# Magnetoresistance of compensated semimetals in confined geometries

## Abstract

Two-component conductors – e.g., semi-metals and narrow band semiconductors – often exhibit unusually strong magnetoresistance in a wide temperature range. Suppression of the Hall voltage near charge neutrality in such systems gives rise to a strong quasiparticle drift in the direction perpendicular to the electric current and magnetic field. This drift is responsible for a strong geometrical increase of resistance even in weak magnetic fields. Combining the Boltzmann kinetic equation with sample electrostatics, we develop a microscopic theory of magnetotransport in two and three spatial dimensions. The compensated Hall effect in confined geometry is always accompanied by electron-hole recombination near the sample edges and at large-scale inhomogeneities. As the result, classical edge currents may dominate the resistance in the vicinity of charge compensation. The effect leads to linear magnetoresistance in two dimensions in a broad range of parameters. In three dimensions, the magnetoresistance is normally quadratic in the field, with the linear regime restricted to rectangular samples with magnetic field directed perpendicular to the sample surface. Finally, we discuss the effects of heat flow and temperature inhomogeneities on the magnetoresistance.

The theory of magnetotransport in solids Pippard (); dau8 () is a mature branch of condensed matter physics. Measurements of magnetoresistance and classical Hall effect are long recognized as valuable experimental tools to characterize conducting samples. Interpreting the experiments within the standard Drude theory Pippard (); abrikos (); Kittel1963 (), one may extract many useful sample characteristics such as the electron mobility and charge density at the Fermi level. However, in materials with more than one type of charge carriers – e.g., semi-metals and narrow band semiconductors – the situation is more complex. Indeed, already in 1928 Kapitsa observed unconventional magnetoresistance in semi-metal bismuth films Kapitza1928 (). More recently, interest in magnetotransport has been revived with the discovery of novel two-component systems including graphene graphene1 (); graphene2 (); graphene3 (); graphene4 (); graphene5 (); graphene6 (), topological insulators topins1 (); topins2 (); topins3 (); topins4 (); topins5 (), and Weyl semimetals weyl1 (); weyl2 (); weyl3 (); weyl4 (); weyl5 (); weyl6 (); weyl7 (); weyl8 (); weyl9 (); weyl10 (); weyl11 (). A common feature of all such systems is the existence of the charge neutrality (or, charge compensation) point, where the concentrations of the positively and negatively charged quasiparticles (electron-like and hole-like, respectively) are equal and the system is electrically neutral.

A fast growing number of experiments on novel two-component materials exhibit unconventional transport properties in magnetic field: (i) linear magnetoresistance (LMR) was reported in graphene and topological insulators close to charge neutrality Friedman2010 (); Singh2012 (); Veldhorst2013 (); Wang2013 (); Gusev2013 (); Weber (); Wiedmann2015 (); Wang2015 (); Vasileva16 () as well as in narrow-gap semiconductors Hu2008 (), bismuth films Yang1999 (); Yang2000 (), and three-dimensional (3D) silver chalcogenides Xu1997 (); Husmann2002 (); Sun2003 () (ii) giant (and sometimes also linear) magnetoresistance was identified in semimetals WTe WTe1 (); WTe2 (); WTe3 (), NbP NbP1 (), LaBi LaBi1 (); LaSb&LaBi (), ZrSiS ZrSiS1 (); ZrSiS2 (), multilayer graphene mghBN () and many others NbSb (); ScPtBi (); CdAs (); NbAs&TaAs (); BiTe (); LuPdBi (); (iii) finally, the widely discussed negative magnetoresistance was found in Weyl semimetals and related materials negativeGR (); negative1 (); negative2 (); negative3 (); negative4 (); negative5 (); negative6 (); negative7 (); negative9 (); negative10 (); negative11 (). Moreover, negative magnetoresistance may by regarded as a “smoking gun” for detecting a Weyl semimetal Burkov2015 (); Sachdev16 (), although experiment Ganichev2001 (); Alekseev2016 () shows the existence of the effect in “non-Dirac” materials as well.

