A Energy relaxation rate

# Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems

## Abstract

The dc magnetoconductivity of the multisubband two-dimensional electron system formed on the liquid helium surface in the presence of resonant microwave irradiation is described, and a new mechanism of the negative linear response conductivity is studied using the self-consistent Born approximation. Two kinds of scatterers (vapor atoms and capillary wave quanta) are considered. Besides a conductivity modulation expected near the points, where the excitation frequency for inter-subband transitions is commensurate with the cyclotron frequency, a sign-changing correction to the linear conductivity is shown to appear for usual quasi-elastic inter-subband scattering, if the collision broadening of Landau levels is much smaller than thermal energy. The decay heating of the electron system near the commensurability points leads to magnetooscillations of electron temperature, which are shown to increase the importance of the sign-changing correction. The line-shape of magnetoconductivity oscillations calculated for wide ranges of temperature and magnetic field is in a good accordance with experimental observations.

###### pacs:
73.40.-c,73.20.-r,73.25.+i, 78.70.Gq

## I Introduction

The discovery of novel microwave-induced oscillations of magnetoresistivity (1) as a function of the magnetic field and so-called zero-resistance states (ZRS) (2); (3) has sparked a large interest in quantum magnetotransport of two-dimensional (2D) electron systems exposed to microwave (MW) radiation. The 1/B-periodic oscillations were observed for quite arbitrary MW frequencies larger than the cyclotron frequency . The period of these oscillations is governed by the ratio . ZRS appear in ultrahigh-mobility GaAs/AlGaAs heterostructures as a result of evolution of the minima of the oscillations with an increase in radiation power.

Recently, MW-induced magnetooscillations and vanishing of the magnetoconductance were observed in the nondegenerate multisubband 2D electron system formed on the liquid helium surface (4); (5). These oscillations have many striking similarities with those observed in semiconductor systems: they are 1/B-periodic, governed by the ratio , and their minima eventually evolve in zero magnetoconductance states nearly at the same values of . The important distinction of these new oscillations is that they are observed only for a MW frequency fixed to the resonance condition for excitation of the second surface subband: (here , and describes the energy spectrum of surface electron states, ).

The ZRS observed in semiconductor systems are shown (6) to be understood as a direct consequence of the negative photoconductivity which can appear with an increase in the amplitude of conductivity oscillations. Regarding the microscopic origin of the oscillations, the most frequently studied mechanism is based on photon-induced impurity scattering within the ground subband, when an electron simultaneously is scattered off impurities and absorb or emit microwave quanta (7); (8). This kind of scattering is accompanied by an electron displacement along the applied dc-electric field whose sign depends on the sign of (here ). Therefore, sometimes this mechanism is termed the ”displacement” mechanism. A different mechanism, called the ”inelastic” mechanism (9), explains conductivity oscillations as a result of oscillatory changes of the isotropic part of the in-plane electron distribution function.

Both microscopic mechanisms of the negative conductivity prosed for semiconductor systems cannot be applied for explanation of similar effects observed in the system of surface electrons (SEs) on liquid helium, because the MW frequency considered in these theories has no relation to inter-subband excitation frequencies . Recently, a new mechanism of negative momentum dissipation relevant to experiments with SEs on liquid helium was briefly reported (10). It cannot be attributed to ”displacement” or ”inelastic” mechanisms. In this theory, the origin of magnetooscillations and negative dissipation is an additional filling of the second surface subband induced by MW irradiation under the resonance condition (), which triggers quasi-elastic inter-subband electron scattering. The ordinary inter-subband scattering, which does not involve photon quanta, is accompanied by electron displacements whose sign depends on the sign of . Usually, this scattering does not lead to any negative contribution to . A correction to proportional to was shown to appear only if , where is the number of electrons at the corresponding subband, and is the electron temperature. It is important that this correction is also proportional to a large parameter equal to the ratio of to the collision broadening of Landau levels.

