R?ssler U. Solid State Theory: An Introduction

Подождите немного. Документ загружается.

208 7 Correlated Electrons

For more realistic descriptions, there exist extension s of the Hubbard

model which also include transfer to next nearest neighbors, more than one

band, and correl ation terms including diﬀerent sites. One of these extensio n s

is the subject of Problem 7.6. These mo dels are used to investigate material

systems whose electronic properties are determined by correlation [

209–213].

Among those are the high-T

cuprate superconductors. Their crystal structure

contains layers with Cu and O atoms in a quadratic lattice. The overlapping

Cu 3d and O 2p orbitals form a narrow p-d hybrid band. Due to the other

constituents of the crystal structure, this band can be partially depopulated

(or ﬁlled with holes). Thus, by comparing photoemission data for diﬀerent

ﬁlling, the spectral weight of the evolving Hubbard bands becomes evident.

The results of this section have uncovered a deﬁciency of the DFT-LDA

concept. Makin g use of an eﬀective single-particle potential, which is the same

for all electrons, irrespective of their orbital, this approach does not account

for the correlation eﬀect outlined in presenting the Hubbard model, i.e., it is

not capable of describing occupation-dependent energy bands. This deﬁciency

is not problematic for energy bands deriving from s or p orbitals, for which the

nearly free electr on or pseudo-potential approach applies as well as the LCAO

method because the orbital character is diminished due to delocalization. In

contrast, for the localized d and f orbitals, this is not the case . In order

to overcome this deﬁciency, the LDA+U concept has been developed [

195].

Its essential idea is to replace the LDA Coulomb energy UN/(N − 1)/2for

d − d interaction, which is assumed to be part of the LDA energy functional

LDA

[n], by the Hubbard correlation by writing

E[n]=E

LDA

[n] −

UN(N − 1) +



i=j

, (7.71)

where N =



is the number of electrons and n

the orbital occupancy.

The orbital energies

∂E[n]

∂n

= ǫ

LDA

+ U

− n

(7.72)

are changed according to their occupancy with respect to the LDA value.

A more recent extension of this concept is the dynamical mean-ﬁeld theory

(DMFT) [

210, 214].

7.3 Fermi Liquids

Classical liquids are known to exist due to particle–particle interaction, when

the average interaction energy cannot be neglected in comparison with the

average thermal energy k

T , and condensation, i.e., the phase transition from

the gaseous to the liquid phase, takes place. In t hi s condensed phase, the ther-

mal motion is comparable with the mean particle separation. By reducing the

7.3 Fermi Liquids 209

temperature, the thermal motion can get so small that the interaction domi-

nates the kinetic energy. In this situation, the phase transition from the liquid

to the solid state, takes place. This systematic, which is based on classical

arguments, does not account, however, for quantum eﬀects a s has become

apparent for He due to its small mass. At suﬃciently low temperatures, liq-

uid He does not condense into the solid state. Instead, when the thermal

deBroglie wavelength λ

=(¯h

/2Mk

T )

1/2

, a quantum mechanical length

scale, becomes comparable to the average particle separation (while at the

same time the energy of the zero-point motion is much larger than the inter-

action energy), it enters a state known as quantum liquid. This phase transition

is ruled by quantum statistics and leads to a Fermi liquid for

He but to a

Bose liquid for

He. As the interacting particles are neutral He atoms, these

two phases are jointly denoted as neutral quantum liquids.

Especially for

He, Land au developed a theory of Fermi liquids [

197], which

comprises the interplay between Fermi statistics and particle interaction. As

it turned out later on, this theory applies as well to interacting electrons in

metals and dop ed semiconductors, whi ch can be classiﬁed as charged Fe rmi

liquids. We note in passing that Landau’s Fermi liquid theory is used also for

neutron stars. In this section, a brief outline is given of this theory, which is

closely connected with the concept of quasi-particles, and we refer for further

reading of the literature [

64, 194, 197–199].

Fermion systems without interaction, as treated in Chap.

4 in the Sommer-

feld model or in Chap.

5 in the independent particle model of electronic band

structure, can be characterized by their ground state and low-energy or ele-

mentary excitations out of the ground state. The former is deﬁned for T =0K

by ﬁlled states up to the Fermi energy, the latter are particle–hole excitations

across the Fermi surface. Here, particle(hole) means an electron(missing elec-

tron) above(below) the Fermi energy. We have noticed already that the eﬀect

of the interaction is to modify the single-particle energy ǫ

kσ

by a self-energy,

which incorporates interaction eﬀects (to an extent depending on the applied

approximation, see e.g., (

7.27)and(7.28)) into the independent particle pic-

ture thus leading to the concept of quasi-particles, which we have addressed

already in Sects.