Conventional Drude-like theories of transport in two-component systems predict parabolic magnetoresistance that saturates in classically strong fields Pippard (); abrikos (); Weiss1954 (); Gant (). Taking into account additional relaxation processes may lead to semiclassical mechanisms of LMR in diverse physical systems including 3D metallic slabs with complex Fermi surfaces and smooth boundaries Azbel (); Kaganov (); strongly inhomogeneous or granular materials Dykhne (); Parish2003 (); Knap2014 (); Weber16 (); short samples Weber16 (); Yoshida (); disordered 3D metals Polyakov (); Song (); and compensated two-component systems Alekseev15 (). Quantum effects result in LMR in strong fields in 3D zero-gap band systems with linear dispersion Abrikosov1969 (); Abrikosov1998 (); Abrikosov2000 (). In weak fields, resistivity of two-dimensional (2D) electron systems acquires an interaction correction ZNA () that is linear in the field.

The extreme quantum limit of Refs. Abrikosov1969, ; Abrikosov1998, ; Abrikosov2000, has been realized in graphene Friedman2010 (), BiTe nanosheets BiTe (), and possibly in the novel topological material LuPdBi LuPdBi (). However, this mechanism is applicable to the specific case of 3D systems with linear dispersion subjected to a strong magnetic field (as usual, is the temperature, is the Planck constant, and is the electron cyclotron frequency), where all electrons are confined to the first Landau level. Recently, this approach has been extended to Weyl semimetals at finite temperatures and with short-range disorder Klier (). However, the above conditions are typically not satisfied by the majority of systems exhibiting unsaturated LMR at high temperatures.

Experiments on strongly inhomogeneous (or strongly disordered) systems are often interpreted using the classical approach of Refs. Dykhne, ; Parish2003, . In particular, the random-resistor model of Ref. Parish2003, was introduced to explain the non-saturating LMR in granular materials such as AgSe Xu1997 (); Husmann2002 (). More recently, this mechanism was used to interpret the behavior of the hydrogen-intercalated epitaxial bilayer graphene Weber (). However, this model (as well as the quantum theory of Refs. Abrikosov1969, ; Abrikosov1998, ; Abrikosov2000, ) does not distinguish between single- and multi-component systems, contradicting the crucial role of the charge neutrality point in many aforementioned experiments. Moreover, both theoretical approaches rely on the presence of disorder and thus cannot be used to interpret the data obtained in ultra-clean, homogeneous samples.

A phenomenological theory of magnetotransport in 2D clean, two-component systems close to charge neutrality was proposed by the present authors in Ref. Alekseev15, . Subjected to a perpendicular magnetic field, such systems exhibit the compensated Hall effect, where the Hall voltages due to positively and negatively charged carriers partially (precisely at charge neutrality – completely) cancel each other. Such compensation of the Hall voltage is accompanied by a neutral quasiparticle flow in the lateral direction relative to the electric current Titov2013 (). In constrained geometries this leads to a nonuniform distribution of charge carriers over the sample area, effectively splitting the sample into the bulk and edge regions. The resistance of the edge region is dominated by the electron-hole recombination, while the bulk of the sample exhibits the usual, essentially Drude resistance. The total resistance of the sample is then obtained by treating the edge and bulk regions as independent, parallel resistors. The linear dependence of the sample resistance on the magnetic field arises due to qualitatively different behavior of the edge region. At charge neutrality, the resulting LMR persists into the range of classically strong fields. Away from the neutrality point, the nonzero Hall voltage leads to the observed saturation of the magnetoresistance. Similar ideas were recently exploited by some of us to explain the phenomenon of the giant magnetodrag in graphene Titov2013 (); Narozhny (). The importance of the electron-hole recombination processes for magnetotransport in narrow-band semiconductors and semimetals has been pointed out earlier by Rashba et. al. in Ref. Rashba76, .

In this paper we present a microscopic theory of magnetotransport in two-component systems. Combining the kinetic equation with the sample electrostatics, we provide a rigorous justification for the phenomenological approach of Ref. Alekseev15, . Furthermore, we extend our theory to 3D systems. We find that although in 3D the magnetoresistance is typically quadratic in the field, there exists a linear regime in rectangular samples with magnetic field directed perpendicular to the sample surface.

