The fine structure of microwave-induced magneto-oscillations in photoconductivity of the two-dimensional electron system formed on a liquid-helium surface

# The fine structure of microwave-induced magneto-oscillations in photoconductivity of the two-dimensional electron system formed on a liquid-helium surface

## Abstract

The influence of the inelastic nature of electron scattering by surface excitations of liquid helium (ripplons) on the shape of magnetoconductivity oscillations induced by resonance microwave (MW) excitation is theoretically studied. The MW field provides a substantial filling of the first excited surface subband which sparks off inter-subband electron scattering by ripplons. This scattering is the origin of magneto-oscillations in the momentum relaxation rate. The inelastic effect becomes important when the energy of a ripplon involved compares with the collision broadening of Landau levels. Usually, such a condition is realized only at sufficiently high magnetic fields. On the contrary, the inelastic nature of inter-subband scattering is shown to be more important in a lower magnetic field range because of the new enhancement factor: the ratio of the inter-subband transition frequency to the cyclotron frequency. This inelastic effect affects strongly the shape of conductivity oscillations which acquires an additional wavy feature (a mixture of splitting and inversion) in the vicinity of the level-matching points where the above noted ratio is close to an integer.

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

## I Introduction

Microwave-induced magnetoconductivity oscillations, whose minima evolve into novel zero-resistance states at high microwave (MW) power, were first observed in degenerate semiconductor two-dimensional (2D) electron systems (1); (2); (3). In these experiments, the magnetic field was directed normally to the electron layer, and the MW frequency was substantially larger than the cyclotron frequency . These observations have attracted much theoretical interest and sparked invention of a wide variety of theoretical mechanisms (4); (5); (6) intended to explain remarkable 1/B-periodic oscillations and the appearance of zero-resistance states.

Recently, similar 1/B-periodic magnetoconductivity oscillations were observed in a particulary simple nondegenerate multi-subband 2D electron liquid formed on the free surface of liquid   (7); (8). Their minima also evolve into zero-resistance states at high MW power. Still, there is an important difference between results reported for surface electrons (SEs) in liquid helium and those obtained for semiconductor systems. In experiments (1); (2); (3), is quite arbitrary: . To observe magneto-oscillations in the electron system formed on the liquid-helium surface it is necessary that the MW frequency be equal to the resonance frequency for electron excitation to the next surface subband. The sequence (here ) describes the energy spectrum of electron subbands formed at the free surface of liquid helium because of the interplay of the attractive image force and the potential barrier at the interface (9); (10). In the limit of low holding electric fields directed normally to the interface, for liquid and for liquid . Thus, in experiments on semiconductor systems, higher electron subbands were not excited by the MW field, whereas in experiments with SEs (7); (8) there is a substantial fraction of electrons occupied the first excited subband because of the MW resonance.

The theory explaining MW-resonance-induced magnetoconductivity oscillations observed for SEs on liquid helium was worked out (11); (12) using the quasi-elastic approximation for electron scattering by ripplons and vapor atoms. This approximation assumes that the energy exchange at a collision is much smaller than the collision broadening of Landau levels.

The origin of magneto-oscillations and negative conductivity effects induced by resonance MW excitation can be seen already from a simple analysis of the energy conservation for usual inter-subband electron scattering (not involving photons) in the presence of the uniform driving electric field . According to this analysis, for the electron spectrum

 εl,n,X=Δl+ℏωc(n+1/2)+eE∥X (1)

(here is the center coordinate of the cyclotron orbit), the decay of an excited subband (, , and ) results in displacements of the orbit center

 X′−X=ℏωceE∥(Δl,l′ℏωc−m∗) (2)

proportional to the quantity . This quantity changes its sign at the level-matching points defined by the condition (here is an integer). At , the displacement and, therefore, the decay of the excited subband is accompanied by electron scattering against the driving force.

Somehow, Eq. (2) resembles the displacement mechanism of the negative conductivity effect discussed broadly for semiconductor systems (13) where in a similar equation the photon quantum stands instead of . In that theory, the sign-changing correction to is due to radiation-induced disorder-assisted current, and the photon energy enters the energy conservation law. The important point is that here photons are not involved in inter-subband scattering directly and the condition found for decay processes is not sufficient for obtaining negative corrections to the momentum relaxation rate. Naturally, a reverse electron scattering from the ground subband to the excited subband results in , and the negative conductivity effect is fully compensated if the electron system is in equilibrium, and fractional occupancies of surface subbands obey the condition (here is the electron temperature).

