ORIGINAL RESEARCH article
Sec. Condensed Matter Physics
Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.640649
Hydrodynamic Approach to Electronic Transport in Graphene: Energy Relaxation
- 1Institute for Theoretical Condensed Matter Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany
- 2National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia
- 3Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe, Germany
- 4Ioffe Institute, St. Petersburg, Russia
In nearly compensated graphene, disorder-assisted electron-phonon scattering or “supercollisions” are responsible for both quasiparticle recombination and energy relaxation. Within the hydrodynamic approach, these processes contribute weak decay terms to the continuity equations at local equilibrium, i.e., at the level of “ideal” hydrodynamics. Here we report the derivation of the decay term due to weak violation of energy conservation. Such terms have to be considered on equal footing with the well-known recombination terms due to nonconservation of the number of particles in each band. At high enough temperatures in the “hydrodynamic regime” supercollisions dominate both types of the decay terms (as compared to the leading-order electron-phonon interaction). We also discuss the contribution of supercollisions to the heat transfer equation (generalizing the continuity equation for the energy density in viscous hydrodynamics).
Electronic hydrodynamics is quickly growing into a mature field of condensed matter physics [1–3]. Similarly to the usual hydrodynamics [4, 5]; this approach offers a universal, long-wavelength description of collective flows in interacting many-electron systems. As a macroscopic theory of strongly interacting systems, hydrodynamics should appear to be extremely attractive for condensed matter theorists dealing with problems where strong correlations invalidate simple theoretical approaches. However, electrons in solids exist in the environment created by a crystal lattice and typically experience collisions with lattice imperfections (or “disorder”) and lattice vibrations (phonons). The former typically dominate electronic transport at low temperatures, while at high temperatures the electron-phonon interaction takes over. In both cases the electron motion is diffusive (unless the sample size is smaller than the mean free path in which case the motion is ballistic) since in both types of scattering the electronic momentum is not conserved. On the other hand, if a material would exist where the momentum-conserving electron-electron interaction would dominate at least in some non-negligible temperature range, then one could be justified in neglecting the momentum non-conserving processes and applying the hydrodynamic theory. In recent years, several extremely pure materials became available with graphene being the most studied [1, 3].
In nearly neutral (or compensated) graphene the electron system is non-degenerate (at least at relatively high temperatures where the hydrodynamic approach is justified) with both the conductance and valence bands contributing on equal footing. Although the electron system is not Lorenz-invariant, the linearity of the Dirac spectrum plays an important role. Firstly, the Auger processes are kinematically suppressed leading to the near-conservation of the number of particles in each band [2, 3, 6, 7]. Secondly, the so-called collinear scattering singularity [7–14] allows for a non-perturbative solution to the kinetic (Boltzmann) equation focusing on the three hydrodynamic modes [13, 15, 16]. As a result, one can determine the general form of the hydrodynamic equations and evaluate the kinetic coefficients [16–18]. To be of any practical value, the latter calculation has to be combined with the renormalization group approach  since the effective coupling constant in real graphene (either encapsulated or put on a dielectric substrate) is not too small,
Next to the conservation laws, the main assumption of the hydrodynamic approach is local equilibrium [4, 22] established by means of interparticle collisions. Neglecting all dissipative processes, this allows (together with the conservation laws) for a phenomenological derivation of hydrodynamic equations [4, 5] that can be further supported by the kinetic theory, where the local equilibrium distribution function nullifies the collision integral in the Boltzmann equation . The resulting ideal hydrodynamics is described by the Euler equation and the continuity equations. This is where the electronic fluid in graphene differs from conventional fluids (both Galilean- and Lorentz-invariant): as in any solid, conservation laws in graphene are only approximate, leaving the collision integrals describing scattering processes other than the electron-electron interaction to be nonzero even in local equilibrium. This leads to the appearance of weak decay terms in the continuity equations.
Two such terms have already been discussed in literature. Firstly, even if the electron-electron interaction is the dominant scattering process in the system, no solid is absolutely pure. Consequently, even ultra-pure graphene samples possess some degree of weak disorder. Disorder scattering violates momentum conservation and hence a weak decay term must appear in the generalized Euler equation [2, 3, 15, 16]. Secondly, conservation of the number of particles in each band is violated by a number of processes (e.g., the Auger and three-particle scattering). Although commonly assumed to be weak, they are manifested in the decay – or recombination – term in the corresponding continuity equation. This was first established in  in the context of thermoelectric phenomena (for the most recent discussion see ). Later, quasiparticle recombination was shown to lead to linear magnetoresistance in compensated semimetals [24–27] as well as to giant magnetodrag [28, 29].