In this work, we perform a systematic theoretical study of negative dissipation phenomena in a multisubband 2D electron system caused by non-equilibrium filling of excited subbands. The magnetotransport theory (10) is generalized in order to include electron scattering by capillary wave quanta (ripplons) which limits SE mobility in experiments (5), where vanishing magnetoconductivity is observed. In order to understand the importance of MW heating at the vicinity of commensurability points, electron energy relaxation is analyzed. A sign-changing correction to the energy relaxation rate similar to the sign-changing correction to the momentum relaxation rate is found for non-equilibrium filling of excited subbands.

## Ii General definitions

Consider a multisubband 2D electron system under magnetic field applied perpendicular. The electron energy spectrum is described by , where represents Landau levels (). For SEs on liquid helium (for review see Ref. (11)) under a weak holding electric field (), , where is the effective Rydberg energy of SE states,

 ER=ℏ22mea2B, \ aB=ℏ2meΛ, \ Λ=e2(ϵ−1)4(ϵ+1), (1)

is the effective Bohr radius, is the electron mass, and is the dielectric constant of liquid helium. The excitation energy is about (liquid ) or (liquid ). It increases with the holding electric field , which allows also to tune in resonance with the MW frequency.

Under typical experimental conditions, the electron-electron collision rate of SEs is much higher than the energy and momentum relaxation rates. Therefore, the electron distribution as a function of the in-plane energy can be characterized by the effective electron temperature,

 fl(ε)=Nl2πl2BAZ∥e−ε/Te, (2)

where , and is the surface area. According to the normalization condition [here is the density-of-state function for the corresponding subband], .

The approach reported here will be formulated in a quite general way to be applicable for any weak quasi-elastic scattering. As important examples, we shall consider interactions which are well established for SEs on liquid helium. Vapor atoms are described by a free-particle energy spectrum with . For electron interaction with vapor atoms, it is conventional to adopt the effective potential approximation

 Missing \left or extra \right (3)

where is proportional to the electron-atom scattering length (12). Ripplons represent a sort of 2D phonons, and the electron-ripplon interaction Hamiltonian is usually written as

 H(r)int=1√A∑e∑qUq(ze)Qq(bq+b†−q)eiq⋅re, (4)

where , is the ripplon spectrum, , is the ripplon momentum, and are the creation and destruction operators, and is the electron-ripplon coupling (11) which has a complicated dependence on .

For both kinds of SE scattering, the energy exchange at a collision is extremely small, which allows to consider scattering events as quasi-elastic processes. In the case of vapor atoms, it is so because . One-ripplon scattering processes are quasi-elastic () because the wave-vector of a ripplon involved is usually restricted by the condition .

For quasi-elastic processes in a 2D electron system under magnetic field, probabilities of electron scattering are usually found in the self-consistent Born approximation (SCBA) (13). Following Ref. (14), we shall express the scattering probabilities in terms of the level densities at the initial and the final states. Then, Landau level densities will be broadened according to the SCBA (13) or to the cumulant expansion method (15),

 Dl(ε)=−A2π2l2Bℏ∑nImGl,n(ε), (5)

where is the single-electron Green’s function. The later method is a bit more convenient for analytical evaluations because it results in a Gaussian shape of level densities

 −ImGl,n(ε)=√2πℏΓl,nexp[−2(ε−εn)2Γ2l,n]. (6)

Here coincides with the broadening of Landau levels given in the SCBA. For different scattering regimes of SEs, equations for are given in Ref. (11). We shall also take into account an additional increase in due to inter-subband scattering.

Effects considered in this work are important only under the condition which is fulfilled for SEs on liquid helium. Therefore, we shall disregard small corrections to caused by collision broadening because they are proportional to . In other equations, sometimes we shall keep terms proportional to , if they provide important physical properties.

