Microscopic stability analysis for homogeneous electric fields in superlattices under quantizing magnetic fields

Physica B 334 (2003) 413–424

Microscopic stability analysis for homogeneous electric fields in superlattices under quantizing magnetic fields$ P. Kleinerta,*, V.V. Bryksinb a

Paul-Drude-Institut fur Hausvogteiplatz 5-7, 10117 Berlin, Germany . Festkorperelektronik, . b Physical Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia Received 1 October 2002; received in revised form 25 March 2003

Abstract The formation of electric-field domains in superlattices subject to a quantizing magnetic field aligned perpendicular to the layers is analysed. Scattering on polar-optical phonons is treated within the density-matrix approach. Both the electron drift velocity and the diffusion coefficient are calculated as functions of the applied bias voltage and the magnetic field. We restrict ourselves to the consideration of the negative differential conductivity region. The appearance of cyclotron-Stark-phonon resonances leads to additional stable branches, in which both a uniform carrier density and a homogeneous electric-field distribution can exist. r 2003 Elsevier Science B.V. All rights reserved. PACS: 72.10.Bg; 73.50.Fq; 73.50.Jt Keywords: Quantum diffusion; Quantizing electric and magnetic fields; Electric field domains

1. Introduction Semiconductor superlattices (SLs) can exhibit an N-shape current–voltage (I–V) characteristics. Under the condition of negative differential conductivity (NDC), a uniform electric-field distribution may become unstable and domain formation is observed. Domains of different electric-field strengths are separated from each other by moving or static domain boundaries. $

The authors acknowledge partial financial support by the Deutsche Forschungs-gemeinschaft. *Corresponding author. Tel.: +49-30-20377-350; fax: +4930-20377-515. E-mail address: [email protected] (P. Kleinert).

Fluctuation phenomena and the system stability have been studied in a number of papers (for a review see Ref. [1]). The related current–voltage instabilities have attracted intense research interests because of its promising device application. Electric field domains result in a multiplicity of branches in the I–V characteristics, when there is enough electric charge inside the SL. In contrast, at sufficiently low carrier density, the electric field is almost spatially uniform. While the drift velocity, which decreases in the NDC region with increasing electric field, is favoring the creation of domains, the carrier diffusion works always against domain formation. The occurrence of two regions with different electric fields and domain-related current oscillations

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(03)00163-7

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P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

have been studied experimentally [2–6]. The physical nature of the experimental findings has been understood quite well on the basis of phenomenological approaches. Most theoretical treatments refer to weakly coupled SLs, which are in the regime of sequential resonant tunneling [7– 9]. The related numerical calculations are based on discrete drift-diffusion models, which were justified recently on a microscopic basis [10]. Unlike this satisfactory development in the field of weakly coupled SLs, there does not seem to be much theoretical work addressing the detailed microscopic analysis of domain formation for strongly coupled SLs. Laikhtman and Miller [11] presented a detailed microscopic theory of current–voltage instabilities in SLs. We will generalize this treatment by proposing a rigorous quantum-mechanical approach and by taking into account a quantizing magnetic field. Interesting results have also been published by Cao et al. [12,13], who exploited the time-dependent hydrodynamic balance equations coupled to the self-consistent field equation in order to study the spatio-temporal evolution of field domains and current oscillations. This promising approach can also be applied to a careful analysis of nonlinear carrier dynamics in wide miniband SLs. Unfortunately, the theory is not designed to cope with the Wannier–Stark (WS) localization and intra-collisional field effects. These disadvantages have been removed in a recent work based on equations of motion for generating functions of electron momentum fluctuations [14]. In this work, a microscopic treatment of the onset of electric-field domains in one-dimensional SLs under the condition of quantizing electric fields was presented. Here, we extend this study of the condition under which a homogeneous electric field distribution becomes unstable by taking into account the influence of a magnetic field in a three-dimensional SL model. The application of a magnetic field perpendicular to the layers leads to the Landau quantization of the lateral energy spectrum and to related cyclotron-Stark resonances [15]. The electric and magnetic field-induced localization of eigenstates gives rise to the formation of a quantum-box SL, in which the quantum character of the carrier statistics plays an essential role [16]. A parallel

