Intraband memory function and memory-function conductivity formula in doped graphene
The generalized self-consistent field method is used to describe intraband relaxation processes in a general multiband electronic system with presumably weak residual electron-electron interactions. The resulting memory-function conductivity formula is shown to have the same structure as the result of a more accurate approach based on the quantum kinetic equation. The results are applied to heavily doped and lightly doped graphene. It is shown that the scattering of conduction electron by phonons leads to the redistribution of the intraband conductivity spectral weight over a wide frequency range, however, in a way consistent with the partial transverse conductivity sum rule. The present form of the intraband memory function is found to describe correctly the scattering by quantum fluctuations of the lattice, at variance with the semiclassical Boltzmann transport equations, where this scattering channel is absent. This is shown to be of fundamental importance in quantitative understanding of the reflectivity data measured in lightly doped graphene as well as in different low-dimensional strongly correlated electronic systems, such as the cuprate superconductors.
pacs:78.67.Wj, 72.80.Vp, 72.10.Di
In condensed matter physics, important information can be obtained about interactions in the electronic subsystem by analyzing relaxation processes associated with the scattering of conduction electrons by static disorder, by phonons, and by other electrons. One of the central questions regarding the relaxation processes is to explain temperature and retardation effects in simple physical terms by using simple enough self-consistent kinetic equations. The memory function is the common name for the - and -dependent relaxation function in such self-consistent kinetic equations Kubo95 (); Forster75 (); Gotze72 (); Allen71 (); Kupcic14 (); Kupcic15 (). The relaxation rate is its imaginary part at zero frequency. The memory function is usually introduced to describe intraband relaxation processes in the dynamical conductivity tensor, in the Raman response functions, as well as in different transport coefficients. It is well known that even in weakly interacting systems the explanation of experimental observations requires a unified diagrammatic representation for the so-called self-energy contributions to the response function in question and the related vertex corrections Mahan90 (); Ziman88 (); Kupcic15 (). Moreover, it is easily seen that more complicated electronic system is longer is the list of requirements that the response functions and the relaxation functions in question must satisfy. The causality principle, the law of conservation of energy, and the charge continuity equation are all of fundamental importance in understanding the relaxation processes. Consequently, they play an important role in analyzing measured transport coefficients and measured reflectivity and Raman scattering spectra by means of such self-consistent kinetic equations.
The stosszahl ansatz in Boltzmann transport equations represents the simplest way to explain qualitatively the temperature dependence of the intraband relaxation rate Kubo95 (); Pines89 (); Ziman72 (); Abrikosov88 (). The part of the relaxation rate associated with the scattering by phonons is proportional to the Bose-Einstain distribution function and the term associated with corresponding quantum fluctuations of the lattice is missing. As a result, the Boltzmann transport equations have serious deficiencies in describing the retardation effects, in particular those associated with the scattering by optical phonons and by other high-energy boson modes.
The generalized Drude formula is the primary tool for investigating retardation effects. It is usually assumed to be a model independent method of analyzing measured reflectivity and Raman scattering spectra in terms of the -dependent memory function Uchida91 (); Degiorgi91 (); Schwartz98 (); Mirzaei13 (); Opel00 (); Basov11 (). However, in most cases of general interest the extraction of from experimental data depends on details in the boson mediated electron-electron interactions, on general properties of the crystal potential, as well as on the very nature of the local field effects. Consequently, such an analysis is usually incomplete and often inadequate. Therefore, to study the interband conductivity, or the excitations across the charge-density-wave (CDW), spin-density-wave (SDW) or superconducting Bardeen-Cooper-Schrieffer (BCS) gap or pseudogap, we need general enough self-consistent kinetic equations and much more sophisticated procedures for solving these equations than that usually used to derive the generalized Drude formula.
Lightly doped graphene is an important weakly interacting two-band system in which the threshold energy for interband electron-hole excitations is of the order of optical phonon energies, the optical phonon energies are quite large, and the intraband and interband contributions to the dynamical conductivity tensor are expected to be decoupled from each other Li08 (); Castro09 (). The structure of the dynamical conductivity is similar to that of typical CDW/SDW pseudogaped systems, and, consequently, lightly doped graphene is a convenient model system for reexamining different open questions regarding electrodynamics of conduction electrons in such multiband electronic systems.
