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Effective Masses
E Effective Masses A L Wasserman, Oregon State University, Corvallis, OR, USA & 2005, Elsevier Ltd. All Rights Reserved.
the edges of the conduction and valence bands. The formulation is then modified for dense Fermi systems in which particle behavior at the Fermi surface is the crucial determinant of electronic properties, including effective masses.
Introduction Electrons in condensed matter interact strongly with their environment. This environment generally consists of a crystalline lattice, phonons, other electrons and an assortment of localized ‘‘defect’’ potentials, as well as applied external fields. As a result of these interactions, free, single particle energies e0 , where e0 ðkÞ ¼
_2 k2 2m
½1
with k the electron wave vector and m the ‘‘bare’’ electron mass, are usually altered in significant ways, which can include simple energy shifts as well as imparting finite lifetimes to the single particle states. It is often possible and convenient, both for experimentalists and theoreticians, to regard the particles plus interactions as a new fermionic entity, referred to as a ‘‘quasiparticle’’ with energy eðkÞ ¼
_2 k2 2meff
½2
where an effective mass meff can parametrize important features of the interacting environment. There is, in fact, a considerable range of experimental phenomena whose results are usefully parametrized in this surprisingly simple way. The extent to which some (or any) of them relate to firstprinciples microscopic interactions are briefly surveyed here. In an attempt to bring some uniformity to the calculation and interpretation of effective masses, one can present a general approach to the problem by deriving a single formulation for effective mass calculations. This formulation is first adapted to semiconductor and insulator problems where effective mass parameters are attributed, in the first order, to interactions with the periodic lattice and, in particular, to electrons whose occupation is confined to
Consequences of Interactions A general expression which describes the effects of interactions is eðkÞ ¼ e0 ðkÞ þ G½k; eðkÞ
½3
where eðkÞ is the fully interaction-modified ‘‘quasiparticle’’ energy, e0 ðkÞ is the noninteracting energy, as given in eqn [1], and G½k; eðkÞ is the quantitative consequence of all the environmental perturbations to the free electron system. It should be noted that G depends on the wave vector k, but may also depend, self-consistently, on eðkÞ, the quasiparticle energy of the state itself. Moreover, G½k; eðkÞ is, in general, a complex quantity: G½k; eðkÞ ¼ D½k; eðkÞ þ ig½k; eðkÞ
½4
the imaginary part usually being responsible for quasiparticle lifetime effects. Although ga0 is often the case, this possibility does not explicitly contribute to the following discussion. The apparent abstractness of eqns [3] and [4] notwithstanding, one finds that a quasiparticle effective mass meff is a useful and intuitive concept.
An Effective Mass Formulation In formulating the general case for an effective mass theory which can embrace simple interactions, such as band structure in semiconductors, as well as phonons and the more subtle electron–electron interactions, one begins with eqns [3] and [4] and assumes that the imaginary part g½k; eðkÞ has a negligible effect in determining meff . The initial step in obtaining an explicit form for the ‘‘quasiparticle’’ energy eðkÞ is most simply done
2 Effective Masses
by applying Newton’s iterative method to eqn [3], which is first rewritten in the form eðkÞ e0 D½k; eðkÞ ¼ 0
½5
Newton’s method consists of first expanding eqn [5] to linear order about some initial value eð0Þ (which will be chosen below) to give eð0Þ e0 ðkÞ Dðk; eð0Þ Þ @Dðk; eÞ þ 1 ½eðkÞ eð0Þ ¼ 0 @e e¼eð0Þ
Band-Structure Effective Mass ½6
Now solving for e, one finds eðkÞ ¼ eð0Þ
½eð0Þ e0 ðkÞ Dðk; eð0Þ Þ ½1 ð@Dðk; eÞ=@eÞe¼eð0Þ
½7
Temporarily assuming, for simplicity, that k is a scalar and expanding to the first order in k2 about k ¼ 0, one has eðkÞ ¼ eð0Þ
½eð0Þ e0 ðkÞ Dð0; eð0Þ Þ ð@Dðk; eð0Þ Þ=@ðk2 ÞÞjk¼0 k2 ½1 ð@Dð0; eÞ=@eÞe¼eð0Þ
½8 Finally, starting the iterations from eð0Þ ¼ 0, after a single iteration one obtains eðkÞ ¼
½e0 ðkÞ þ Dð0; 0Þ þ ð@Dðk; 0Þ=@ðk2 ÞÞjk¼0 k2 ½1 @Dð0; eÞ=@ee¼0
½9
Assuming the constant energy shift Dð0; 0Þ ¼ 0 and from the above equation e0 ðkÞ ¼ _2 k2 =2m; one can obtain ( ) _2 1 þ ð2m=_2 Þ½@Dðk; 0Þ=@ðk2 Þk¼0 2 eðkÞ ¼ k 2m ð1 ð@Dð0; eÞ=@eÞÞe¼0
½10
Therefore, an effective mass can be defined by ( ) m 1 þ ð2m=_2 Þ½@Dðk; 0Þ=@ðk2 Þk¼0 ¼ ½11 meff ð1 ð@Dð0; eÞ=@eÞÞe¼0 In the case that k is not a scalar, one should expand instead to the second order and similarly find an effective mass tensor
m meff
(
¼ mn
dmn þ ðm=_2 Þ½@ 2 Dðk; 0Þ=@km @kn k¼0 ð1 ð@Dð0; eÞ=@eÞÞe¼0
where E(k) is the band dispersion relation and the k-expansion of eqn [8] is taken about any band extremal point km (the Fermi gas case is discussed below). Once the relevant perturbing terms are known, eqn [11] or [12] may be straightforwardly applied.