The important role of the resonant MW field is to provide an extra filling of the excited subband to break the balance of inter-subband scattering mentioned above. For example, the decay of the first excited subband () to the ground subband () caused by quasi-elastic scattering is possible only if is close to the level-matching points , because is restricted by the magnetic length. If is substantially away from these level-matching points, the quasi-elastic decay is impossible and because of MW excitation. Under a weak driving field, the width of regions of the magnetic field near , where the excited subband decays quasi-elastically, is determined by the collision broadening of the corresponding Landau levels. Within these regions, the momentum relaxation rate of SEs caused by inter-subband scattering (11); (12)

 νinter∝−(ω2,1ωc−m∗)(¯n2−e−Δ2,1/Te¯n1). (3)

Therefore, the condition is crucial for the appearance of negative corrections to the momentum relaxation rate and to magnetoconductivity . An increase in caused by trivial heating of SEs obviously cannot lead to the sign-changing term.

The above noted analysis assumes that the ripplon energy can be neglected as compares to typical electron energies. In the absence of the magnetic field, this is the conventional approximation because for ripplons involved in scattering events. Under a magnetic field applied perpendicular to the electron layer, there is an additional energy parameter which describes the width of the single-electron density of states: the collision broadening of Landau levels . For SEs on liquid helium, Landau levels are extremely narrow: . Wave vectors of ripplons involved in electron scattering [here is the magnetic length], and the energy exchange increases with faster than which is approximately proportional to . Therefore, in a high magnetic field range, depending on temperature (), becomes comparable with , and electron scattering is suppressed.

Experimental observation (14); (15) indicates that the inelastic effect on the quantum magnetotransport becomes substantial at and , and the suppression of is the stronger the higher magnetic field is applied. Simple estimates allow to assume that similar conditions can be realized in an experiment on magneto-oscillations in photoconductivity of SEs on liquid , because the corresponding excitation frequency is high, and the level-matching points , if . In this case, the inelastic effect can cause additional variations of the shape of magnetoconductivity oscillations in the vicinity of the level-matching points , which could be used for experimental identification of the microscopic mechanism of zero-resistance states and the resonant photovoltaic effect discovered recently (16).

In this work, we report the theory of magnetoconductivity oscillations induced by resonance MW excitation which takes into account the inelastic nature of the decay of excited subbands caused by electron-ripplon interaction. We show that for inter-subband scattering the inelastic effect displays differently as compared to the equilibrium magnetotransport in a single subband. In our treatment, the maximum of the decay rate of an excited subband is naturally split near the level-matching points because of one-ripplon creation and destruction processes. The unusual thing is that this inelastic effect increases with , and therefore extends itself into the lower magnetic field range .

We show also that the inelastic effect on the momentum relaxation rate of the electron layer caused by inter-subband scattering cannot be reduced to simple splitting similar to that of the decay rate. In magnetoconductivity curves, this effect displays itself as a combination of splitting and inversion. As a result, develops a new remarkable wavy feature in the vicinity of the level-matching points. The influence of both mutual electron interaction and electron heating on the new fine structure of MW-induced magneto-oscillations is also analyzed.

## Ii Inelastic inter-subband scattering and momentum relaxation

The most interesting features of MW-induced magnetoconductivity oscillations such as zero-resistance states are observed (7); (8) in the low temperature range () where SEs are predominantly scattered by capillary wave quanta (ripplons). Ripplons represent a sort of 2D phonons with an unusual spectrum , where and are the surface tension and mass density of liquid helium, correspondingly. Therefore, the Hamiltonian of electron-ripplon interaction is similar to the Hamiltonian of electron-phonon interaction in solids

 H(e−R)int=∑e∑qUq(ze)Qq(bq+b†−q)eiqre, (4)

where and are destruction and creation operators, , and is the electron-ripplon coupling whose properties and matrix elements are well defined in the literature (15); (17).

To describe quantum magnetotransport of a 2D electron gas it is conventional to use the self-consistent Born approximation (SCBA) theory (18) or the linear response theory (19) with the proper approximation for the electron density-of-states function. Unfortunately, these approaches cannot be applied directly to the SE system under resonance MW excitation. In these theories, conductivity is an equilibrium property of the system, whereas here we need to describe conductivity of the system which is far away from its equilibrium state. In our conductivity treatment, we intend also to include strong Coulomb interaction between electrons whose average potential energy can be much higher then the average kinetic energy. For this purposes, it is necessary to use an extension of the SCBA theory applicable for arbitrary subband occupancies and .