In this paper, we report the derivation of the third weak decay term in the hydrodynamic theory in graphene due to weak violation of energy conservation. Indeed, the electron-phonon interaction may lead not only to the loss of electronic momentum (responsible for electrical resistivity in most metals at high temperatures), but also to the loss of energy. Although subdominant in the hydrodynamic regime, the electron-phonon interaction should be taken into account as one of the dissipative processes. In graphene, the linearity of the Dirac spectrum once again plays an important role: as the speed of sound is much smaller than the electron velocity
Our arguments are based on the kinetic theory approach to electronic transport. In the spirit of , we assume the possibility of deriving the hydrodynamic equations from the kinetic equation in the weak coupling limit ,
with the Liouville’s operator (in the left-hand side)
and the collision integrals describing the electron-electron interaction (
Hydrodynamics is the macroscopic manifestation of the conservation of energy, momentum, and the number of particles. In a two-band system, the latter comprises excitations in both bands. In the conductance band these are electron-like quasiparticles with the number density (
while in the valence band the quasiparticles are hole-like
with the total “charge” (or “carrier”) density being
Assuming the numbers of particles in the conduction and valence bands are conserved independently, we can also define the total quasiparticle (or “imbalance” ) density
Global charge conservation (or gauge symmetry) can be expressed in terms of the usual continuity equation. This can be obtained from Eq. 1 by performing a summation over all quasiparticle states upon which all three collision integrals vanish .
As a result, the continuity equation has the usual form
where the corresponding current is defined as
The rest of the conservation laws in graphene are approximate as manifested in the collision integrals not vanishing upon corresponding summations. The continuity equation expressing momentum conservation (i.e., the Euler equation) is obtained by multiplying the kinetic equation by the quasiparticle momentum
Weak disorder scattering is typically described within the simplest τ-approximation .
where the momentum density is defined as
The last equality reprsents the fact that in graphene the momentum density is proportional to the energy density (due to the properties of the Dirac spectrum Eq. 2).
Supercollisions contributing to the recombination collision integral also violate momentum conservation, however, in comparison to the above weak disorder scattering, this is a second-order process. Moreover, the disorder mean free time
The remaining two continuity equations – energy and quasiparticle imbalance – are unaffected by the electron-electron interaction and weak disorder scattering. Indeed, the electron-electron interaction conserves energy and – neglecting the Auger processes – particle number in each band:
Same applies to the (Elastic) disorder scattering
However, supercollisions do not conserve both quantities and hence lead to weak decay terms in the two continuity equations. Contribution of the recombination collision integral to the continuity equation for the quasiparticle imbalance (for derivation see Section 4) is given by
The same scattering process contributes a weak decay term to the continuity equation for the energy density. Defining the decay coefficients similarly to Eq. 11, we may present the result in the form
Here the equivalence of the two forms of the decay term stems from the fact that
Finally, once the dissipative processes due to electron-electron interaction are taken into account, one usually replaces the continuity equation for the energy density by the equivalent equation for the entropy density, the so-called “heat transfer equation”  (for derivation seeSection 4; here
The obtained heat transfer equation is the final hydrodynamic equation in graphene. Together with the generalized Navier-Stokes equation derived in  and the continuity equation Eqs. 4b and 12, the heat transfer equation forms a complete set of macroscopic equations describing electronic transport in graphene in hydrodynamic regime.
Supercollisions are not the only scattering process contributing to both quasiparticle recombination and energy relaxation. Clearly, direct (not impurity-assisted) electron-phonon interaction contributes to energy relaxation as well as to quasiparticle recombination (in the case of intervalley scattering) [6, 13, 14, 29, 32]. In addition, optical phonons may also contribute [30, 31], although within the hydrodynamic approach these contributions were discussed only at the level of dissipative (viscous) hydrodynamics . The contribution of the direct [37, 38] and impurity assisted electron-phonon scattering to energy relaxation was compared in , where it was argued that at high enough temperatures,
At charge neutrality and in the hydrodynamic regime, the coefficients
The order of magnitude of
The obtained Eq. 2 should be compared to the corresponding equations in [3, 6, 31], where energy relaxation due to supercollisions was not taken into account. All other terms are present in all four equations with the following exceptions. The Eq. 54 of  is written in the relativistic notation omitting the imbalance mode, quasiparticle recombination, and disorder scattering, all of which are discussed separately elsewhere in . Ref.  was the first to focus on the imbalance mode with the Eq. 2.6 containing all the terms of Eq. 2 except for the viscous term (and energy relaxation). Finally, the Eq. 1c of  contains all of the terms in Eq. 2 except for energy relaxation and in addition contains a term describing energy relaxation due to electrons scattering on the optical phonon branch that is neglected here (generalization of the resulting theory is straightforward).
To summarize, we have considered supercollisions as a mechanism of quasiparticle recombination and energy relaxation in graphene and derived the corresponding decay terms in the hydrodynamic continuity equations. Since the same scattering mechanism is responsible for both effects, one has to take into account energy relaxation while considering quasiparticle recombination. The latter is an indespensible feature of electronic hydrodynamics in graphene in constrained geometries, where homogenious solutions violate the boundary conditions .