The remainder of the paper is organized as follows. First, we discuss the qualitative physics of magnetotransport in two-component systems. In the technical part of the paper we present a Boltzmann equation approach to magnetotransport in finite-size 2D and 3D systems. In the latter case, we focus on the rectangular sample geometry to simplify the analysis of the sample electrostatics. We conclude the paper by discussing the experimental relevance of our results.

## I Qualitative discussion

Let us first recall the results of the classical linear response theory Pippard (); abrikos (); Kittel1963 (); Weiss1954 (); Gant () applied to two-component systems. A system of charge carriers subjected to a homogeneous external electric field, exhibits an electrical current. The current density, , is proportional to the applied field, , where is the conductivity tensor. In two-component systems, one can define currents for each individual carrier subsystems, which we will refer to as electron and hole quasiparticle flows, and , respectively. The electric current is then given by their difference, .

In external magnetic field, the system exhibits the classical Hall effect: a voltage is generated across the system in the transverse direction to the electric current. In a typical transport measurement, external leads are attached to the sample in such a way, that no current is allowed to flow in the direction of the Hall voltage. Theoretical description of the effect is most transparent in isotropic systems, where . If we associate the -axis with the electric current and the -axis with the magnetic field, then the Hall voltage is generated in the direction, while . In two-component systems, the latter condition leads to a field-dependent longitudinal resistivity Weiss1954 (); Gant ()

(1) |

where is the magnetic field, and stand for the equilibrium electron and hole densities, and and are the electron and hole mobilities. Within the standard Drude theory Pippard (); abrikos (); Kittel1963 (), the conductivity can be expressed in terms of the quasiparticle densities and mobilities as

whereas the additional parameter is

In the presence of the electron-hole symmetry, the mobilities of the two types of carriers coincide, , and the resistivity (1) simplifies to

(2) |

where we have introduced quasiparticle and charge densities, and , respectively.

The results (1) and (2) yield a positive magnetoresistance that is quadratic in weak magnetic fields and saturates in classically strong fields. The two exceptions are provided by neutral systems (, ), where the quadratic magnetoresistance is non-saturating, and single-component systems (e.g. for purely electronic transport , ), where the longitudinal resistivity is independent of the magnetic field Pippard (); abrikos (); Kittel1963 ().

Previously Titov2013 (); Alekseev15 (), we have pointed out an inconsistency that appears when the above classical theory is applied to finite-sized samples. Indeed, even partially compensated Hall effect is accompanied by a neutral quasiparticle flow in the direction transversal to that of the electrical current, see Figs. 1 and 2. As the quasiparticles cannot leave the sample, this flow leads to quasiparticle accumulation near the sample boundaries. The excess quasiparticle density is controlled by inelastic recombination processes that are excluded from the classical theory. The typical length scale characterizing such processes, , hereafter referred to as the recombination length, determines the size of the boundary region with excess density of quasiparticles. Here we consider rectangular samples with the length being the longest length scale in the systemfootnote (),

(3) |

The classical results are applicable if the boundary regions are small as compared to the sample width, . If, on the other hand, is comparable with , then the behavior of the system may strongly deviate from the predictions of the classical theory.

Treating the bulk and boundary regions as parallel conductors, we estimate the sheet resistance of the sample Alekseev15 ()

(4) |

In the bulk, the lateral quasiparticle flow leads to the so-called “geometric” magnetoresistance Lakeou (); Chang2014 ()

where we have used Eq. (2) in the limit of classically strong magnetic fields, .

In the boundary regions, the quasiparticle flows are mostly directed along the external electric field, see Figs. 1 and 2, and the geometric enhancement does not take place. Instead, the field dependence of the edge contribution to the sample resistance,

is due to the recombination length, . In homogeneous samples, the simplest estimate Titov2013 (); Alekseev15 () yields that is inverse proportional to in classically strong fields

(5) |

where is the zero-field recombination length determined by the diffusion coefficient and the characteristic recombination time .

The asymptotic behavior (5) of the recombination length may be qualitatively understood as follows. In classically strong magnetic fields, , the charge carriers move over a typical distance (the cyclotron radius) during a typical diffusion time . Since the quasiparticle life-time is determined by the recombination processes, the overall distance covered by the electron during the time may not exceed , which yields the estimate for the size of the boundary regions.