Firstly, we note that the well-known results of the SCBA theory and Kubo equations for magnetoconductivity of the 2D electron gas can be reproduced in a quite direct way by simple evaluation of the momentum gained by scatterers (15), if scattering probabilities of the Born approximation are taken in the proper form which includes the contribution from high order terms (self-energy effects). This kind of probabilities were actually given already in the Kubo theory (19). Here we express these probabilities through a quite general correlation function of the multi-subband 2D electron system which preserves basic equilibrium properties of the in-plane motion and, at the same time, is independent of subband occupancies.

Consider the average probability of both intra and inter-subband scattering () which is accompanied by the momentum exchange caused by ripplon destruction and creation . Conventional Born approximation yields

 ¯ν(−)l,l′(q)=2πℏAu2l,l′⟨∑n′,X′∣∣(eiqre)l,X;l′,X′∣∣2×
 ×δ(εn−εn′+Δl,l′+ℏωq+eE∥X−eE∥X′)⟩in, (5)

where represents Landau levels, means averaging over initial in-plane states for the given surface subbad , and we have introduced

 \ u2l,l′(xq)=Aℏ2NqQ2q∣∣(Uq)l,l′∣∣2≃TL2B4αℏ2xq∣∣(Uq)l,l′∣∣2 (6)

as the function of the dimensionless parameter . For , the distribution function of ripplons . In the following, the surface area will be set to unity. It is well known that can be written as , where is a function of the absolute value of the 2D wave-vector. The exact expression for is given in the literature (for recent examples, see (20); (17)).

According the relationship the quantity to be averaged in Eq. (5) does not depend on , and, therefore, Eq. (5) actually contains the averaging over discrete Landau numbers only. It is natural to assume that a weak dc driving field does not change electron distribution over Landau levels, and one can use the distribution function for the averaging operation. This is quite clear in the absence of scatterers, because under the magnetic field a driving electric field can be eliminated by a proper choice of the inertial reference frame: . Moreover, if there is no a driving electric field in the laboratory frame, it appears in any other inertial reference frame. At a low drift velocity of the electron system, quasi-elastic scattering cannot change the population of Landau levels. The above given statement is also verified by the comparison of the results obtained in the treatment considered here with the well-known results of the conventional SCBA at zero MW power.

Following the procedure described in the linear response theory (19), and taking into account that (here is the absolute value of the Hall velocity), Eq. (5) can be transformed into the form containing level densities of the initial and final states

 \ \ \ ¯ν(−)l,l′(q)=u2l,l′Sl,l′(q,ωl,l′+ωq+qyVH),\ \ (7)

where

 Sl,l′(q,Ω)=2πℏZ∥∑n,n′J2n,n′(q)×
 ×∫dεe−ε/TeImGl,n(ε)ImGl′,n′(ε+ℏΩ). (8)

Here is the single-electron Green’s function of the corresponding subband whose imaginary part is a substitute of . We retain the index keeping in mind further broadening due to interaction with scatterers because its strength is different for different surface subbands. Similar equation can be found for ripplon creation processes:

 ¯ν(+)l,l′(q)=eℏωq/Tu2l,l′Sl,l′(q,ωl,l′−ωq+qyVH). (9)

At the function coincides with the dynamic structure factor (DSF) of a nondegenerate 2D electron gas. It should be noted that the above given equations resemble scattering cross-sections of thermal neutrons and X-rays in solids (21). Here scatterers (ripplons) play the role of particle fluxes whereas the electron layer represents a target.

The self-energy effects (high order terms), which are very important for 2D electron systems under a quantizing magnetic field, are taken into account by inclusion of the collision broadening of Landau levels of a given surface subband () according to the cumulant approach (22):

 −ImGl,n(ε)=√2πℏΓl,nexp[−2(ε−εn)2Γ2l,n]. (10)

Thus, similarly to the Kubo presentation (19), the average probabilities of electron scattering with the momentum exchange are expressed in terms of density-of-state functions of the initial and final states broadened because of interaction with scatterers. For the Gaussian shape of level densities, the integral entering the definition of can be evaluated analytically (12). Moreover, this kind of a level density represents a good starting point for obtaining an analytical form of for the multi-subband 2D Coulomb liquid (17).