magnetic field effectively reduces the dimensionality of the SL by quantizing the lateral carrier spectrum and influences the carrier transport along the electric-field direction via elastic and inelastic scattering. The expected effects of a magnetic field on the domain formation has been treated in a number of experimental studies [17–20]. Additional magnetic-field induced stable branches in the sawtooth-like I–V characteristics have been observed [18] and analysed on the basis of a rate-equation model [21]. Compared with the experimental efforts, there are only a few systematic theoretical studies of the domain formation under both quantizing electric and magnetic fields. Many challenging problems are still to be addressed. In this paper, we will focus on the magnetic-field dependent onset of the instability of a homogeneous electric field in semiconductor SLs. The instability is approached from a regime, where a homogeneous electric field distribution still exists. Under this condition, an infinite periodic SL is treated. Special boundary effects of real finite SLs become relevant within the instability region. To perform the linear stability analysis, both the drift velocity and the diffusion coefficient have to be calculated. While the drift velocity has been treated in the literature, there is no comparable study of the field-dependent diffusion coefficient. The development of a microscopic approach for the electric- and magnetic-field dependent diffusion coefficient is one of the main tasks of our paper. The stability analysis is applicable for weakly and strongly coupled SLs and takes into account quantum effects. Theoretical approaches in the literature exploited rate equations, in which the field-mediated quantization is not taken into account, or focused on one-dimensional SLs, in which a parallel magnetic field has no effect.

2. Basic theory In this paper, we study the onset of the electricfield-domain formation in SLs, to which a magnetic field is applied parallel to the SL axis. The critical point, at which a homogeneous electric-field distribution becomes unstable is

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

approached from the electric and magnetic field region, for which a uniform field distribution is still realized. The condition for the onset of the domain formation is obtained by considering the effect of small charge and field fluctuations nðr; tÞ ¼ n0 þ dnðr; tÞ;

Eðr; tÞ ¼ E0 þ dEðr; tÞ

ð1Þ

around the respective uniform steady-state values n0 ; E0 on the expression for the current density # j ¼ enðr; tÞv  eDrnðr; tÞ ð2Þ

rjþe

s-0

@nðr; tÞ ¼ 0: ð3Þ @t

Here, nðr; tÞ and D# denote the carrier density and the diffusion tensor, respectively. e represents the average dielectric permittivity of the SL. It is assumed that the carrier-density and the electricfield fluctuations along the field direction take the form of a plane wave

D ¼ lim

1 /ðxðtÞ  xð0ÞÞ2 S: 2t

ð7Þ

Here, first the thermodynamic limit (total number N of lattice sites goes to infinity at a constant density of states) and after that the limit t-N must be taken. xðtÞ denotes the position of the particle at time t: The Laplace transformed second quantized version of Eq. (7) has the form Dnn0 ¼ lim

s-þ0

s2 X 0 m;m0 ðm  mÞn ðm0  mÞn0 Mm;m 0 ðsÞ; 2 m;m0 ð8Þ

dnðz; tÞ ¼ dn0 expðiqz z  iotÞ; dEðz; tÞ ¼ dE0 expðiqz z  iotÞ:

ð6Þ

m;m0

In addition, we need the diffusion coefficient to determine the onset of domain formation according to the criterion in Eq. (5). For a particle moving along the x axis, the general expression for the diffusion coefficient has the form t-N

and the Maxwell equations er  Eðr; tÞ ¼ 4penðr; tÞ;

four-point function according to [22] X m;m0 ðm  m0 ÞMm;m v ¼  lim s2 0 ðsÞ:

415

ð4Þ

The internal electric field becomes unstable, i.e. small fluctuations rapidly grow out, when the imaginary part of o becomes positive. According to Eqs. (1) to (4) this happens, when the inequality
 4pen0 dvz 2 Im o ¼  Dzz qz þ >0 ð5Þ e dE is satisfied. In Eq. (5), only the z components of the vectors v; q and the diffusion tensor D# appear. It is seen from inequality (5) that NDC ðdvz =dEo0Þ favors the development into an instability, whereas diffusion works always against this tendency. To analyse the linear stability criterion in detail, it is necessary to calculate both the electron drift velocity and the diffusion coefficient on a microscopic basis [14]. The respective quantities are obtained from the twoparticle correlation function M: Let us treat a SL system, which reaches the steady state after a sufficiently long time interval (t-N or s- þ 0; where s is the variable of the Laplace transformation). We will restrict the consideration to the one-electron picture. Within the Wannier representation, the drift velocity is calculated from the

where m; m0 denote lattice vectors. In Eq. (8), it is understood that a diverging contribution, which arises in the limit s- þ 0; is compensated by the squared drift velocity. The Laplace transformed two-particle correlation function is defined by Z N 1 0 ;m3 Mmm21;m ðsÞ ¼ dt est Sp½eH =kB T eiHt=_ 4 Z0 0 iHt=_ þ

aþ am1 am2 ; m 4 am 3 e

ð9Þ

where Z0 is the partition function, aþ m ; am are Fermion field operators, and the Hamiltonians H; H 0 are defined below. The correlation function M does not depend on the carrier density. We will treat the onset of charge-density instabilities and the related formation of field domains due to strong electric and magnetic fields applied along the z-axis, i.e., perpendicular to the SL layers. It is assumed that almost all carriers occupy the lowest miniband. The extension of the approach to SLs with more general carrier distributions is possible and will be treated in future work. The considered electric-field strengths are sufficiently high so that the miniband splits into a WS ladder. In this NDC regime, an accurate treatment of the electric and magnetic fields beyond perturbation theory is