In this paper, we use the generalized self-consistent field method [usually called the generalized random-phase approximation (RPA)] to derive the memory-function conductivity formula for the intraband conductivity and to determine the structure of the intraband memory function in heavily doped and lightly doped graphene. The results are compared to both the results of the common variational method for the dc conductivity Ziman72 () and to the results of a more accurate approach based on the quantum kinetic equation Kupcic13 (); Kupcic15 (). It is shown that the scattering by phonons leads to the redistribution of the conductivity spectral weight over a wide energy range in a way consistent with the partial transverse conductivity sum rule. The intraband memory function has the same structure as that obtained by means of the quantum kinetic equation.
The paper is organized as follows. In Sec. II, we briefly describe all elements in the total Hamiltonian for conduction electrons in a general multiband case. To make the reading of the paper easier, we give in Secs. III and IV an overview of both the macroscopic identity relations among the exact elements of the real-time RPA irreducible response tensor and the microscopic version of the same identity relations. The partial effective mass theorem and the related transverse conductivity sum rule are shown to play an essential role in determining the proper structure of the memory-function conductivity formula. This transverse conductivity sum rule can also be useful in reexamining gauge invariance of the conductivity formula obtained by means of the common current-current approach Ando02 (); Peres08 (); Carbotte10 () or by different charge-charge approaches Hwang07 (); Despoja13 (). In Secs. IV and V, we discuss general properties of the generalized self-consistent RPA equations and the quantum transport equations. These two equations are used in Sec. VI to derive the intraband memory-function conductivity formula and the leading contributions to the intraband memory function. The relation between the memory-function conductivity formula and the generalized Drude conductivity formula is briefly discussed in Sec. VII. In Sec. VIII, the numerical results for the real and imaginary parts of the intraband memory function are presented for heavily doped graphene for typical values of the model parameters. In Sec. IX, we consider the two-band conductivity in lightly doped graphene. In this section, the emphasis is on the appropriate parametrization of the low-energy intraband conductivity tensor and on the connection between the effective generalized Drude formula obtained in this way and the aforementioned partial transverse conductivity sum rule. Section X contains concluding remarks.
Ii Model Hamiltonian
In electronic systems with multiple bands in the vicinity of the Fermi level, conduction electrons are described by the Hamiltonian Kupcic15 ()
The bare electronic contribution
represents noninteracting electrons in such a multiband case. Here, is the bare electron dispersion measured with respect to the chemical potential in the band labeled by the band index . is the bare phonon Hamiltonian
given in terms of the phonon field , and the conjugate field , is the bare phonon frequency, is the phonon branch index, and is the corresponding effective ion mass.
The electron-phonon coupling Hamiltonian can be shown in the following way
where and . This expression includes the scattering by acoustic and optical phonons. On the other hand, the scattering by static disorder is given by
Finally, the electron-electron interaction Hamiltonian
describes all nonretarded electron-electron interactions.
( in a general three-dimensional case). Here, and are, respectively, the Fourier transforms of the external scalar and vector potentials, while the corresponding screened potentials are labeled by and . The total charge density operator in the coupling Hamiltonian (7) is
The structures of the corresponding current density operator and the bare diamagnetic density operator are similar. Finally, , , and are the bare vertex functions in question. Hereafter, the dispersions and all these vertex functions are taken as known functions (for doped graphen see, for example, Ref. Kupcic13 ()).
Iii Kubo formula for conductivity tensor
Electrodynamic properties of multiband electronic systems are naturally described in terms of the screened dynamical conductivity tensor
This relation is known as the Kubo formula for conductivity Kubo95 (). The conductivity tensor is simply the RPA irreducible part of . In those multiband electronic systems in which Lorentz local field effects are absent (the two-band model for electrons in graphene from Sec. VIII being an example), the result is
This form of holds in the single-band case as well, because there are no local field effects in this case.