)
½12
The most common application of effective mass theories is in parametrizing the features of a band structure. This is especially practical in semiconductors and insulators where most experimental features arise from electron occupation near the extremal points of band maxima (valence bands) and band minima (conduction bands). In the energy band approximation, it is first noted that @Dð0; eÞ=@e ¼ 0. Recasting eqn [12] to reflect the energy band approximation (see eqn [13]), one obtains
1 meff
1 @ 2 Eðk; 0Þ ¼ 2 _ @km @kn k¼km mn
where E(k,0) is the electron energy band dispersion relation and km are wave vectors of any conductionband minima or valence-band maxima. Since the derivative in eqn [14] is related to the ‘‘curvature’’ of the dispersion relation, bands of large curvature correspond to small effective masses (light ‘‘quasiparticles’’) while flatter bands, that is, bands with small curvature, are ‘‘heavier’’ quasiparticles. Moreover, since valence bands have maxima at km, their curvature will have negative values and, therefore, be quasiparticles with negative effective masses. This inadmissible condition is resolved by assigning to these quasiparticles a positive charge, referring to them as holes and thereby restoring proper signs to transport properties.
Effective Mass of a Degenerate Fermi System In the case of a degenerate Fermi system, Fermi– Dirac statistics usually restrict dominant behavior to those electrons at the Fermi surface where kF , the Fermi wave vector, is neither a band maximum nor a minimum. Therefore, the noninteracting energy band relation e0 ðkÞ is expressed as an expansion about k ¼ kF : Since kF is not an extremal point, one can obtain an expression which has a term linear in (k kF):
In the general semiconductor band case, e0 ðkÞ þ Dðk; oÞ-Eðk; 0Þ
½13
½14
e0 ðkÞ ¼ e0 ðkF Þ þ
_2 kF ðk kF Þ þ ? m
½15
Effective Masses 3
Similarly, eqn [7] is expanded about k ¼ kF instead of k ¼ 0 giving, to linear order, eðkÞ ¼ eð0Þ f½1 ð@DðkF ; eÞ=@eÞe¼eð0Þ g1 ½eð0Þ e0 ðkF Þ DðkF ; eð0Þ Þ ð_2 kF =mÞð½1 þ ðm=_2 kF Þ ð@Dðk; eð0Þ Þ=@kÞjk¼kF ðk kF Þ þ ?
½16
Iterating once again, starting with eð0Þ ¼ 0, one obtains eðkÞ ¼ f½1 ð@DðkF ; eÞ=@eÞe¼0 g1 ½e0 ðkF Þ þ DðkF ; eð0Þ Þ þ ð_2 kF =mÞ½1 þ ðm=_2 kF Þ ð@Dðk; eð0Þ Þ=@kÞjk¼kF ðk kF Þ þ ?