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

416

necessary. Within our envisaged one-electron picture, Coulomb interaction is not taken into account. Scattering on polar-optical phonons is treated within the simple bulk-phonon model. The Hamiltonian of the SL is written in the form H ¼ H 0 þ HE ;

H 0 ¼ He þ Hph þ Hint ;

ð10Þ

with the field-dependent free-electron contributions X HE ¼ eE  maþ m am ; He ¼

X

m 0

Eðm  m0 ÞeiAðm Þm aþ m am 0 :

ð11Þ

m;m0

EðmÞ denotes the tight-binding dispersion relation and AðmÞ the vector potential of the magnetic field in the symmetric gauge. Hph and Hint refer to free phonons and the electron–phonon interaction, respectively. The Hamiltonian in Eq. (10) is expressed in the Wannier representation. Alternatively, we may switch to plane waves by a Fourier transformation 1 X ikm am ¼ pffiffiffiffiffi e ak ; ð12Þ N k which introduces field operators in k space. The advantage of such a transformation to k space is at first glance not obvious, since under the action of electric and magnetic fields quasi-momentum conservation no longer holds. Nevertheless, one can identify a remaining symmetry of the SL system against ‘‘electro-magnetic’’ translations, which allows a major simplification of the treatment by switching to k space. Let us consider the following canonical transformation of the Hamiltonian (cf. Ref. [22]): TðmÞ ¼ Tph ðmÞTe ðmÞTH ðmÞ;

ð13Þ

which is composed of a number of operators defined by ! X þ Tph ðmÞ ¼ exp im  qbql bql ; ð14Þ ql

Te ðmÞ ¼ exp im 

X k

! kaþ k ak

;

ð15Þ

TH ðmÞ ¼ exp im 

!

X

0

Aðm

Þaþ m 0 am 0

:

ð16Þ

m0

The canonical transformation leads to the following equivalent form: 0

T þ ðmÞam0 TðmÞ ¼ amþm0 eiAðm Þm ;

ð17Þ

T þ ðmÞHTðmÞ ¼ H þ eE  m; T þ ðmÞH 0 TðmÞ ¼ H 0 :

ð18Þ

ðbþ ql Þ

In Eq. (14), bql denotes the annihilation (creation) operator for phonons of wave vector q in the branch l: Applying this canonical transformation to the two-particle correlation function M; we arrive at the symmetry relation ;m3 Mmm21;m ðsÞ 4 þm;m3 þm ðsÞ ¼ Mmm21þm;m 4 þm

expfiAðmÞ  ðm1  m2  m3 þ m4 Þg;

ð19Þ

which expresses the invariance under combined ‘‘electro-magnetic’’ translations. In the absence of a magnetic field ðA ¼ 0Þ; spatial translational invariance is restored, when the eigenenergies are shifted accordingly ðH-H þ eE  mÞ: When both an electric and a magnetic fields are applied to the SL, one can exploit Eq. (19) to introduce the so-called Wigner transformed functions, which depend only on three quasi-momentum vectors as in the case, when there are no external fields. This symmetryadapted function Mðk; k0 ; k j sÞ introduced by ;m3 ðsÞ Mmm21;m 4

¼ eiAðm1 Þm2 iAðm3 Þm4

1 X Mðk; k0 ; k j sÞ N 2 k;k0 ;k

h

exp ik  ðm2  m1 Þ  ik0  ðm4  m3 Þ i k þ i  ðm1 þ m2  m3  m4 Þ 2

ð20Þ

automatically satisfies the symmetry relation in Eq. (19). The Wigner transformed correlation functions are the objects we will concentrate on in our approach, since they solve the equations of motion, which are a natural generalization of equations valid in the case of vanishing external fields E ¼ 0; A ¼ 0: The two-particle correlation function M is calculated by a diagrammatic