One usually uses the definition relation (LABEL:eq10) and the two basic relations from macroscopic electrodynamics,
to show in terms of the elements of the real-time RPA irreducible response tensor
( in graphene, and in a general three-dimensional case) and the real-time current-dipole correlation function , rather than in terms of the correlation functions . The result is Kubo95 (); Kupcic14 ()
Here, we have introduced the notation , where is the dipole density operator and is the corresponding dipole vertex function Kupcic13 (). Equation (14), for example, shows that the conductivity tensor , divided by , is nothing but the second-order coefficient in the Taylor expansion of the charge-charge correlation function with respect to .
Since is a non-singular function of and for all and , the elements of the response tensor are expected to have the properties
These relations are the usual starting point for hydrodynamic formulation of electrodynamics of conduction electrons Forster75 (); Gotze72 (); Vollhardt80 (). They prove useful in systematic microscopic studies of as well Kupcic15 (); Kupcic16 ().
iii.1 Partial transverse conductivity sum rule
For transverse electromagnetic fields polarized along the axis, we can write
After performing the Kramers-Kronig analysis Kubo95 (), the transverse conductivity sum rule becomes a function of the static current-current correlation function ,
From the multiband version of the Ward identity relation Kupcic14 (), it follows that
For long wavelengths, the effective number can be rewritten in the alternative form, in terms of the dimensionless reciprocal effective mass tensor . In this limit, the total effective number becomes
Therefore, the sum rule (20) is in accordance with the partial effective mass theorem (24) linking the bare diamagnetic vertex with the reciprocal effective mass tensor and the interband current vertices .
Here, is the Fermi-Dirac distribution function, is the single-electron Green’s function, and is the corresponding spectral function. is the Matsubara Fourier transform of .
The sum rule (20) must not be confused with the usual form of the transverse conductivity sum rule, which can be found in the literature Pines89 (); Mahan90 (). The latter represents the generalization of Eq. (20) to the case with infinite number of valence bands. In this case, the effective number reduces to the nominal concentration of conduction electrons [ for the conduction band, in this case].
The partial version of the sum rule holds for any electronic system with finite number of valence bands which is decoupled from the rest of the band structure. Evidently the partial transverse conductivity sum rule is much more useful in investigations of tight-binding systems with a few bands [where is usually very different from ] than its common textbook version. In this case, the left-hand side and the right-hand side of Eq. (20) can be calculated independently providing the direct test of the conductivity formula used in the calculations.
Iv Theoretical approaches
iv.1 Bethe-Salpeter equations
In realistic electronic systems with multiple bands, the microscopic structure of the conductivity tensor is usually determined by using the Matsubara finite-temperature formalism Abrikosov75 (); Fetter71 (); Ziman88 (); Mahan90 (). In this approach, the correlation functions from Eqs. (14)(16) are obtained by analytical continuation of (), where is the Matsubara Fourier transform of
According to Fig. 1, is shown in terms of the exact single-electron Green’s function and the exact RPA irreducible four-point interaction . The single-electron Green’s function satisfies the Dyson equation, and the RPA irreducible four-point interaction the corresponding Bethe-Salpeter equation Abrikosov75 (); Fetter71 (); Ziman88 (); Mahan90 (). For many purposes, it is helpful to rewrite this Bethe-Salpeter expression for in terms of and the exact renormalized vertex function . The Bethe-Salpeter equation for is closely related to that for the RPA irreducible four-point interaction. Finally, it is also possible to show as a function of and , the quantity which is usually called the auxiliary electron-hole propagator Kupcic13 (); Kupcic15 (); Kupcic16 (), the three-point electron-hole propagator, or the three-point susceptibility Bergeron11 ().