½17
Comparing this result with eqn [15], an expression for an effective mass is given by " # 1 1 1 @Dðk; eð0Þ Þ þ ¼ meff m _2 kF @k k¼kF 1 @DðkF ; eÞ ½18 1 @e e¼0 In the case where the band structure is of dominant interest in interpreting an effective mass, one can write (see eqn [14]) " # 1 1 1 @Eðk; 0Þ @DðkF ; eÞ 1 ¼ 2 meff @e _ kF @k k¼kF e¼0
eqn [20] as required gives the nonsensical result: meff ¼ 0
½21
which can be interpreted as an indication of just how flawed the Hartree–Fock approximation is for the degenerate Fermi gas. In the case of exotic heavy fermion alloys, for example, CeB6 , CeCu2 , which have effective masses 10–100 times those of bare electron masses, the mixture of strong on-site f-state correlations and hybridization between s-states and f-states requires a description in terms of strongly hybridized bands. These hybridized bands have extremely large ‘‘slopes’’ at the Fermi surface due to the large f-state interactions having forced open a small energy gap just above the Fermi energy. This results in flattening the occupied conduction band to an almost horizontal conformation at kF , so that according to eqn [19], an unusually large effective mass is to be expected.
Density-of-States Effective Mass Although optical experiments are capable of probing a band structure with considerable detail and can even determine effective mass tensor components, thermodynamic experiments (e.g., specific heat) do not have this capability. For example, the low-temperature specific heat in Fermi systems is linear in temperature (its signature):
½19 which is inversely proportional to the ‘‘slope’’ of E(k,0), the electron dispersion at the Fermi surface. Here, DðkF ; eÞ represents all perturbing terms beyond the band approximation (phonons, etc.). This should be compared to the semiconductor case in which proportionality to the inverse ‘‘curvature’’ at the band extrema determines the effective mass.
Electron Correlations: Hartree–Fock The simplest case for calculating an effective mass arising from electron–electron interactions is to consider the contribution to G½k; eðkÞ in the Hartree– Fock approximation. The widely known Hartree– Fock result has only a real part _2 e2 kF ðk2F k2 Þ k þ kF Dðk; 0Þ ¼ 2þ ln ½20 kkF 2p k kF
CV ¼
p2 2 k DðEF ÞT 3 B
½22
and proportional to DðEF Þ, the ‘‘density of electronic states’’ at the Fermi energy EF, where T is temperature in K and kB is the Boltzmann constant. The density of states is able to make no distinctions about the detailed geometry of the electron energy bands and thereby presents an effective mass interpretation which, at best, averages the mass tensor components in a specific way. The density of electronic states is determined by ‘‘sifting’’ through all the electrons in a band and counting those that have a particular energy. The sifting and counting process is repeated for all bands. This process is succinctly described by the expression X DðeÞ ¼ 2 d½e EðkÞ ½23 k
This approximation, although far from an adequate picture of electrons in metals, provides an example to which one can apply eqn [18]. Differentiating
Note how the Dirac d-function combs through the band electrons and registers a count when it encounters one with energy e. It then continues to
4 Effective Masses
comb and count. The factor of 2 accounts for spin degeneracy. This is the simple meaning of the density of states. Now, to see how the effective mass enters into the density of states, evaluate eqn [23] for a single, simple isotropic band such as represented by eqn [2]. Using the fact that in three dimensions X k
-
Z
V ð2pÞ3
2p
df
Z
0
p
dy sin y
0
Z
N
dk k2
½24
0
eqn [23] becomes Dc ðeÞ ¼
Z
2V ð2pÞ
3
2p
df
0
Z
p
dy sin y
0
"
k2 d e
Z
which defines the averaging of effective mass tensor components imposed by the density-of-states effective mass. For an ellipsoidal valence band, one obtains Dv ðeÞ pffiffiffiðm> m> mN Þ1=3 pffiffiffiffiffiffi ¼ 2 e; V p 2 _3
It is now interesting to note that the thermodynamic implication of the Hartree–Fock result of eqn [23] is that CV (Hartree–Fock) ¼ 0, which continues to point to the inadequacy of that approximation.