Average scattering probabilities of SEs on liquid helium and even the effective collision frequency can be expressed in terms of the dynamical structure factor (DSF) of the 2D electron liquid (11) . This procedure somehow reminds the theory of thermal neutron (or X-ray) scattering by solids, where the scattering cross-section is expressed as an integral form of a DSF. Without MW irradiation, most of unusual properties of the quantum magnetotransport of SEs on liquid helium are well described by the equilibrium DSF of the 2D electron liquid (16); (11). A multisubband electron system is actually a set of 2D electron systems. Therefore, the single factor is not appropriate for description of inter-subband electron scattering. Luckily, for non-interacting electrons, we can easily find an extension of which could be used in expressions for average scattering probabilities of a multisubband system:

 Sl,l′(q,ω)=2πℏZ∥∑n,n′∫dεe−ε/TeJ2n,n′(xq)× ×ImGl,n(ε)ImGl′,n′(ε+ℏω), (7)

where

 J2n,n′(x)=[min(n,n′)]![max(n,n′)]!x|n−n′|e−x[L|n−n′|min(n,n′)(x)]2,

, and are the associated Laguerre polynomials. The factor contains the level densities at the initial and the final states, and it includes averaging over initial in-plane states. At , this factor coincides with the DSF of a nondegenerate 2D system of non-interacting electrons. Generally, is not the dynamical structure factor of the whole system, nevertheless this function is very useful for description of dissipative processes in presence of MW irradiation.

As a useful example, consider the average inter-subband scattering rate caused by quasi-elastic scattering, which is important for obtaining subband occupancies under the MW resonance (17). Using the damping theoretical formulation (14) and the SCBA (13), can be represented in the following form

 ¯νl→l′=ℏmeA∑qχl,l′(q)Sl,l′(q,ωl,l′), (8)

where () describes electron coupling with scatterers. For SEs on liquid helium, we have two kinds of scatterers: ripplons and vapor atoms. Therefore, . Electron-ripplon scattering gives

 χ(r)l,l′(q)=meℏ3Q2q2Nq∣∣(Uq)l,l′∣∣2≃meTαℏ3q2∣∣(Uq)l,l′∣∣2, \ (9)

where , and . For electron scattering at vapor atoms,

 χ(a)l,l′(q)=ν(a)0pl,l′, \ \ \ ν(a)0=men(3D)a(V(a))2ℏ3B1,1 (10)

where

 \ pl,l′=B1,1Bl,l′, \ \ \ B−1l,l′=L−1z∑Kz∣∣(eiKzze)l′,l∣∣2,

is the height above the liquid surface, is the density of vapor atoms, and is the projection of the vapor atom wave-vector. The represents the SE collision frequency at vapor atoms for .

The generalized factor of a multi-subband 2D electron system will be used throughout this work because its basic property

 Sl,l′(q,−ω)=e−ℏω/TeSl′,l(q,ω) (11)

allows us straightforwardly to obtain terms responsible for negative dissipation. This property follows from the detailed balancing for quasi-elastic processes, , and also directly from the definition of Eq. (7). Using Gaussian level shapes of Eq. (6), one can find

 Sl,l′(q,ω)=2π1/2ℏZ∥∑n,n′J2n,n′(xq)Γl,n;l′,n′e−εn/TeIl,n;l′,n′(ω), (12)

where , and

 Il,n;l′,n+m(ω)= =exp⎡⎢⎣−(ℏω−mℏωc−Γ2l,n/4TeΓl,n;l′,n+m)2+Γ2l,n8T2e⎤⎥⎦. (13)

The , as a function of frequency, has sharp maxima when  equals the in-plane excitation energy . The parameter describes broadening of these maxima. Eqs. (12) and (13) satisfy the condition of Eq. (11). Terms of the order of entering the argument of Eq. (13) could be omitted, as it was done for , because even the linear in term provides us the necessary condition of Eq. (11). Anyway, our final results will be represented in forms which allow to disregard even the linear in term entering .

Consider the decay rate of the first excited subband . Under typical experimental condition, is much smaller than . Therefore, most of terms entering are exponentially small and can be disregarded. The exceptional terms satisfy the condition , where is an integer nearest to . In this notation,

 \ ¯ν2→1=∞∑n=0e−εn/TeZ∥ℏωcβ2,n;1,n+m∗π1/2Γ2,n;1,n+m∗× ×exp{−ℏ2(ω2,1−m∗ωc)2Γ22,n;1,n+m∗}, (14)

where

 βl,n;l′,n+m=∫∞0χl,l′J2n,n+m(xq)dxq.