4 Materials and Methods
4.1 Collision Integral due to Supercollisions
An electron in the upper (conductance) band may scatter into an empty state in the lower (valence) band – effectively recombining with a hole – emitting a phonon (which carries away the energy) and losing its momentum to the impurity. Within the standard approach to the electron-phonon interaction, this process is described by the collision integral
Similarly, an electron in the lower band may absorb a phonon and scatter into the upper band – effectively creating an electron-hole pair – while still losing its momentum to the impurity
The collision Integral (14) conserves the total charge
(see Eq. 4) and vanishes in global equilibrium
where the quasiparticle distribution is described by the Fermi function. This should be contrasted with local equilibrium described by
In global equilibrium
Now we show, that in local equilibrium, i.e. for nonzero
4.2 Derivation of the Weak Decay Terms in Continuity Equations
To the leading order, we can describe the difference between the local equilibrium distribution function
Now we re-write the collision integral (14) with the help of the relations
and find (to the leading order in h)
Consider now the contribution of the recombination collision integral to the continuity equation for the quasiparticle imbalance
To the leading order, the deviation
The remaining integral has dimensions of particle density divided by time and therefore the result can be written in the form Eq. 11. The same scattering process contributes a weak decay term to the continuity equation for the energy density. Indeed, multiplying the collision integral Eq. 15 by the quasiparticle energy and summing over all states, we find after similar algebra
Defining the decay coefficients similarly to Eq. 11 above, we may present the result in the form Eq. 13.
4.3 Derivation of the Heat Transfer Equation
The entropy density of a system of fermions is defined in terms of the distribution function as
Treating this integral as
any derivative of s can be represented in the form
Consider now each term of the kinetic equation multiplied by the derivative
The gradient term yields similarly
where the quantity
can be interpreted as the entropy current.The electric field term vanishes as the total derivative
while the Lorentz term vanishes for rotationally invariant systems upon integrating by parts [justified by the fact that
The last equality follows from
Similar approach was used in  to derive the continuity equations (as outlined above). Combining all four terms, we conclude that integration with the factor
Equation 25 is valid for an arbitrary distribution function.
Denoting the integral of the right-hand side of the kinetic equation by
we arrive at the “continuity equation for the entropy”
In the usual hydrodynamics  the only contribution to the collision integral is given by particle-particle scattering, i.e. the processes assumed to be responsible for establishing local equilibrium such that at
For the Local Equilibrium Distribution Function
Substituting this expression into Eq. 26, we find that the remaining integration is very similar to the above derivation of the continuity equations. The integral with the quasiparticle energy yields exactly the above Eq. 13. The integral with
The decay terms in Eq. 14 appear already at local equilibrium. To complete the heat transfer equation one has to take into account disispation. In graphene, this is most conveniently done by considering the classical limit of relativistic hydrodynamics since the form of dissipative corrections is determined by the symmetries of the quasiparticle spectrum. The result has been already reported in literature, therefore we combine the dissipative corrections with Eq. 14 and write the heat transfer equation in graphene in the form Eq. 14.
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Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Both authors have equally contributed to the reported research.
This work was supported by the German Research Foundation DFG within FLAG-ERA Joint Transnational Call (Project GRANSPORT), by the European Commission under the EU Horizon 2020 MSCA-RISE-2019 program (Project 873028 HYDROTRONICS), by the German-Israeli Foundation for Scientific Research and Development (GIF) Grant no. I-1505-303.10/2019 (IVG) and by the Russian Science Foundation, Grant No. 20-12-00147 (IVG). BNN acknowledges the support by the MEPhI Academic Excellence Project, Contract No. 02.a03.21.0005.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors are grateful to P. Alekseev, A. Andreev, U. Briskot, A. Dmitriev, L. Golub, V. Kachorovskii, E. Kiselev, A. Mirlin, J. Schmalian, M. Schütt, A. Shnirman, K. Tikhonov, and M. Titov for fruitful discussions.
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Keywords: hydrodynamics, graphene, electronic transport, energy relaxation, supercollisions, quasiparticle recombination
Citation: Narozhny BN and Gornyi IV (2021) Hydrodynamic Approach to Electronic Transport in Graphene: Energy Relaxation. Front. Phys. 9:640649. doi: 10.3389/fphy.2021.640649
Received: 11 December 2020; Accepted: 11 February 2021;
Published: 23 April 2021.
Edited by:Alessandro Principi, The University of Manchester, United Kingdom
Reviewed by:Ke-Qiu Chen, Hunan University, China
Giovanni Vignale, University of Missouri, United States
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*Correspondence: B. N. Narozhny, email@example.com