N
dk
Electron–Phonon Interactions
_2 k2 2meff
!#
½25
After explicitly evaluating the density of states for an isotropic band structure, the constant volume specific heat for a Fermi gas, eqn [22], becomes
for the choice of an isotropic conduction band in an effective mass approximation, _2 k2 2meff
½26
where Egc is a bandgap energy. Using the Dirac identity d½f ðkÞ ¼
½31
0
Egc þ
Ec ðk; 0Þ ¼ Egc þ
eo0
X dðk kR Þ @f ðkÞ kR @k
½27
k¼kR
where the values of kR are determined by the solutions of f ðkR Þ ¼ 0, one gets the density of states per unit volume: 3=2 pffiffiffi Dc ðeÞ meff 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ½28 e Egc ; e4Egc V p 2 _3 This isotropic case does not, of course, expose any of the averaging that will take place in the anisotropic case. To understand better what kind of averaging this is, the ellipsoidal band case is taken as ! 2 2 k2z _2 kx þ ky EðkÞ ¼ Eg þ þ ½29 2 m> mN Following eqn [23] but this time integrating in cylindrical coordinates, one gets, for a conduction band, pffiffiffi Dc ðeÞ ðm> m> mN Þ1=3 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e Egc ; e4Egc ½30 V p2 _3
CV ¼
k2B meff kF T _2
½32
As is well known, measured specific heats generally exceed the free electron values even for simple metals in the alkali series, which have nearly free electron bands. By determining an effective mass which arises from electron–phonon interactions, it can be seen that a fraction of this excess specific heat can be attributed to electron–phonon interactions. One can use eqns [11] and [12] to explore this case by referring to the well-known result that for electron–phonon interactions in a Fermi gas, De2p ðkF ; eÞ ¼ le
½33
where e is the self-consistent ‘‘quasiparticle’’ energy that is generally characteristic of Dðk; eÞ and l¼2
Z
N
do 0
a2 ðoÞFðoÞ o
½34
is a measure of the electron–phonon coupling strength, with a2 ðoÞ the electron–phonon coupling function and FðoÞ the phonon density of states. Returning to eqn [18] (ignoring band structure effects), eðkÞ ¼
e0 ðkF Þ _2 k2F ¼ ½1 ð@Dep ðkF ; eÞ=@eÞe¼0 2mð1 þ lÞ
½35
so that an electron–phonon enhanced effective mass may be identified as meff ¼ mð1 þ lÞ
½36
Effective Masses 5
Other Effective Mass Parametrizations An effective mass is used to parametrize interacting electron behavior in other measurements. For example,
which there is no rigorously applicable or meaningful theory. For example, the plasma frequency o2p ¼
1. Electrical conductivity s¼
ne2 t meff
½37
where t is a transport relaxation time. 2. The theory of excitons and interband optical absorption, in which a reduced effective mass m is defined: 1 1 1 ¼ þ m meff;c meff;h
½38
in order to parametrize the excitonic Rydberg states. Here, meff;c is a conduction band effective mass and meff;h is a valence band effective mass. 3. The de Haas–van Alphen magneto-oscillations, where the temperature-dependent amplitude of the rth harmonic r ¼ 1; 2; y; is Ar ¼
2 1 2p rkB meff T sinh e_B
½39
The de Haas–van Alphen effective mass is almost identical to the density-of-states effective mass which appears in the specific heat: meff ðCV Þ ¼ mð1 þ lep þ ?Þ
½40
except that the de Haas effective mass refers to only one extremal areal slice of the Fermi surface perpendicular to the magnetic field direction, whereas the specific heat effective mass is averaged over the entire Fermi surface. There are experiments for which some kind of effective mass description seems appropriate, but for
Elastic Behavior
ne2 meff e0
which can be determined from optical or electron stopping power experiments (and a Hall measurement to determine the carrier concentration n) is parametrized by an effective mass. But the plasma frequency is a collective excitation of all available mobile electrons so that effective mass calculations of the type described above have little relevance. In those cases, phenomenology rules. Comparing effective masses from different types of experiments may have to be done with great care and with the knowledge of the theoretical origins of the parameter in each specific case. See also: Electron–Phonon Interactions and the Response of Polarons.
PACS: 71.18. þ y
Further Reading Abrikosov AG, Gor’kov LP, and Dzyaloshinskii IY (1965) Quantum Field Theoretical Methods in Statistical Physics. Oxford: Pergamon. Ibach H and Luth H (1990) Solid-State Physics. Berlin: Springer. Kittel C (1987) Quantum Theory of Solids. New York: Wiley. Kittel C (1996) Introduction to Solid State Physics, 7th edn. New York: Wiley. Knox RS (1963) Theory of Excitons. New York: Academic Press. Marder MP (2000) Condensed Matter Physics. New York: Wiley. Shoenberg D (1984) Magnetic Oscillations in Metals. Cambridge: Cambridge University Press. Wasserman A and Springford M (1996) Advances in Physics 45: 471–503. Ziman J (1964) Principles of the Theory of Solids. Cambridge: Cambridge University Press.
See Mechanical Properties: Elastic Behavior.