The Eq. (8) is a useful generalization of the DSF for the multi-subband 2D electron system because it preserves the important property of the equilibrium of the in-plane motion

 Sl′,l(q,−Ω)=e−ℏΩ/TeSl,l′(q,Ω), (11)

and, at the same time, it does not depend on , which allows to describe momentum relaxation for arbitrary subband occupancies. The property of Eq. (11) simplifies evaluations of the momentum relaxation rate. For example, using this property average probabilities for the reverse scattering processes discussed in the Introduction can be transformed into the same quantities of the direct processes:

 ¯ν(+)l′,l(q)=e−Δl,l′/Teeℏωq(1/T−1/Te)eℏqyVH/Te¯ν(−)l,l′(−q), (12)
 ¯ν(−)l′,l(q)=e−Δl,l′/Teeℏωq(1/Te−1/T)eℏqyVH/Te¯ν(+)l,l′(−q). (13)

We shall use these relationships in the following analysis.

The introduced above quantities represent useful assemblies to construct major relaxation rates of the multi-subband 2D electron system under a quantizing magnetic field such as the decay rate of an excited subband and the momentum relaxation rate due to inter-subband scattering. It is important that they preserve peculiarities of quantum magnetotransport in two-dimensions. They include the self-energy effects eliminating magnetoconductivity singularities, the effect of the driving electric field and, after the following generalization, can even include strong Coulomb forces acting between electrons.

### ii.1 The many-electron effect

Even for the lowest electron areal density (about ) used in the experiments on SEs in liquid helium (7); (8), Coulomb interaction between SEs cannot be neglected. For example, at the average interaction energy per an electron is much larger than the average kinetic energy (). The generalized DSF of the multi-subband 2D electron system applicable for such conditions was found in Ref. (17):

 Sl,l′(Ω)=2√πℏZ∥∑n,n′J2n,n′~Γl,n;l′,n′exp[−εnTe−Dl,n;l′,n′(Ω)], (14)

where

 Dl,n;l′,n′=ℏ2(Ω−m∗ωc−Γ2l,n+xqΓ2C4Teℏ)2~Γ2l,n;l′,n′−Γ2l,n8T2e , (15)
 ~Γ2l,n;l′,n′(xq)=Γ2l,n+Γ2l′,n′2+xqΓ2C , (16)

, and . The quantity represents the typical quasi-uniform electric field of other electrons acting on a given electron because of thermal fluctuations (23). Here and in some following equations we do not show explicitly the dependence on of functions , etc., in order to shorten lengthy equations. In the limiting case , Eq. (14) transforms into the generalized DSF of the multi-subband 2D system of noninteracting electrons.

The function has sharp maxima when is close to Landau excitation energies . These maxima are broadened because of electron interaction with scatterers and because of the fluctuational electric field . It is important that Coulomb broadening of the DSF is not equivalent to the collision broadening because it depends on through the dimensionless parameter . The fluctuational field does not broaden the single-electron level densities, because, as noted above, it can be eliminated by a proper choice of the inertial reference frame moving along the layer (15).

Small frequency shifts in the general expression for play very important role: they provide the basic property of Eq. (11). The small shift can be neglected only for substantially positive values of . Therefore, it is convenient to transform terms containing with negative into forms with positive employing the relationship of Eq. (11). It worth noting also that the Coulomb shift in the frequency argument of the DSF increases with and faster than the Coulomb broadening which curiously affects positions of magnetoconductivity extremes (17). Therefore, we shall retain it in even for substantially positive values of the frequency argument.

### ii.2 The decay rate of an excited subband

The decay rate of an excited subband due to electron scattering down to a lower subband is easily expressed in terms of and :

 ¯νl→l′=∑q[¯ν(+)l,l′(q)+¯ν(−)l,l′(q)]. (17)

Here, we can neglect the small corrections in the frequency argument of the generalized DSF entering Eqs. (7) and (9). Then, using Eqs. (12) and (13), one can see that for inelastic scattering the detailed balancing is fulfilled only if the electron temperature coincides with the temperature of the environment. Anyway, because of the condition discussed above, the detailed balancing is approximately valid even at high electron temperatures.