Combining the above arguments, we arrive at the following expression for the sheet resistance (4) in classically strong magnetic fields, ,

(6) |

The sheet resistance (6) exhibits all qualitative features of the magnetoresistance in nearly compensated two-component systems.

In wide samples, , magnetotransport is dominated by the bulk and can be described by the classical theory, see Eqs. (1) and (2) and the subsequent discussion. We consider such samples as essentially infinite.

Deviations from the classical behavior (1) and (2) occurs in finite-size samples of the width belonging to the intermediate interval determined by the magnetic field,

In this case, the sheet resistance of compensated (neutral, ) systems is linear in the magnetic field

(7) |

Away from charge neutrality, LMR appears only in an intermediate range of magnetic fields. In strong fields, , magnetoresistance saturates.

In narrow samples, , recombination is ineffective and the above physical picture breaks down. In this case, the two carrier subsystems behave as two independent single-component systems. As a consequence, classical magnetoresistance is absent Pippard (); abrikos (); Kittel1963 ().

The sheet resistance (6) is illustrated in Fig. 3 where it is plotted in a wide range of classically strong magnetic fields in the above three regimes. Panel (a) shows for a symmetric system at charge neutrality, while panels (b), (c), and (d) illustrate our results for asymmetric systems at (solid curves) and away from (dashed curves) the compensation point.

The above semiclassical mechanism of LMR in finite-size, nearly compensated two-component system was first suggested in Ref. Titov2013, in the context of Coulomb drag Narozhny (). The results (4)-(7) were derived rigorously in graphene hydrolin () on the basis of a microscopic transport theory. Subsequently, the macroscopic equations derived in graphene were generalized to a generic compensated two-component system using a phenomenological approach Alekseev15 ().

In this paper, we justify the phenomenological approach of Ref. Alekseev15, and derive the LMR for a wide range of systems using the Boltzmann kinetic equation. The key point that makes our theory so general, is the simple fact that in an magnetic field charge carriers driven through the system by the external electric field experience a lateral drift in the direction () defined by the electric and magnetic fields. The ultimate cause of this drift is the Lorenz force that acts on all charge carriers independently of their density, mobility, details of the spectrum, and additional quantum numbers. The second essential feature of our theory is the presence of the boundary leading to accumulation of the excess quasiparticle density in the narrow regions near the sample edges. Again, this is a completely general feature since all samples used in laboratory (as well as all industrial electronic devices) have a finite size. The width of the boundary regions (and hence, the degree of macroscopic inhomogeneity in the system, see Fig. 2) is controlled by the quasiparticle recombination length. The particular dependence (5) of on the magnetic field is crucial for the resulting LMR, given by Eq. (7). The original estimate Alekseev15 () (5) is not universal Vasileva16 () insofar that the coefficient of the inverse proportionality (in classically strong fields) is system (or model) dependent. In a sense, the technical goal of the microscopic theory presented in this paper is to calculate the field dependence of the effective recombination length.

In our qualitative arguments, we have tacitly assumed that the energy transfer plays no role in formation of the macroscopic inhomogeneities of the quasiparticle currents and densities. At the microscopic level, this means energy relaxation (and hence, thermalization) in the system is much faster than quasiparticle recombination. As a result, the temperatures of both carrier subsystems are uniform within the sample (and are, in fact, identical).

The theory of Refs. Titov2013, ; hydrolin, ; Alekseev15, , as well as the present qualitative discussion and the microscopic theory of Sec. II, is focused on 2D systems. Similar behavior can be found also in 3D samples. In particular, if cyclotron orbits do not remove the carriers from a plane parallel to one of the sample faces, a linear regime similar to Eq. (7) may be observed. In this paper, we make the first steps towards a full microscopic understanding of magnetotransport in 3D two-component systems, see Sec. III.

## Ii Transport theory of 2D two-component systems

In this Section we show that the linear dependence of resistivity on the sufficiently strong magnetic field is a generic effect for two-component systems at charge neutrality. For brevity, we employ the natural system of units where .

The usual starting point for developing a microscopic transport theory is the kinetic equation abrikos (). For a generic two-component electronic system, the kinetic equation has the standard form

(8) |

The semiclassical distribution functions describe the positively and negatively charged quasiparticles (“holes” and “electrons”, respectively, distinguished by the index ) with the energies and velocities . The system is subjected to the external electric and magnetic fields and .