Elasticity in Lattices
½41
See Lattice Dynamics: Structural Instability and Soft Modes.
the edges of the conduction and valence bands. The formulation is then modified for dense Fermi systems in which particle behavior at the Fermi surface is the crucial determinant of electronic properties, including effective masses.
Introduction Electrons in condensed matter interact strongly with their environment. This environment generally consists of a crystalline lattice, phonons, other electrons and an assortment of localized ‘‘defect’’ potentials, as well as applied external fields. As a result of these interactions, free, single particle energies e0 , where e0 ðkÞ ¼
_2 k2 2m
½1
with k the electron wave vector and m the ‘‘bare’’ electron mass, are usually altered in significant ways, which can include simple energy shifts as well as imparting finite lifetimes to the single particle states. It is often possible and convenient, both for experimentalists and theoreticians, to regard the particles plus interactions as a new fermionic entity, referred to as a ‘‘quasiparticle’’ with energy eðkÞ ¼
_2 k2 2meff
½2
where an effective mass meff can parametrize important features of the interacting environment. There is, in fact, a considerable range of experimental phenomena whose results are usefully parametrized in this surprisingly simple way. The extent to which some (or any) of them relate to firstprinciples microscopic interactions are briefly surveyed here. In an attempt to bring some uniformity to the calculation and interpretation of effective masses, one can present a general approach to the problem by deriving a single formulation for effective mass calculations. This formulation is first adapted to semiconductor and insulator problems where effective mass parameters are attributed, in the first order, to interactions with the periodic lattice and, in particular, to electrons whose occupation is confined to
Consequences of Interactions A general expression which describes the effects of interactions is eðkÞ ¼ e0 ðkÞ þ G½k; eðkÞ
½3
where eðkÞ is the fully interaction-modified ‘‘quasiparticle’’ energy, e0 ðkÞ is the noninteracting energy, as given in eqn [1], and G½k; eðkÞ is the quantitative consequence of all the environmental perturbations to the free electron system. It should be noted that G depends on the wave vector k, but may also depend, self-consistently, on eðkÞ, the quasiparticle energy of the state itself. Moreover, G½k; eðkÞ is, in general, a complex quantity: G½k; eðkÞ ¼ D½k; eðkÞ þ ig½k; eðkÞ
½4
the imaginary part usually being responsible for quasiparticle lifetime effects. Although ga0 is often the case, this possibility does not explicitly contribute to the following discussion. The apparent abstractness of eqns [3] and [4] notwithstanding, one finds that a quasiparticle effective mass meff is a useful and intuitive concept.
An Effective Mass Formulation In formulating the general case for an effective mass theory which can embrace simple interactions, such as band structure in semiconductors, as well as phonons and the more subtle electron–electron interactions, one begins with eqns [3] and [4] and assumes that the imaginary part g½k; eðkÞ has a negligible effect in determining meff . The initial step in obtaining an explicit form for the ‘‘quasiparticle’’ energy eðkÞ is most simply done
2 Effective Masses
by applying Newton’s iterative method to eqn [3], which is first rewritten in the form eðkÞ e0 D½k; eðkÞ ¼ 0
½5
Newton’s method consists of first expanding eqn [5] to linear order about some initial value eð0Þ (which will be chosen below) to give eð0Þ e0 ðkÞ Dðk; eð0Þ Þ @Dðk; eÞ þ 1 ½eðkÞ eð0Þ ¼ 0 @e e¼eð0Þ
Band-Structure Effective Mass ½6
Now solving for e, one finds eðkÞ ¼ eð0Þ
½eð0Þ e0 ðkÞ Dðk; eð0Þ Þ ½1 ð@Dðk; eÞ=@eÞe¼eð0Þ
½7
Temporarily assuming, for simplicity, that k is a scalar and expanding to the first order in k2 about k ¼ 0, one has eðkÞ ¼ eð0Þ
½eð0Þ e0 ðkÞ Dð0; eð0Þ Þ ð@Dðk; eð0Þ Þ=@ðk2 ÞÞjk¼0 k2 ½1 ð@Dð0; eÞ=@eÞe¼eð0Þ
½8 Finally, starting the iterations from eð0Þ ¼ 0, after a single iteration one obtains eðkÞ ¼
½e0 ðkÞ þ Dð0; 0Þ þ ð@Dðk; 0Þ=@ðk2 ÞÞjk¼0 k2 ½1 @Dð0; eÞ=@ee¼0
½9
Assuming the constant energy shift Dð0; 0Þ ¼ 0 and from the above equation e0 ðkÞ ¼ _2 k2 =2m; one can obtain ( ) _2 1 þ ð2m=_2 Þ½@Dðk; 0Þ=@ðk2 Þk¼0 2 eðkÞ ¼ k 2m ð1 ð@Dð0; eÞ=@eÞÞe¼0
½10
Therefore, an effective mass can be defined by ( ) m 1 þ ð2m=_2 Þ½@Dðk; 0Þ=@ðk2 Þk¼0 ¼ ½11 meff ð1 ð@Dð0; eÞ=@eÞÞe¼0 In the case that k is not a scalar, one should expand instead to the second order and similarly find an effective mass tensor
m meff
(
¼ mn
dmn þ ðm=_2 Þ½@ 2 Dðk; 0Þ=@km @kn k¼0 ð1 ð@Dð0; eÞ=@eÞÞe¼0
where E(k) is the band dispersion relation and the k-expansion of eqn [8] is taken about any band extremal point km (the Fermi gas case is discussed below). Once the relevant perturbing terms are known, eqn [11] or [12] may be straightforwardly applied.