For electron scattering at vapor atoms, which coincides with. In the case of electron-ripplon scattering, has a more complicated expression due to a particular form of entering the definition of . The is a 1/B-periodic function. It has sharp maxima when equals an integer. In the argument of the exponential function of Eq. (14), we have disregarded terms which are small for .

Transition rates determine subband occupancies under the MW resonance. At low electron temperatures, the two-subband model is applicable, and the rate equation gives

 ¯n2¯n1=r+e−Δ2,1/Te¯ν2→1r+¯ν2→1, \ (15)

where is the stimulated absorption (emission) rate due to the MW field, and . Thus, under the MW resonance, magnetooscillations of lead to magnetooscillations of subband occupancies and . For further analysis, it is important that MW excitation provides the condition , which is the main cause of negative momentum dissipation.

## Iii Magnetoconductivity under resonance MW irradiation

Consider now an infinite isotropic multisubband 2D electron system under an in-plane dc-electric field, assuming arbitrary occupancies of surface subbands induced by the MW resonance. In the linear transport regime, the average friction force acting on electrons due to interaction with scatterers is proportional to the average electron velocity . This relationship can be conveniently written as , where the proportionality factor represents an effective collision frequency which depends on and, generally, on electron density. The is balanced by the average Lorentz force , which yields the usual Drude form for the electron conductivity tensor , where the quasi-classical collision frequency is substituted for  (16); (11).

The effective collision frequency can be obtained directly from the expression for the average momentum gained by scatterers per unite time. Usually, to describe momentum relaxation, one have to obtain deviations of the in-plane electron distribution function from the simple form of Eq. (2) induced by the dc-electric field. For the highly correlated 2D system of SEs on liquid helium under magnetic field, this problem was solved in a general way, assuming that in the center-of-mass reference frame the electron DSF has its equilibrium form . In the laboratory frame, its frequency argument acquires the Doppler shift due to Galilean invariance (16); (11). This approach is similar to the description of electron transport by a velocity shifted Fermi-function of the kinetic equation method, where is found from the momentum balance equation. The same properties can be ascribed to the generalized factor (10) .

Here, we consider a different way, taking into account that , as well as the momentum gained by scatterers, can be evaluated in any inertial reference frame. We choose the reference frame fixed to the electron liquid center-of-mass, because in it the in-plane distribution function of highly correlated electrons () has its simplest form of Eq. (2), and the generalized factor has it equilibrium form of Eq. (7). Then, can be considered as the drag due to moving scatterers. At the same time, distribution functions of scatterers which are not affected by external fields can be easily found according to well-known rules.

In the electron liquid center-of-mass reference frame, the in-plane spectrum of electrons is close to the Landau spectrum, because the driving electric field is nearly zero, at least for . At the same time, in this frame the ripplon excitation energy is changed to , because the gas of ripplons moves as a whole with the drift velocity equal to . The same Doppler shift correction appears for the energy exchange in the case of electron scattering at vapor atoms, even for the limiting case (impurities which are motionless in the laboratory reference frame). In the electron center-of-mass reference frame, vapor atoms move with the velocity and hit electrons which results in the energy exchange .

Describing electron-ripplon scattering probabilities in terms of the equilibrium factor , as discussed above, contributions to the frictional force from creation and destruction processes can be found as

 Missing or unrecognized delimiter for \left (16)

It is clear that disregarding the Doppler-shift correction in this equation yields zero result. This correction enters the ripplon distribution function and the frequency argument of the factor . In the linear transport regime, the Doppler-shift correction entering the ripplon distribution function is unimportant. This can be seen directly from Eq. (16): setting in the frequency argument of gives zero result for . Therefore, in this equation one can substitute for defined in Eq. (9). We can also disregard in the frequency argument of . Then, interchanging the running indices of the second term in the square brackets, and using the basic property of given in Eq. (11), Eq. (16) can be represented as

 Fscat=−Neℏ2meA∑l,l′∑qℏqχl,l′(q)Sl,l′(q,ωl,l′+q⋅Vav) ×(¯nl−¯nl′e−Δl,l′/Tee−ℏq⋅Vav/Te), (17)

where was defined in Eq. (9). This equation has the most convenient form for expansion in .