The collision integral in the right hand side of Eq. (8) comprises contributions from impurity, electron-phonon, and electron-electron scattering. We will describe these scattering processes by the typical time scales , , and . The impurity and electron-phonon scattering contribute to momentum relaxation, while the electron-electron and electron-phonon interactions determine the thermalization properties of the system, as well as quasiparticle recombination. The traditional transport theory Pippard (); abrikos (); Kittel1963 () assumes that in the absence of external fields the system is in equilibrium. The electric current (or more generally, the quasiparticle flows) appears as a response to the applied fields. Within linear response, the system experiences no heating and remains thermalized. In this (and the following) Section we work under the same assumptions.

Finding a general solution to the kinetic equation (8) is a complicated task that is best accomplished numerically. In the special case of Dirac fermions in graphene, the solution is facilitated by the so-called collinear scattering singularity hydrolin (). Otherwise, an analytical solution can be found in the two paradigmatic limiting cases, known as the “disorder-dominated” and “hydrodynamic” regimes abrikos (); Narozhny (), which can be distinguished by comparing the scattering rates for elastic and inelastic processes:

(i) in the disorder-dominated regime, the fastest scattering process in the system is due to potential disorder,

(9) |

Since the electron-electron scattering time is typically inverse proportional to temperature,

the relation (9) implies

which means that the motion of the charge carriers is diffusive. In this case, most of the transport coefficients can be expressed in terms of the diffusive constant. As a result, qualitative features of the physical observables are independent of the microscopic details, such as the precise form of the single-particle spectrum.

(ii) in the hydrodynamic regime, the fastest process is due to electron-electron interaction

(10) |

Now, the relation between the temperature and the impurity scattering time is reversed,

so that the motion of charge carriers is ballistic. In this limit, the system of charged quasiparticles behaves similarly to a fluid and is described by the hydrodynamic equations.

Remarkably, in both regimes the resistance of 2D two-component systems close to charge neutrality exhibits linear dependence on the orthogonal magnetic field (in sufficiently strong fields).

### ii.1 Disorder-dominated regime

#### Symmetric, parabolic bands at charge neutrality

We begin with the simplest case of the symmetric parabolic spectrum with the band gap ,

(11) |

where the quasiparticle velocity is proportional to the momentum

Furthermore, we will assume the energy-independent momentum relaxation time

At charge neutrality, the equilibrium state of the system is described by the Fermi distribution function with the zero chemical potential

Since the single-particle spectrum (11) depends only on the momentum, the equilibrium quasiparticle density is given by

(12) |

where is the degeneracy factor reflecting other possible quantum numbers, such as spin, valley, etc.

External fields drive the system out of equilibrium, giving rise to deviations of the quasiparticle densities from the equilibrium value (12)

(13) |

and the corresponding flow densities :

(14) |

The nonequilibrium densities and currents are related by the continuity equations that can be derived by integrating the kinetic equation (8)

(15) |

Here denotes the quasiparticle recombination time. The recombination processes typically involve electron-phonon scattering, although in certain circumstances electron-electron Narozhny () and three-particle hydro1 () collisions may also contribute. A calculation of the recombination time using a particular microscopic model is beyond the scope of the present paper.

Macroscopic equations hydrolin () for the flow densities (14) can be obtained by multiplying the kinetic equation (8) by the quasiparticle velocity and summing over all single-particle states. As a result, we find hydro1 (); hydrolin ()

(16) |

where are the carrier cyclotron frequencies .

Comparing the integral in Eq. (16) with the flow density (14), we find it natural to split the distribution functions into the “isotropic” and “anisotropic” parts,

(17) |

The isotropic term depends only on the quasiparticle energy and hence does not contribute to the currents (14). On the contrary, the anisotropic term is an odd function of the momentum. It is this part of the distribution function that determines the currents (14), but at the same time, it does not contribute to the integral in Eq. (16).

Within linear response, deviations of the isotropic function from the equilibrium distribution can either reflect deviations of the local electronic temperature from the equilibrium value determined by the lattice, or the local fluctuations of the chemical potential .