)
½12
The most common application of effective mass theories is in parametrizing the features of a band structure. This is especially practical in semiconductors and insulators where most experimental features arise from electron occupation near the extremal points of band maxima (valence bands) and band minima (conduction bands). In the energy band approximation, it is first noted that @Dð0; eÞ=@e ¼ 0. Recasting eqn [12] to reflect the energy band approximation (see eqn [13]), one obtains
1 meff
1 @ 2 Eðk; 0Þ ¼ 2 _ @km @kn k¼km mn
where E(k,0) is the electron energy band dispersion relation and km are wave vectors of any conductionband minima or valence-band maxima. Since the derivative in eqn [14] is related to the ‘‘curvature’’ of the dispersion relation, bands of large curvature correspond to small effective masses (light ‘‘quasiparticles’’) while flatter bands, that is, bands with small curvature, are ‘‘heavier’’ quasiparticles. Moreover, since valence bands have maxima at km, their curvature will have negative values and, therefore, be quasiparticles with negative effective masses. This inadmissible condition is resolved by assigning to these quasiparticles a positive charge, referring to them as holes and thereby restoring proper signs to transport properties.
Effective Mass of a Degenerate Fermi System In the case of a degenerate Fermi system, Fermi– Dirac statistics usually restrict dominant behavior to those electrons at the Fermi surface where kF , the Fermi wave vector, is neither a band maximum nor a minimum. Therefore, the noninteracting energy band relation e0 ðkÞ is expressed as an expansion about k ¼ kF : Since kF is not an extremal point, one can obtain an expression which has a term linear in (k kF):
In the general semiconductor band case, e0 ðkÞ þ Dðk; oÞ-Eðk; 0Þ
½13
½14
e0 ðkÞ ¼ e0 ðkF Þ þ
_2 kF ðk kF Þ þ ? m
½15
Effective Masses 3
Similarly, eqn [7] is expanded about k ¼ kF instead of k ¼ 0 giving, to linear order, eðkÞ ¼ eð0Þ f½1 ð@DðkF ; eÞ=@eÞe¼eð0Þ g1 ½eð0Þ e0 ðkF Þ DðkF ; eð0Þ Þ ð_2 kF =mÞð½1 þ ðm=_2 kF Þ ð@Dðk; eð0Þ Þ=@kÞjk¼kF ðk kF Þ þ ?
½16
Iterating once again, starting with eð0Þ ¼ 0, one obtains eðkÞ ¼ f½1 ð@DðkF ; eÞ=@eÞe¼0 g1 ½e0 ðkF Þ þ DðkF ; eð0Þ Þ þ ð_2 kF =mÞ½1 þ ðm=_2 kF Þ ð@Dðk; eð0Þ Þ=@kÞjk¼kF ðk kF Þ þ ?