4.4, 5.2,and7.1.

Lo oking at the quantum numb ers, momentum p =¯hk and spin σ (the

band index is suppressed here), we have found a one-to-one relation between

an independent particle and the corresponding quasi-particle, a l th ough via

the self-energy, the quasi-particle energy ǫ

kσ

] becomes a functional of the

occupation numbers. Thereby, the spin is not changed and the quasi-particles

remain fermions. The self-e n ergy quantiﬁes virtual excitations of electron–hole

pairs, which represent charge or spin density waves. Consequently, a quasi-

particle is the bare particle of the n oninteracting system dressed by a cloud

of virtually excited density waves. The lifetime o f a quasi-particle, deﬁned by

the imaginary part of its self-energy, is determined by the scattering processes

which take place under the constraints of energy and momentum conserva-

tion. At T = 0, this leads to an inﬁnite lifetime for quasi-particles at the Fermi

210 7 Correlated Electrons

energy, because the available phase space for scattering shrinks to zero, while

the scattering rate increases quadratically with the energy separation from

the Fermi energy (or at ﬁnite T from the chemical potential) [

122, 200]. Thus,

quasi-particles are well deﬁned for low-energy excitations. This leads to the

concept of Fermi liquids: it is based on the assumption that the excitation

spectrum of the interacting Fermi system is similar to that of the noninter-

acting Fermi system and that the particles (or states) of the latter evolve

one-to-one into the quasi-particles (or states) of the former without changing

the quantum numbers when the interaction is adiabatically switched on. This

concept is supported by the observation that in a certain temperature range

some properties (speciﬁc heat, spin su sceptibili ty) of many metals (like those

He) correspond, in their temperature dependence, to those of the nonin-

teracting Fermi system, however, with changed kinematic properties such as

the particle mass.

Elementary excitations can be described as changing the occupation of

states around th e chemical potential (or the Fermi energy) with respect to

the ground state occupation n

kσ

δn

kσ

= n

kσ

− n

kσ



+1 |k| >k

−1 |k|≤k

(7.73)

The total energy of the system is a functional of the occupation numbers n

kσ

E = E[n

kσ

] but E =



kσ

, (7.74)

because the quasi-particle energy ǫ

kσ

= δE/δn

kσ

depends on the occupation

due to the self-energy and has in general a nonvanishing variational derivative

δǫ

kσ

δn

′

δn

kσ

δn

′

=: f (kσ; k

′

) =0. (7.75)

We may express the total energy as a Taylor series with respect to the ele-

mentary excitations (i.e., the changes i n the occupation numbers) about the

ground state energy E

E[n

kσ

]=E



kσ

]δn

kσ



kσ

′

f(kσ; k

′

)δn

kσ

δn

′

+ O(δn

). (7.76)

Denoting the quasi-particle energies for the ground state distribution ǫ

kσ

]

by ǫ

kσ

, the quasi-particle energies can b e expressed as

kσ

δE

δn

kσ

≃ ǫ

kσ



′

f(kσ; k

′

)δn

′

. (7.77)

7.3 Fermi Liquids 211

The adequate thermodynamic potential for the grand-canonical ensemble

with varying occupation N =



kσ

is the free energy F = E −μN which

under elementary excitations with N −N



kσ

δn

kσ

,whereN

is the total

o ccupation in the ground state, changes by

F − F

= E − E

− μ(N − N

)



kσ



kσ

− μ



δn

kσ



′

f(kσ; k

′

)δn

′

δn

kσ

. (7.78)

Note that we consider a situation where the quasi-particle energies are close

to the chemical potential μ and that ǫ

kσ

− μ =0onlyforkσ with δn

kσ

=0;

thus, the ﬁrst term is of the order (δn

kσ

)

. As for the noninteracting particles,

the free energy is stationary for the equilibrium distribution function

kσ

=[1+exp(β(ǫ

kσ

− μ))]

−1

, (7.79)

with the dispersion of the independent particles replaced by that of the

quasi-particles. It has the form of the Fermi–Dirac distribution function but

is an implicit equation for n

kσ

due to the functional dependence of the

quasi-particle energies ǫ

kσ

on the occupation.