Thermalization between the electronic system and the lattice is achieved by means of electron-phonon coupling. While the same coupling is also responsible for quasiparticle recombination, the latter is a much slower process and does not affect the local temperature. Relegating a more detailed discussion of this issue to a future publication, hereafter we assume that the relation

allows us to neglect local temperature fluctuations

As a result, the isotropic part of the distribution function may only depend on the local fluctuations of the chemical potential

(18) |

This implies the proportionality between the local density fluctuations (13) and :

(19) |

where (cf. Ref. hydrolin, )

(20) |

with being the density of states [ has dimensions of ].

Since the equilibrium distribution is independent of , we can express the integral in Eq. (16) as

and introduce the diffusion coefficient in Eq. (16)

(21) |

The diffusion coefficient is the same for the electrons and holes:

(22) |

At charge neutrality the averages in the expression for the diffusion coefficient can be evaluated analytically:

(23) |

The macroscopic equations (15) and (21) allow us to find transport coefficients of the system, as well as the carrier density and current profiles. These equations are semiclassical in the sense that the effects of quantum interference ZNA () and Landau quantization abrikos (); Abrikosov1969 (); Abrikosov1998 (); Abrikosov2000 () are neglected.

In this paper we are interested in solving the macroscopic transport equations (15) and (21) in confined geometries (in fact, that is why we have considered the nonuniform distributions). For simplicity, we consider a rectangular sample with the length that is much larger than the width , as well as any correlation length in the system. In this case, all physical quantities depend only on the transversal coordinate (). If no contacts are attached to the side edges of the sample, the quasiparticle flows have to vanish at the edges

(24) |

Combining the carrier densities (13) into the charge density, , and total quasiparticle density , and introducing the corresponding currents, and , we may represent the macroscopic equations (15) and (21) in the form Alekseev15 ()

(25a) | |||

(25b) | |||

(25c) |

Looking for solutions independent of the coordinate and keeping in mind the hard-wall boundary conditions (24), we find

Moreover, we note that the equations (25) preserve the direction of the applied electric field if choose it to be

Then we can use Eq. (25c) to exclude the quasiparticle density and simplify Eqs. (25a) and (25b) as

(26a) | |||

(26b) |

where is the electric current in the absence of magnetic field.

The second-order differential equation (26a) with the hard-wall boundary conditions (24) admits the solution Alekseev15 ()

(27) |

where the quasiparticle recombination length in magnetic field is

The quasiparticle current (27) and the corresponding electric current are illustrated in Fig. 1. The nonuniform nature of the currents does not allow for establishing a meaningful resistivity in our system. Instead, we may define the sheet resistance Alekseev15 ()

(28) |

The resulting value of is given by

(29) |

The sheet resistance (29) was previously obtained in Ref. Alekseev15, using a phenomenological approach. Depending on the sample width , recombination length , and magnetic field, one may identify three types of asymptotic behavior Alekseev15 ():

(i) in wide samples, , the resistance (29) is a non-saturating, quadratic function of the field Weiss1954 ()

(30a) | |||

The resistance (30a) exhibits geometric enhancement that is a consequence of the compensated hall effect, where the Hall voltage is absent despite the tilt of the carrier trajectories. | |||

(ii) in narrow samples, , quasiparticle recombination is ineffective, all currents flow along the -axis, and hence the geometric enhancement factor is absent | |||

(30b) | |||

(iii) samples of intermediate width, , in classically strong magnetic fields, , exhibit a linear behavior Alekseev15 (); hydrolin () | |||

(30c) |

shown in Eq. (7) above (note that ).

The results of this section provide the microscopic justification to the phenomenological approach of Ref. Alekseev15, . Similar results were previously obtained for monolayer graphene hydrolin (). In the following sections we generalize our theory to the case of arbitrary quasiparticle spectrum and prove that LMR in classically strong fields is a generic feature of compensated, two-component systems.