½17
Comparing this result with eqn [15], an expression for an effective mass is given by " # 1 1 1 @Dðk; eð0Þ Þ þ ¼ meff m _2 kF @k k¼kF 1 @DðkF ; eÞ ½18 1 @e e¼0 In the case where the band structure is of dominant interest in interpreting an effective mass, one can write (see eqn [14]) " # 1 1 1 @Eðk; 0Þ @DðkF ; eÞ 1 ¼ 2 meff @e _ kF @k k¼kF e¼0
eqn [20] as required gives the nonsensical result: meff ¼ 0
½21
which can be interpreted as an indication of just how flawed the Hartree–Fock approximation is for the degenerate Fermi gas. In the case of exotic heavy fermion alloys, for example, CeB6 , CeCu2 , which have effective masses 10–100 times those of bare electron masses, the mixture of strong on-site f-state correlations and hybridization between s-states and f-states requires a description in terms of strongly hybridized bands. These hybridized bands have extremely large ‘‘slopes’’ at the Fermi surface due to the large f-state interactions having forced open a small energy gap just above the Fermi energy. This results in flattening the occupied conduction band to an almost horizontal conformation at kF , so that according to eqn [19], an unusually large effective mass is to be expected.
Density-of-States Effective Mass Although optical experiments are capable of probing a band structure with considerable detail and can even determine effective mass tensor components, thermodynamic experiments (e.g., specific heat) do not have this capability. For example, the low-temperature specific heat in Fermi systems is linear in temperature (its signature):
½19 which is inversely proportional to the ‘‘slope’’ of E(k,0), the electron dispersion at the Fermi surface. Here, DðkF ; eÞ represents all perturbing terms beyond the band approximation (phonons, etc.). This should be compared to the semiconductor case in which proportionality to the inverse ‘‘curvature’’ at the band extrema determines the effective mass.
Electron Correlations: Hartree–Fock The simplest case for calculating an effective mass arising from electron–electron interactions is to consider the contribution to G½k; eðkÞ in the Hartree– Fock approximation. The widely known Hartree– Fock result has only a real part _2 e2 kF ðk2F k2 Þ k þ kF Dðk; 0Þ ¼ 2þ ln ½20 kkF 2p k kF
CV ¼
p2 2 k DðEF ÞT 3 B
½22
and proportional to DðEF Þ, the ‘‘density of electronic states’’ at the Fermi energy EF, where T is temperature in K and kB is the Boltzmann constant. The density of states is able to make no distinctions about the detailed geometry of the electron energy bands and thereby presents an effective mass interpretation which, at best, averages the mass tensor components in a specific way. The density of electronic states is determined by ‘‘sifting’’ through all the electrons in a band and counting those that have a particular energy. The sifting and counting process is repeated for all bands. This process is succinctly described by the expression X DðeÞ ¼ 2 d½e EðkÞ ½23 k
This approximation, although far from an adequate picture of electrons in metals, provides an example to which one can apply eqn [18]. Differentiating
Note how the Dirac d-function combs through the band electrons and registers a count when it encounters one with energy e. It then continues to
4 Effective Masses
comb and count. The factor of 2 accounts for spin degeneracy. This is the simple meaning of the density of states. Now, to see how the effective mass enters into the density of states, evaluate eqn [23] for a single, simple isotropic band such as represented by eqn [2]. Using the fact that in three dimensions X k
-
Z
V ð2pÞ3
2p
df
Z
0
p
dy sin y
0
Z
N
dk k2
½24
0
eqn [23] becomes Dc ðeÞ ¼
Z
2V ð2pÞ
3
2p
df
0
Z
p
dy sin y
0
"
k2 d e
Z
which defines the averaging of effective mass tensor components imposed by the density-of-states effective mass. For an ellipsoidal valence band, one obtains Dv ðeÞ pffiffiffiðm> m> mN Þ1=3 pffiffiffiffiffiffi ¼ 2 e; V p 2 _3
It is now interesting to note that the thermodynamic implication of the Hartree–Fock result of eqn [23] is that CV (Hartree–Fock) ¼ 0, which continues to point to the inadequacy of that approximation.