In contrast to a microscopic theory, which aims at a calculation of the

quasi-particle energies, the Fermi liquid theory replaces the interaction by

parameters and relies on the one-to-one corr e spondence between independent

(or bare) particles and quasi-par ticles including their statistics. This is out-

lined in the following by, assuming for better transparency, an isotropic Fermi

liquid and a spin degenerate dispersion. The Fermi velocity is deﬁned by

k,F

¯h

∇

kσ

k=k

¯hk

∗

, (7.80)

where m

∗

denotes the eﬀective mass of the quasi-particle at the Fermi energy.

Let us a ssume the quasi-particle dispersion ǫ

kσ

to be a suﬃciently smooth

function in the vicinity of the Fermi energy E

(or the chemical potential μ).

Then, we can write

kσ

− μ ≃

∇

kσ

k=k

(k − k

¯h

∗

(k − k

) (7.81)

as depicted in Fig.

7.5. The same relation holds for the independent particle,

however, with a mass m instead of m

∗

. This diﬀerence in the masses is due to

the fact that the quasi-particle consists of the bare particle and a cloud of den-

sity ﬂuctuations around it, which moves along with the particle and reduces

its mobility, i.e., m

∗

>m. One immediate consequence is the enhancement of

the density of states at the Fermi energy

D(E



kσ

δ(ǫ

kσ

− E

∗

¯h

, (7.82)

212 7 Correlated Electrons

kσ

Fig. 7.5. Quasi-particle dispersion (solid) and linear approximation around the

chemical potential (dashed)

which implies an enhancement of all quantities which are proportional to the

D(E

) as e.g., the particle contribution to the speciﬁc heat. The eﬀect of

the interaction is considered here in the parameter m

∗

, which is accessible by

measuring the Sommerfeld coe ﬃcient (see Sect.

4.2).

Let us now take into account the interaction between quasi-particles repre-

sented by f (kσ; k

′

). For the isotropic Fermi liquid and dominating exchange

interaction (as in Sect.

6.2), this can be written [64]as

f(kσ; k

′

)=f

(k, k

′

)+σ · σ

′

(k, k

′

). (7.83)

Being close to the Fermi energy, we have |k|, |k

′

|≃k

and because of the

isotropy, the f

(k, k

′

) depend only on the angle θ between k and k

′

. Therefore,

we may expand these functions in terms of Legendre polynomials (normalized

for practical reasons by the density of states D(E

))

(k, k

′

D(E

)

∞



(cos θ). (7.84)

The coeﬃcients F

are the phenomenological Fermi liquid parameters wh ich

have to be determined by comparison with experimental data. As it turns out

[

64, 122], the quasi-particle eﬀective mass is given by

∗

= m (1 + F

/3) . (7.85)

Thus, F

could be determined from the low-temperature behavior of the

speciﬁc heat (provided the system is isotropic). The Pauli spin susceptibility

is enhanced due to interactions and can be expressed as

spin

∗

m(1 + F

)

spin

, (7.86)

with the spin susceptibility χ

spin

of free electrons. Its experimental value pro-

vides the parameter F

, if the eﬀective mass is already known from speciﬁc

heat data. Fermi liquid parameters are rep o rted so far only for He [

64,199, 215].

7.4 Luttinger Liquids 213

kσ

Fig. 7.6. Particle (dashed) and quasi-particle (solid) distribution at T =0K.The

discontinuity at k

represents the weight Z

Finally, we look at the momentum distribution n

kσ

, which can be obtained

from the Green function with the help of the dissipation–ﬂuctuation theorem

kσ

= c

†

kσ

 = −

¯hπ



+∞

−∞

ImG(kσ; ǫ)

1+e

−βǫ

dǫ, (7.87)

with G(kσ; ǫ)from(

7.31). The denominator simpliﬁes for T =0Kandthe

integration can be extended to a contour in the upper complex plane. This

includes only the quasi-particle poles for k<k

(which are in the upper half

plane) but not those for k>k

(which are in the lower half plane) and results

in a jump of n

at k = k

, which equals th e weight Z(k

) of the quasi-particle

pole (Fig.

7.6). The calculation of this jump is the subject of Problem 7.7.