#### Symmetric bands with arbitrary spectrum

In this section, we generalize our kinetic theory to the case of the arbitrary quasiparticle spectrum, , and energy-dependent momentum relaxation time, . For simplicity, we only consider rotationally invariant spectra, , . The cyclotron frequency is now also energy-dependent

(31) |

while the velocity and momentum are given by the usual relations

(32) |

The energy dependence of the velocity and momentum relaxation time makes the derivation of the macroscopic transport equations rather tedious. Instead, we use the kinetic equation (8) to relate the two parts of the distribution function (17). The anisotropic part of the kinetic equation reads

(33) |

where the angle describes direction of the velocity. Solving Eq. (33) for , we find

(34) |

where the indices indicate the 2D vector components. The tensor is given by

(35) |

Now we can use Eq. (33) to express the carrier flow densities (14) in terms of the isotropic part of the distribution function. Instead of the direct momentum integration, we now evaluate the currents (14) in two steps. Firstly, we average over the direction of the velocity. This yields the energy-dependent currents

(36) |

where . Secondly, we integrate over the energy using the explicit form (18) of the distribution function. The expression (18) is still valid, since none of the assumptions of the previous section relied on the particular shape of the quasiparticle spectrum. Substituting Eq. (18) into Eq. (36) we find

(37) |

Integrating Eq. (37) over the energy, we obtain

(38) |

with the averaged “diffusion tensor” is

(39a) | |||

The individual matrix elements of are given by | |||

(39b) | |||

(39c) |

For the energy-independent and the matrix simplifies to

(40) |

where is given by Eq. (23).

The expression (38) generalizes the above macroscopic equation (21) for the case of an arbitrary quasiparticle spectrum and energy-dependent momentum relaxation rate [for the parabolic spectrum, we recover Eq. (21) with the help of the identity ]. The corresponding continuity equations are still given by Eq. (15), where now stands for the mean value of the recombination time. Again, in this paper we do not study microscopic details of the recombination processes and, in particular, the energy dependence of the recombination rate.

At charge neutrality, the densities of electrons and holes coincide, . Similarly to the case of the parabolic spectrum, the hard-wall boundary conditions (24) ensure that the electric field does not deviate from its direction along the the -axis, . This allows us to re-write Eq. (38) in the form

(41a) | |||

(41b) |

Combining the currents (41) with the continuity equation (15), we find a second-order differential equation for

(42) |

The equations (41) and (42) are completely equivalent to Eqs. (26). The only difference is the precise definition of the diffusion coefficients. Hence, it is not surprising that the solution to Eq. (42) with the hard-wall boundary conditions (24) is similar to Eq. (27)

(43) |

Finally, we use the solution (43) and Eqs. (41) to find the averaged electric current and sheet resistance (28)

(44) |

Qualitatively, the result (44) is similar to Eq. (29), see also Fig. 4. Most importantly, the dependence of on the magnetic field and sample geometry is given by the same function . Therefore, we can identify the same three types of behavior as in Eqs. (30).

(i) in the limit of a wide sample the contribution of the function may be neglected. The resulting magnetoresistance is quadratic and unsaturating, .

(ii) the limit of a narrow sample corresponds to the approximation . In this case, the sheet resistance (44) is not strictly speaking a constant, but exhibits weak, quickly saturating dependence on the magnetic field, .

(iii) the limit of an intermediate sample size exists in classically strong magnetic fields, where we may approximate and neglect the field-independent term in Eq. (44). This leads to the linear magnetoresistance similar to Eq. (30c). The parameter range for this regime is similar to that of the previous section: . The resulting resistance is

(45) |

The result (45) may be simplified if we formally assume the limit . Then the elements of the diffusion matrix are

The recombination length is inverse proportional to the magnetic field

and hence the resistance is linear in the -field

(46) |

#### Asymmetric bands

Now we discuss a generic two-component system without electron-hole symmetry. For simplicity, we will consider the parabolic spectra (as we have seen above, changing the shape of the quasiparticle spectrum does not lead to qualitatively new physics)

(47) |

In addition, the system may be doped away from charge neutrality, i.e. the equilibrium chemical potential may be shifted from the middle of the band gap. Nevertheless, we may repeat the derivation of the continuity equations (15) and macroscopic equations (21) and arrive at the following description of the system

(48) |

The electrons and holes are described by their respective densities , masses , momentum relaxation times , cyclotron frequencies , and diffusion coefficients

(49) |

Here the averaging over energies is similar to Eq. (20), but with the different equilibrium distribution functions for electrons and holes, :

(50) |

where is the corresponding density of states.