As long as these building blocks of are exact, all Kubo-Ward relations from the previous section are exactly fulfilled. This means that, in this case, the correlation functions have a form which is gauge invariant by definition, and the charge continuity equation is exactly satisfied. However, any approximation used to determine the structures of and leads to some extent to the violation of the charge continuity equation. As a consequence, we are usually forced to take care of the charge continuity equation explicitly when solving the Dyson and Bethe-Salpeter equations.
iv.2 Generalized self-consistent RPA equations
In weakly interacting systems, we can also use the alternative approach which represents an obvious generalization of the common self-consistent RPA equation. In this approach, we consider the Heisenberg equation for the density operator Kupcic14 (); Platzman73 (),
In the general case, the Hamiltonian is given by Eq. (1). Therefore, we can use this approach to study the scattering of conduction electrons by static disorder, by phonons, as well as by other electrons. For example, for the relaxation processes associated with the scattering by phonons, a straightforward calculation leads to
. Here, is again the macroscopic electric field, the are the intraband and interband dipole vertex functions, and . To obtain the self-consistent structure of these equations, we have to determine the right-hand side expressions in the equations
and retain only the contributions proportional either to or to . The former contributions will be referred to as the self-energy contributions and the latter ones as the vertex corrections. When the electron does not change the band when it is scattered by phonons and , then the result is the self-consistent equation for the induced density of the form
is a useful abbreviation.
It must be recalled that is the nonequilibrium part of the nonequilibrium distribution function in question Kupcic14 (); Platzman73 (). Therefore, the induced current density can be shown in terms of the current-dipole correlation function in the following way
Similarly, the induced charge density is given by
V Bethe-Salpeter expressions for
Let us now restrict our attention to a single-band case and explain how the simultaneous treatment of the Dyson equation, the Bethe-Salpeter equations, and the charge continuity equation mentioned in Sec. IV A works in typical approximate schemes. In this paper, the correlation functions are shown in terms of and , and instead of the Bethe-Salpeter equation for , we use the corresponding quantum kinetic equation
. This equation is equivalent to the original Bethe-Salpeter equation, and also represents the generalization of the intraband part of self-consistent equation (30). This equation is an integral equation of a complicated kind. For simplicity we omit here explicit reference to the conduction band index.
The relation between the renormalized vertex function and the electron-hole propagator is thus
On the other hand, the second expression in the third row leads to
For long wavelengths, the charge vertex is a constant and the current vertex is proportional to the electron group velocity . This means that the electron-hole propagator can be shown as a sum of four contributions of different symmetries,
where , resulting in
Evidently for electromagnetic fields polarized along the axis, there are only two components in Eq. (LABEL:eq39), i.e.,
Here, is the analytically continued form of
and the are the components of the four-component wave vector . Similarly, the charge continuity equation from Eq. (15) leads to
Finally, it is important to notice that the same symmetry based analysis holds for the intraband contributions in Sec. IV B as well. The relation between the two notations is the following
v.1 Common Fermi liquid theory
Here, is the solution of the Landau-Silin kinetic equation, which is simplified version of the equations (30) and (34) Platzman73 (); Kupcic15 (). In this theory, the main simplification is in the way how vertex corrections are taken into account. Namely, for electromagnetic fields polarized along the axis, we can insert the assumption (LABEL:eq41) into Eq. (34), separate all contributions which are odd functions of from the even contributions, and use the ansatz for the sum of the second and third term on the right-hand side of the kinetic equation which makes the sum of the even contributions identical to the charge continuity equation (42). In this way, Eq. (34) reduces to two coupled equations for and ; the first one is the charge continuity equation and the second one is the transport equation Pines89 (); Kupcic14 (). After retaining only the leading contributions to the self-energy and the related contributions to the irreducible four-point interaction, we obtain the well-known textbook expression for . This conductivity formula is known to describe well the Thomas-Fermi static screening, the collective modes of the electronic subsystem as well as the dc and dynamical conductivity.
Let us now present the formal derivation of both the memory-function conductivity formula [Eq. (51)] and its simplified form in which the issue of the Thomas-Fermi static screening is taken aside [Eq. (54)]. These expressions reduce to the well-known Fermi-liquid expressions when the memory function is approximated by its imaginary part [here is the usual notation for the relaxation rate, which depends on and on the polarization index ].
Vi Memory-function conductivity formula
The present derivation of the memory-function conductivity formula follows the same general path as the textbook derivation of the transport coefficients in the Fermi liquid theory Pines89 (). We consider the quantum kinetic equation for in the presence of the electromagnetic field polarized along the axis, and use the ansatz
for the sum of the last two terms on the right-hand side of the equation. In this way, this integral equation transforms into an ordinary equation