As we have seen by these consid erations, the Fermi surface introduced

in Chap. 4 for the nonintera cting electrons exists also in the presence of

the electron–electron interaction, as can b e seen from the discontinuity

in the momentum distribution (Fig.

7.6). It is the signature of existing

quasi-particles. However, when replacing the independent particles by quasi-

particles, the kinematic properties and consequently also physical quanti-

ties change as compared with the independent particle result. This can be

exploited to determine the parameters of the Fermi liquid theory.

The incorporation of particle interaction in the quasi-particle depends

essentially on the phase space available for interaction processes close to the

Fer mi energy. While in three and two dimensions, this phase space is a sphere

or a circle, resp ectively, it shrinks to two points for a one-dimensional sys-

tem. The dimensionality eﬀect can be demonstrated for the Lindhard function

(Problem 7.8). We shall see in the next section how this will dramatically alter

the situation when we consider one-dimensiona l electron systems.

7.4 Luttinger Liquids

The dimensionality and its inﬂuence on solid state properties have been men-

tioned already in several sections. Obviously the two -dimensional systems

are the surface o f a solid (Sect.

3.6) and the interface between two diﬀerent

214 7 Correlated Electrons

solids (a heterostructure). We have learnt about semiconductor heterostruc-

tures that they can accommodate a two-dimensional electron gas (Sect.

5.6).

In this section the one-dimensional electron systems will be the focus of inter-

est. There are several r ealizations, which have stimulated the investigation

of such systems. Among those are special molecular crystal structures such

as inorganic and organic linear chain compounds that allow band formation

by overlapping atomic orbitals in one spatial direction only [

201]. Another

example is conducting polymers, for which energy bands arise from repeated

conjugated bonds along the strand [

111, 112]. But one may start also from the

2D electron systems in heterostructur es to prepare by etching, cleaved edge

overgrowth, or depletion via top gates a 1D channel (a quantum wire) along

which electrons can move freely [

216–218]. The youngest child in this family is

carbon na notubes, which can be understood as a graphite monolayer rolled up

to a cylinder with a diameter of a few nm [

21, 113, 114]. All these systems have

been and are still under investigation due to their peculiar properties deter-

mined by the low dimensionality which do not ﬁt into the framework of Fermi

liquid theory. The essential point here is the breakdown of the quasi- particle

concept [

22, 122, 193, 219–222].

Free electrons in one dimension would be characterized by a qua d ratic

dispersion rela tion which cuts the Fermi energy at k = ±k

as depicted in

Fig.

7.7. B eing interested in elementary excitations around the Fermi energy,

it is advantageous to linearize the dispersion relation as in the previous section

by writing (relative to the chemical potential)

±,k

≃

¯h

(±k − k

)=¯hv

(±k − k

) , (7.88)

with the Fermi velocity v

. This spectrum consists of two branches (linear

in k) that correspond to electrons traveling left and right along the extension

of the 1D system. We can immediately write down the corresponding single-

particle part of the Hamiltonian in terms of fermion operators ({c

kα

†

′

} =

α,α

′

k,k

′

)

ε°

–k

Fig. 7.7. Dispersion relation for a 1D fermion system with linear approximation

around the Fermi energy

7.4 Luttinger Liquids 215

=¯hv



k,α=±

(αk − k

)



†

kα

−c

†

kα





. (7.89)

Here, the ground state expectation value c

†

kα



of the number opera-

tor is subtracted to prevent a divergence of the ground state energy due to

o ccupation of states with negative energy.

In the following, we assume a system length L and apply periodic boundary

conditions with the consequence of discretizing k in multiples of 2π/L.Number

ﬂuctuations would be described here by

qα



kα



†

k+qα

kα

− δ

q,0

c

†

kα





, (7.90)

which according to their commutation ru le

qα

−q

′

]=αδ

α,α

′

q,q

′

2π

(7.91)

are boson operators. Moreover, we have

qα

]=α¯hv

qα

, (7.92)

indicating that the number ﬂuctuations created by n

qα

are eigenstates of H

with the eigenvalue α¯hv

q. This leads to an alternative formulation of H

π¯hv

⎛

⎝



q=0,α

qα

−qα



0α

⎞

⎠

, (7.93)

now in terms of boson operators. A closer inspection shows that the existence

of these two equivalent formulations of H

7.89)and(7.93) is characteristic

for the 1D case and the linearized dispersion relation.