A similar equation for can be found considering electron scattering at vapor atoms. Evaluating momentum relaxation rate, one can disregard which represents the energy exchange at a collision in the laboratory reference frame. In the center-of-mass reference frame, Doppler-shift corrections enter the vapor atom distribution function and the frequency argument of the factor due to the new energy exchange at a collision . The correction entering  is unimportant because of the normalization condition: . Therefore, we have

 Fscat=−Neℏν(a)0meA∑l,l′¯nlpl,l′× ×∑qℏqSl,l′(q,ωl,l′+q⋅Vav). (18)

In order to obtain the form of Eq. (17), we represent the right side of Eq. (18) as a sum of two identical halves and change the running indices in the second half: and . Then, the basic property of yields Eq. (17)  with , where is from Eq. (10).

Thus, Eq. (17) is applicable for both scattering mechanisms. In the general case, . The effective collision frequency under magnetic field can be found expanding Eq. (17) in up to linear terms. We shall represent as a sum of two different contributions: . The normal contribution originates from the expansion of the exponential factor . In turn,  can be represented as a sum of contributions from intra-subband and inter-subband scattering . The sums of take account of all . It is useful to rearrange terms with () by interchanging the running indices , and using the basic property of . Then, we have

 νN,intra=ℏω2c4πTe∑l¯nl∫∞0xqχl,l(q)Sl,l(q,0)dxq, (19)
 νN,inter=ℏω2c4πTe∑l>l′(¯nl+¯nl′e−Δl,l′/Te)× ×∫∞0xqχl,l′(q)Sl,l′(q,ωl,l′)dxq. (20)

The is always positive. In the limiting case of a one-subband 2D electron system (), Eq. (19) reproduces the known relationship between the effective collision frequency and the electron DSF (11). In the parentheses of Eq. (20), the first term is due to scattering from to , while the second term describes the contribution of scattering back from to . It should be noted that the forms of Eqs. (19) and (20) allow to simplify of Eq. (12) by disregarding small corrections proportional to and entering defined by Eq. (13).

The anomalous contribution to the effective collision frequency can be found from Eq. (17) expanding in , and setting in the parentheses. In this case, to rearrange terms with (), we shall use the property

 S′l′,l(q,−ω)=−e−ℏωTeS′l,l′(q,ω)+ℏTee−ℏωTeSl,l′(q,ω)≃
 ≃−e−ℏωTeS′l,l′(q,ω). (21)

Here , and the last transformation assumes that . Interchanging the running indices of terms with and using Eq. (21), can be found as

 \ \ \ νA=ω2c2π∑l>l′(¯nl−¯nl′e−Δl,l′/Te)× ×∫∞0xqχl,l′(q)S′l,l′(q,ωl,l′)dxq. (22)

As compared to of the normal contribution, here the second term in parentheses has the opposite sign. Therefore, for usual Boltzmann distribution of subband occupancies, . The anomalous contribution appears only when , which occurs under the MW resonance condition .

In the form of Eq. (22), it is possible to use a simplified expression

 S′l,l′(q,ωl,l′)≃−2π1/2ℏZ∥∑n,n′J2n,n′(xq)Γl,n;l′,n′e−εn/Teexp⎧⎨⎩−[Δl,l′−(n′−n)ℏωc]2Γ2l,n;l′,n′⎫⎬⎭2ℏ2[ωl,l′−(n′−n)ωc]Γ2l,n;l′,n′, (23)

which disregards terms proportional to and . From Eqs. (22) and (23) one can see that at , the sign of is opposite to the sign of . Therefore, when the magnetic field is slightly lower the commensurability condition (here is an integer), which agrees with the experimental observation for minima of .