N
dk
Electron–Phonon Interactions
_2 k2 2meff
!#
½25
After explicitly evaluating the density of states for an isotropic band structure, the constant volume specific heat for a Fermi gas, eqn [22], becomes
for the choice of an isotropic conduction band in an effective mass approximation, _2 k2 2meff
½26
where Egc is a bandgap energy. Using the Dirac identity d½f ðkÞ ¼
½31
0
Egc þ
Ec ðk; 0Þ ¼ Egc þ
eo0
X dðk kR Þ @f ðkÞ kR @k
½27
k¼kR
where the values of kR are determined by the solutions of f ðkR Þ ¼ 0, one gets the density of states per unit volume: 3=2 pffiffiffi Dc ðeÞ meff 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ½28 e Egc ; e4Egc V p 2 _3 This isotropic case does not, of course, expose any of the averaging that will take place in the anisotropic case. To understand better what kind of averaging this is, the ellipsoidal band case is taken as ! 2 2 k2z _2 kx þ ky EðkÞ ¼ Eg þ þ ½29 2 m> mN Following eqn [23] but this time integrating in cylindrical coordinates, one gets, for a conduction band, pffiffiffi Dc ðeÞ ðm> m> mN Þ1=3 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e Egc ; e4Egc ½30 V p2 _3
CV ¼
k2B meff kF T _2
½32
As is well known, measured specific heats generally exceed the free electron values even for simple metals in the alkali series, which have nearly free electron bands. By determining an effective mass which arises from electron–phonon interactions, it can be seen that a fraction of this excess specific heat can be attributed to electron–phonon interactions. One can use eqns [11] and [12] to explore this case by referring to the well-known result that for electron–phonon interactions in a Fermi gas, De2p ðkF ; eÞ ¼ le
½33
where e is the self-consistent ‘‘quasiparticle’’ energy that is generally characteristic of Dðk; eÞ and l¼2
Z
N
do 0
a2 ðoÞFðoÞ o
½34
is a measure of the electron–phonon coupling strength, with a2 ðoÞ the electron–phonon coupling function and FðoÞ the phonon density of states. Returning to eqn [18] (ignoring band structure effects), eðkÞ ¼
e0 ðkF Þ _2 k2F ¼ ½1 ð@Dep ðkF ; eÞ=@eÞe¼0 2mð1 þ lÞ
½35
so that an electron–phonon enhanced effective mass may be identified as meff ¼ mð1 þ lÞ
½36
Effective Masses 5
Other Effective Mass Parametrizations An effective mass is used to parametrize interacting electron behavior in other measurements. For example,
which there is no rigorously applicable or meaningful theory. For example, the plasma frequency o2p ¼
1. Electrical conductivity s¼
ne2 t meff
½37
where t is a transport relaxation time. 2. The theory of excitons and interband optical absorption, in which a reduced effective mass m is defined: 1 1 1 ¼ þ m meff;c meff;h
½38
in order to parametrize the excitonic Rydberg states. Here, meff;c is a conduction band effective mass and meff;h is a valence band effective mass. 3. The de Haas–van Alphen magneto-oscillations, where the temperature-dependent amplitude of the rth harmonic r ¼ 1; 2; y; is Ar ¼
2 1 2p rkB meff T sinh e_B
½39
The de Haas–van Alphen effective mass is almost identical to the density-of-states effective mass which appears in the specific heat: meff ðCV Þ ¼ mð1 þ lep þ ?Þ
½40
except that the de Haas effective mass refers to only one extremal areal slice of the Fermi surface perpendicular to the magnetic field direction, whereas the specific heat effective mass is averaged over the entire Fermi surface. There are experiments for which some kind of effective mass description seems appropriate, but for
Elastic Behavior
ne2 meff e0
which can be determined from optical or electron stopping power experiments (and a Hall measurement to determine the carrier concentration n) is parametrized by an effective mass. But the plasma frequency is a collective excitation of all available mobile electrons so that effective mass calculations of the type described above have little relevance. In those cases, phenomenology rules. Comparing effective masses from different types of experiments may have to be done with great care and with the knowledge of the theoretical origins of the parameter in each specific case. See also: Electron–Phonon Interactions and the Response of Polarons.
PACS: 71.18. þ y
Further Reading Abrikosov AG, Gor’kov LP, and Dzyaloshinskii IY (1965) Quantum Field Theoretical Methods in Statistical Physics. Oxford: Pergamon. Ibach H and Luth H (1990) Solid-State Physics. Berlin: Springer. Kittel C (1987) Quantum Theory of Solids. New York: Wiley. Kittel C (1996) Introduction to Solid State Physics, 7th edn. New York: Wiley. Knox RS (1963) Theory of Excitons. New York: Academic Press. Marder MP (2000) Condensed Matter Physics. New York: Wiley. Shoenberg D (1984) Magnetic Oscillations in Metals. Cambridge: Cambridge University Press. Wasserman A and Springford M (1996) Advances in Physics 45: 471–503. Ziman J (1964) Principles of the Theory of Solids. Cambridge: Cambridge University Press.
See Mechanical Properties: Elastic Behavior.
Elasticity in Lattices
½41
See Lattice Dynamics: Structural Instability and Soft Modes.