin splittings of electronic
states in systems (semiconductors and low–dimensional structures) without spatial inversion symmetry, and ii) the simultaneous existence of spin–
and orbital moments, their coupling and influence on magnetic properties.
There are good reasons for emphasizing these topics. The spin–splitting
of semiconductor bands affects the optical properties, and the analysis of
the spin states is important in the fields now called spintronics, involving
transport of spins through semiconductors. This may lead to construction
of technologically very interesting new devices such as spin diodes [172,173]
or spin memory [174] elements. If sufficiently long spin coherence times can
be achieved for the spins injected in semiconductors as well as transport
over long distances, such coherent properties may enable quantum computational operations in the solid state, see Refs. [175,176]. In that respect
also a better understanding of magnetic metals (injection) as well as nonmagnetic and magnetic semiconductors is essential. And, in order to return
to the statement that relativistic effects are ”small” in solids, it should be
recalled that spin splittings in GaSb and CdTe for some states reach 250
and 310 meV, respectively. This is not small at all when compared to
typical optical transition energies.
The second topic, spin-orbit interactions in magnetic metals, relates to
this, but in addition there are several other questions of fundamental nature
48
which need to be answered, not least for materials with strongly correlated
electron systems and complex spin–orbital interactions. Where can we use
an LSDA (relativistic) formalism, what are the limits of LDA+U, SIC,
orbital polarizations, dynamical mean-field theory? — etc. Applications
are manyfold of new magnetic structures and materials. Magnetooptics,
circular dichroism, magnetoelastic effects a.o. are important as experimental tools and in device applications. Transport properties in layered
magnetic structures have been, and are, parts of a rapidly growing field
with almost immediate applications in recording and data storage devices
(GMR effect, for example). An excellent review of perspectives of the giant
magnetoresistance was presented recently by Tsymbal and Pettifor in Ref.
[177].
Acknowledgements. The author wishes to thank his collaborators in
the joint research referred to in this chapter. In particular he has benefitted from O.K. Andersen’s expertise on his linear methods (LMTO, LAPW),
from the extensive collaboration with M. Cardona on the electronic structures on semiconductors, not least the spin-splitting problems. R.C. Albers, M. Boring and G. Zwicknagl are thanked for their collaborations
on the relativistic electronic structures of heavy-fermion materials, and A.
Svane and L. Petit for several discussions and valuable information on their
progress in the description of strongly correlated electron systems.
The present chapter has addressed some subjects of solid state theory
which are different from those treated in quantum chemistry. On the other
hand, there are indeed large areas of common interest of science as well as
concerning methodology. In that respect communication with P. Pyykkö
has been very fruitful.
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