Electron–electron interaction can take place within each branch of the

spectrum or between the two diﬀerent branches. For small momentum transfer,

correspon d i n g to forward scattering, this can be written in terms of boson

operators as

int



q,α

qα

−qα,

+ n

qα

−q−α

) , (7.94)

where v

quantiﬁes the strength of this scattering. The system Hamiltonian

H = H

+ H

int

is a bilinear expression in the boson operato rs and can be

diagonalized by a Bogoliubov transformation

˜n

qα

= n

qα

cosh(φ(q)) + n

q−α

sinh(φ(q)), (7.95)

with the interaction parameter

2φ(q)

π¯hv

−1/2

=: K(q). (7.96)

216 7 Correlated Electrons

hω

Fig. 7.8. Particle–hole excitations in the 1D fermion system. Note the missing of

lo w-energy excitations for 0 < |k| < 2k

The diagonal form of H reads

H =



q=0

˜n

qα

˜n

−qα



+ v



, (7.97)

where v

=¯hv

/K(q),v

= v

/¯hv

, and v

=¯hv

, while, N = n

0−

and J = n

− n

0−

describe charge and current excitations respectively.

Ultimately, by introducing normalized boson operators

†

2π

L|q|

(θ(q)˜n

+ θ(−q)˜n

q−

) , (7.98)

with the step function θ(q), the Hamiltonian takes the form

H =



q=0

¯hω

†



+ v



(7.99)

with the dispersion ¯hω(q)=v

|q|. It is the model Hamiltonian for interacting

1D electrons named after Tomonaga

[223] and Luttinger

[224].

The remarkable property of this Hamiltonian is that it is expressed in terms

of collective excitations, which are the low-energy excitations of the system

similar to the vibrations of a string. This means that low-energy electron–hole

pair excitations are absent her e , which indicates also the absence of quasi-

particles for interacting electrons in 1D. This ﬁnding can be conﬁrmed by

looking at the excitation spectrum in Fig.

7.8. At low energies, excitations

are possible only between states close to the two Fermi po ints (see Fig.

7.7),

but there are no particle–hole excitation s and, therefore, no quasi-particles for

0 < |k| < 2k

. This nonexistence of quasi-particles can b e made explicit by

Sin-itiro Tomonaga 1906–1979.

Joaquin Luttinger 1923–1997.

7.5 Heavy Fermion Systems 217

Fig. 7.9. Schematic v iew of charge and spin separation in a linear spin chain with

antiferromagnetic order: a charge excitation (hole) is created (upper part) and moves

away by hopping leaving a spin excitation behind (second and third line)

calculating the momentum distribution, which for the Tomonaga–Luttinger

model does not exhibit the quasi-particle discontinuity of the Fermi liquids at

(see Fig.

7.6) but instead an inﬁnite slope.

Including the spin, the model Hamiltonian H would contain additional

terms corresponding to collective spin excitations. As they are additive, the

propagation of collective charge and spin excitations takes place with diﬀerent

velocities. This separation of charge and spin is another characteristic feature

of a Luttinger liquid. It can be visualized for a spin chain with antiferromag-

netic order (Fig.

6.6), as described by a 1D Hubbard model with half-ﬁlling

and negative exchange coupling which is a lattice variant of the Luttinger

liquid. A charge excitation corr esponds to removing an electron with its spin

(or to create a hole). This hole is surrounded by two parallel spins (upper part

of Fig.

7.9). Its motion (to the left, see lower parts of Fig. 7.9)isruledbythe

transfer matrix element and not connected with spin-ﬂips thus, the moving

hole is always surrounded by a pair of up/down spins, while the parallel spins

remain at the site, where the hole was created.

7.5 Heavy Fermion Systems

Metallic systems with the Fermi energy in a range of the spectrum, where

strongly localized states deriving from f electrons hybridize with delocal ized

states with s, p,ord character, show Fermi liquid behavior, however, with a

strong enhancement of the Sommerfeld coeﬃcient γ in the speciﬁc heat and

also of the Pauli spin susceptibility χ

Pauli

(see Sect.

7.3). In an independent

particle description, both quantities are proportional to the density of states

at the Fermi energy, which for isotropic systems is proportional to the eﬀec-

tive mass m

∗

of the quasi-particles. Thus, the observed enhancement can be

understood as a dramatic increase of the eﬀective mass and these materials

have been named, therefore, heavy fermion systems [

110, 122, 225].