For further analysis, it is convenient to introduce

 \ \ λl,n;l′,n′=∫∞0xqχl,l′(q)J2n,n′(xq)dxq .\ \ (24)

When referring to a particular scattering mechanism, we shall use a superscript, . Consider a two-subband model which is valid at low enough electron temperatures. Using the new definitions given above, the normal contribution to the effective collision frequency can be represented as

 νN,intra=∞∑n=0e−εn/Te(ℏωc)22√πTeZ∥[¯n1λ1,n;1,nΓ1,n+¯n2λ2,n;2,nΓ2,n], (25)
 νN,inter=(¯n2+¯n1e−Δ2,1/Te)(ℏωc)22√πTe∞∑n=0e−εn/TeZ∥× ×λ2,n;1,n+m∗Γ2,n;1,n+m∗exp[−ℏ2(ω2,1−m∗ωc)2Γ22,n;1,n+m∗], (26)

where is the function of defined in the previous Section. The and have magnetooscillations of two kinds. Oscillations of are quite obvious, because quasi-elastic inter-subband scattering increases sharply at the commensurability condition: . The shape of these peaks is symmetrical with respect to the point . It is formed by the interplay of the exponential factor, having for the broadening parameter, and the line-shapes of the subband occupancies. It should be noted that at low electron temperatures, is exponentially small. The intra-subband scattering contribution oscillates with in an indirect way because of oscillations in level occupancies and induced by oscillations in the decay rate , according to Eqs. (14) and (15). These oscillations have also a symmetrical shape whose broadening is affected by the relation between and .

Magnetooscillations of have a completely different shape:

 \ νA=−(¯n2−¯n1e−Δ2,1/Te)(ℏωc)2√π× ×∞∑n=0e−εn/TeZ∥λ2,n;1,n+m∗Γ22,n;1,n+m∗× ×exp[−ℏ2(ω2,1−m∗ωc)2Γ22,n;1,n+m∗]2ℏ(ω2,1−m∗ωc)Γ2,n;1,n+m∗, (27)

In the ultra-quantum limit , terms with entering Eq. (27) can be omitted, which allows to describe magneto-oscillations of in an analytical form. In contrast with oscillations of the normal contribution , in the vicinity of the commensurability condition, is an odd function of .

Thus, the effective collision frequency and magnetoconductivity of SEs are found for any given electron temperature. In order to obtain as a function of the magnetic field, it is necessary to describe energy relaxation of SEs for arbitrary subband occupancies.

## Iv Energy dissipation

It is instructive to analyze another important example of negative dissipation which can be induced by the MW resonance. Consider the energy loss rate of a multisubband 2D electron system due to interaction with scatterers. In this case, there are no complications with the dc-driving electric field or with the Doppler shifts which can be set to zero. This analysis will be important also for description of electron heating due to decay of the SE state excited by the MW.

The energy loss rate per an electron due to one-ripplon creation and destruction processes can be represented in terms of quite straightforwardly:

 ˙W=−1ℏ2A∑qℏωqQ2q(Nq+1)∑l,l′∣∣(Uq)l,l′∣∣2×
 ×[¯nlSl,l′(q,ωl,l′−ωq)−¯nle−ℏωq/TSl,l′(q,ωl,l′+ωq)]. (28)

Interchanging the running indices () in the second term, and using the basic property of given in Eq. (11), the terms entering the square brackets can be rearranged as

 Sl,l′(q,ωl,l′−ωq)[¯nl−¯nl′e−Δl,l′/Tee−ℏωq(1/T−1/Te)]. (29)

Since the processes considered here are quasi-elastic, we can expand this equation in and represent as a sum of two different contributions: . The normal energy loss rate is proportional to , which is a measure of deviation from the equilibrium,

 ˙WN=−(Te−T)ℏmeA∑q∑l,l′¯nl′e−Δl,l′/Te~χ(r)l,l′Sl,l′(q,ωl,l′), (30)

Here

 ~χ(r)l,l′=meq2ρℏTe∣∣(Uq)l,l′∣∣2.

This contribution originates from expansion of the exponential function in .

It is conventional to represent the energy loss as , where