The recombination rates are generally different for electrons and holes and may be approximated as

(51) |

where the coefficient is the function of and depends on a particular model of electron-hole recombination.

In the absence of the electron-hole symmetry, the classical Hall effect is no longer completely compensated and the Hall voltage is formed. The corresponding lateral component of the electric field can be related to the nonuniform charge density across the sample. In principle this can be done by solving the Poisson equation with the sample-specific boundary conditions. This electrostatic problem can be rather complicated and may admit only numerical solutions. While one may have to solve the electrostatic problem to describe the behavior of any particular sample quantitatively, qualitative physics is independent of such complications. Here, we will consider the simplest case of a gated sample. If the distance between the 2D electron system and the gate electrode is much smaller than any typical length scale describing inhomogeneity of the charge density and carrier flows, then the system is in the strong screening limit, where the electric field is related to the charge density as Alekseev15 ()

(52) |

where is the external field, is the gate-to-channel capacitance per unit area, is the the distance to the gate, is dielectric constant, and is the charge density.

The macroscopic equations (48) are linear differential equations that can be solved similarly to the above case of the symmetric bands. Before presenting the general solution, we discuss two particular limiting cases, (i) the Boltzmann limit away from charge neutrality, and (ii) the fast Maxwell relaxation.

#### Boltzmann limit

First, we consider the low-temperature (Boltzmann) limit, , where the effective number of charge carriers in both bands is small. In the simplest case, the carriers have the same mass, , and momentum relaxation time, . Consequently, the two cyclotron frequencies also coincide, . The electron-hole symmetry is broken by the non-zero chemical potential, . The above parameters can be combined into the “Drude conductivities” of the electrons and holes

(53) |

In this limit, the equilibrium distribution functions have the simple form

(54) |

allowing for the explicit expressions for the equilibrium carrier concentrations (with being the density of states for the 2D parabolic spectrum)

(55) |

Furthermore, with exponential accuracy the diffusion coefficients (49) can be approximated by

(56) |

The above simplifications allow for a straightforward solution of the macroscopic equations (48). The averaged sheet resistance (28) is given by

(57) |

where and the magnetic-field dependent recombination length is

(58) |

At charge neutrality, , we recover the previous result (29). The magnetoresistance is shown in Fig. 3 for several values of .

Since the system is doped away from charge neutrality, the classical Hall effect is no longer fully compensated. This can be seen in the solution to the equations (48), where the electric field acquires a constant component in the lateral direction

(59) |

leading to the nonzero Hall voltage, . The corresponding Hall sheet resistance

is given by

(60) | |||

#### Fast Maxwell relaxation

A more general situation with unequal carrier masses and momentum relaxation times also allows for a simple solution under the assumption of fast Maxwell relaxation,

In this limit, charge fluctuations in the two-component system relax much faster than the usual diffusion.

Formally taking the limit in Eqs. (48) and (52), we recover the balance between the nonequilibrium density fluctuations of the electrons and holes

(61) |

Note, that this does not imply charge neutrality, since these fluctuations occur on the background of nonzero equilibrium charge density .

Now, we can express the quasiparticle flows (14) in terms of the density perturbation (61) and electric field

(62) |

where

(63) |

Here the cyclotron frequency, , has the opposite sign for electrons and holes.

The hard-wall boundary conditions (24) imply the equality . Excluding the -component of the electric field from Eqs. (62), we can express the currents in terms of the quasiparticle density . This allows us to re-write the continuity equations in the form of the second-order differential equation on , same as Eq. (42), which we reproduce here for convenience,

(64) |

In contrast to Eq. (42), the effective recombination length is now given by

(65) |

where the two-component quasiparticle system is characterized by the field-dependent conductivity matrix

(66a) | |||

and the field-dependent diffusion matrix | |||

(66b) |

The solution to Eq. (64), which satisfies the boundary conditions, differs from the previous result (43) by the normalization factor that is dictated by the relation (62) between the density and quasiparticle flows. In the present case we find

(67) |

Substituting Eq. (67) into Eq. (62), we express the inverse sheet resistance in the form

(68) |

where is the resistivity of an infinitely large system

(69) |

and