When evaluating in the ultra-quantum limit () one can use the approximate expression for the generalized DSF,

 Sl,l′(q,ωl,l′±ωq)≃2√πℏ∞∑m=0xmqe−xqm!~Γl,0;l′,mI(±)l,l′;m(xq), (18)

applicable for positive values of the frequency argument. Here we introduce functions

 I(±)l,l′;m(xq)=exp{−[R(±)l,l′;m(xq)]2}, (19)

and

 R(±)l,l′;m(xq)=(ℏωl,l′±ℏωq−mℏωc−xqΓ2C/4Te)~Γl,0;l′,m, (20)

It should be noted that in Eqs. (19) and (20) we have omitted small frequency shifts of the order , because the frequency argument of in Eq. (18) is substantially positive for decay processes.

Employing the above given notations can be transformed into

 ¯νl→l′=T4√παℏ∞∑m=11m!∫∞0∣∣(Uq)l,l′∣∣2~Γl,0;l′,mxm−1qe−xq×
 ×[I(−)l,l′;m(xq)+I(+)l,l′;m(xq)]dxq. (21)

From this equation it is quite clear that the inelastic effect splits the decay maximum of the elastic theory into two maxima when becomes comparable with the broadening . It is very important that the position of the maximum of the function entering the integrand of Eq. (21) increases strongly with the level-matching number , whereas is nearly independent of if electron density is sufficiently low. This leads to unexpected enhancement of the inelastic effect in the low magnetic field range where is larger.

The above stated is illustrated in Fig. 1 where is shown as the function of for . Conditions of the figure are chosen to be such ( and ) that the splitting of the decay maximum is absent at (), however, it appears near some lower level-matching points . At the lowest (), the splitting disappears again because of the Coulombic correction to which also increases strongly with .

### ii.3 Subband occupancies

Under the condition of the MW resonance , the stimulated photon absorption (emission) rate , where is the half-width of the resonance, and is the Rabi frequency proportional to the amplitude of the MW field . In dynamic equilibrium, the fractional occupancies are found from the time-independent rate equation. In the framework of the two-subband model (), the solution of the rate equation for the relative occupancy has the following form (24)

 ¯n2¯n1=rmw+e−Δ2,1/Te¯ν2→1rmw+¯ν2→1. (22)

According to this equation the -periodic dependence of the decay rate with sharp maxima in the vicinity of the level-matching points induces a -periodic dependence of the fractional occupancies and .

In experiments (7); (8), the half-width of the MW resonance was limited by the inhomogeneous broadening (). We shall use this estimate in the following numerical evaluations. For typical , the results of calculation of are presented in Fig. 2. Variations of are shown in the vicinity of the level-matching point vs the parameter . Far away from the level-matching point, is nearly zero and, therefore, . In the vicinity of , the occupancy drops according to the sharp increase in the decay rate. The inelastic effect broadens the minima, and leads to small local maxima at the level-matching points. The local peak at becomes more pronounced with cooling, as shown in this figure by the dash-dotted line.

For SEs on liquid helium at , there is a spectrum crowding: at . Therefore, at low holding fields, there is a good chance to meet the MW resonance condition for three surface subbands simultaneously: where is substantially larger than . In wide ranges between the level-matching points introduced above for the first excited subband, the decay rate is very small, and the occupancy of the third subband can also satisfy the condition necessary for the appearance of the sign-changing correction to . Since , the new level-matching condition defined for the decay from to can be rewritten as

 ω2,1ωc=m∗/2. (23)

Thus, oscillatory features of could appear also at fractional values of the ratio .

### ii.4 Magnetoconductivity variations induced by the MW resonance

Using the quantities and the frictional force acting on the whole electron system can be evaluated directly: at first we multiply these average probabilities by , and then perform summation over all and the subband numbers and . Since we intend to obtain the conductivity of interior electrons ignoring edge effects, consider an uniform and infinite electron layer. In this case, the kinetic friction is antiparallel and proportional to the current (25); (26); (15), which can be written as , where is the average electron velocity. The proportionality factor represents an effective collision frequency, because balancing  and the average Lorentz force yields the conventional Drude form for the conductivity tensor with standing instead of the semi-classical collision frequency. It should be emphasized that here is not a semi-classical quantity because it depends on and .

The simplest way to obtain is to consider the component and to assume that the magnetic field is strong enough (): . Then, we can represent as

 νeff=1meVH∑l,l′¯nl∑qℏqy[¯ν(+)l,l′(q)+¯ν(−)l,l′(q)]. (24)

According to (7) and (9), the right side of this relationship is actually a nonlinear function of . Therefore, one have to expand it in . From the first glance at given in Eq. (14) one may conclude that is the main expansion parameter. Still, an accurate analysis employing the relationship of Eq. (11) indicates that for intra-subband scattering at the actual expansion parameter equals . Therefore, before proceeding with the expansion of the right side of Eq. (24) we shall transform it into the form containing with positive frequency arguments only.

For intra-subband scattering , the basic property of Eq. (11) applied to yields

 νintra=1meVH∑l¯nl∑qu2l,lℏqySl,l(ωq+qyVH)×
 ×[1−eℏωq(1/T−1/Te)e−ℏqyVH/Te]. (25)

Here we have changed the sign of the summation index in the term with . At and low , the expression in the square brackets is proportional to . Therefore, one can neglect in the frequency argument of the DSF. This confirms the above given statement that for intra subband scattering is the main expansion parameter.

In the general case (), one have to expand the exponential function in and the DSF in as well. This gives two terms ():

 ν(0)intra=1meTe∑l¯nl∑qu2l,lℏ2q2ySl,l(q,ωq), (26)
 ν(1)intra=−Te−TmeTTe∑l¯nl∑qu2l,lℏ2q2yωqS′l,l(q,ωq). (27)

Here and below . The term coincides with the well-known result obtained previously for intra-subband scattering (15). In the limit , it transforms into the result of the conventional SCBA theory (18). The second term appears only for when the scattering is substantially inelastic. Therefore, it is not an equilibrium property of the system. In the absence of the MW field, it can appear only as a nonlinear correction.

Consider now the contribution from inter-subband scattering. In Eq. (24), one can transform terms with negative () into the forms with positive using Eqs. (12) and (13). Thus, we have

 νinter=ℏmeVH∑l>l′∑qqy×
 {[¯nl−¯nl′e−Δl,l′/Teeℏωq(1/Te−1/T)e−ℏqyVH/Te]¯ν(+)l,l′(q)+
 +[¯nl−¯nl′e−Δl,l′/Teeℏωq(1/T−1/Te)e−ℏqyVH/Te]¯ν(−)l,l′(q)}. (28)

The sign ”-” of the second term in the square brackets appears because of the change of the summation index for terms containing and . Expanding this equation in we find that linear in terms of the square brackets yield a positive (normal) contribution

 ν(N)inter=ℏ2meTe∑l>l′¯nl′e−Δl,l′/Te∑qu2l,l′q2y×
 ×{Sl,l′(ωl,l′−ωq)+Sl,l′(ωl,l′+ωq)}. (29)

Here we used the condition . If the electron system is not heated high (), this contribution is exponentially small.

Under MW excitation, the major contribution to comes from the expansion of the DSF entering and :

 ν(A)inter=1me∑l>l′∑qℏq2yu2l,l′×
 {[¯nl−¯nl′e−Δl,l′/Teeℏωq(1/Te−1/T)]×
 ×eℏωq/TS′l,l′(ωl,l′−ωq)+
 +[¯nl−¯nl′e−Δl,l′/Teeℏωq(1/T−1/Te)]S′l,l′(ωl,l′+ωq)}. (30)

These anomalous terms are proportional to the derivative of . The Eq. (30) can be simplified considering the two-subband model with and . Then, we obtain

 ν(A)inter=ℏme(¯n2−¯n1e−Δ2,1/Te)×
 ×∑qu22,1q2y[S′2,1(ω2,1−ωq)+S′2,1(ω2,1+ωq)]. (31)

From this equation, one can see that the anomalous contribution is proportional to and to the sign-changing terms , as expected from the qualitative analysis given in the Introduction.

In the elastic theory, changes its sign once in the vicinity of each . When the inelastic effect is substantial, the expression in the square brackets is a derivative of the function which has two maxima and one minima near each . Therefore, in the vicinity of a level-matching point, the anomalous contribution caused by inelastic inter-subband scattering changes its sign three times.

It should be noted that here we use slightly different definitions of the normal and anomalous contributions than that given before (12); (17). Now we apply labels N and A to the corresponding expressions which are transformed into the form containing the summation over only. For such definition, becomes substantially smaller, and does not contain small terms which are not proportional to with . In the limiting case , the sum of