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

approach. We introduce the vertex G; which collects all diagrams that remain finite in the limit s- þ 0 and which characterizes the initial correlations [22]. All other diagrammatic contributions, which are essential for the calculation of the drift velocity and the diffusion coefficient, are incorporated into a two-particle function P: The decomposition of the correlation function M X m ;m ;m3 3 Mmm21;m ðsÞ ¼ Gm12 ;m0 ðsÞPm;m ð21Þ m0 ;m4 ðsÞ; 4 m;m0

simplifies considerably by a transformation to the Wigner representation X Mðk; k0 ; k j sÞ ¼ Gðk; k1 ; k j sÞPðk1 ; k0 ; k j sÞ: ð22Þ k1

In order to expand P in powers of 1=s; the Bethe– Salpeter equation is treated X m1 ;m0 m1 ;m3 1 ;m3 Pm2 ;m10 ðsÞ Pm m2 ;m4 ðsÞ ¼ Rm2 ;m4 ðsÞ þ 2

m0i

m0 ;m0

m0 ;m

Wm01;m03 ðsÞRm30 ;m34 ðsÞ; 2

4

ð23Þ

4

where W denotes the scattering probability related to Hint : The two-particle function R refers to a reference system, in which scattering is absent ðHint ¼ 0Þ; i.e. it is defined by Z N m1 ;m3 Rm2 ;m4 ðsÞ ¼ dt est /0 j am2 eiðHe þHE Þt=_

result. The solution P of the respective kinetic equation # 0 ; kÞPðk; k0 ; k j sÞ ½s þ Iðk X Pðk; k1 ; k j sÞW ðk1 ; k0 ; k j sÞ ¼ dk;k0 þ must satisfy the sum rule [22] X 1 Pðk; k0 ; k ¼ 0 j sÞ ¼ : s k0

# 0 ; kÞRðk; k0 ; k j sÞ ¼ dk;k0 ; ½s þ Iðk where the differential operator h # 0 ; kÞ ¼ eE  rk0  i E k0 þ k þ Aðirk0 Þ Iðk _  _ 2 i k  E k0   Aðirk0 Þ 2

ð25Þ

ð26Þ

has been introduced. A similar equation of motion is obtained for the correlation function P by transforming Eq. (23) according to Eq. (20) and by applying the operator s þ I# on both sides of the

ð28Þ

The calculation of the drift velocity and the diffusion coefficient relies on the quantities

@ P1 ðk; k0 j sÞ ¼ Pðk; k0 ; k j sÞ

; @kz

k¼0

@2 0 0 ð29Þ P2 ðk; k j sÞ ¼ 2 Pðk; k ; k j sÞ

; @kz k¼0 which can be calculated from Eqs. (27) and (28). Below, similar definitions are introduced for the functions W and G: The formal analytic solutions of the corresponding equations of motion have the form ( X i 0 ½vz ðk1 þ Aðirk1 ÞÞ P1 ðk; k j sÞ ¼ 2 k 1

þ vz ðk1  Aðirk1 ÞÞPðk; k1 j sÞ þ

ð24Þ

An equation of motion for this quantity is easily obtained by an integration by parts and by switching to the Wigner representation. We obtain

ð27Þ

k1

0

iðHe þHE Þt=_ þ

aþ am1 j 0S: m 4 am 3 e

417

X

)

Pðk; k2 j sÞW1 ðk2 ; k1 j sÞ

k2

Pðk1 ; k0 j sÞ;

0

P2 ðk; k j sÞ ¼

X

ð30Þ

( i½vz ðk1 þ Aðirk1 ÞÞ

k1

þ vz ðk1  Aðirk1 ÞÞP1 ðk; k1 j sÞ i @ þ ½vz ðk1 þ Aðirk1 ÞÞ 4 @k1z  vz ðk1  Aðirk1 ÞÞPðk; k1 j sÞ X P1 ðk; k2 j sÞW1 ðk2 ; k1 j sÞ þ2 k2

þ

X

) Pðk; k2 j sÞW2 ðk2 ; k1 j sÞ

k2

Pðk1 ; k0 j sÞ

ð31Þ

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

418

with vz ðkÞ ¼ ð1=_Þ@EðkÞ=@kz being the z component of the drift velocity. Solutions (30) and (31) are used to derive general expressions for the drift velocity and the diffusion coefficient. The Wigner transformed version of Eq. (6) has the form

X

v ¼ i lim s2 rk Mðk; k0 ; k j sÞ ð32Þ

s-þ0 0 k;k

k¼0

the z component of which is further evaluated by inserting Eq. (22) in this result. We obtain for the z component X X vz ¼  i lim s2 Gðk; k1 j sÞ s-þ0

k;k0

k1

0

P1 ðk1 ; k j sÞ;

ð33Þ

where a term B@G=@kz jk¼0 has been neglected, which does not contribute in the limit s- þ 0: Next, P1 is replaced by the formal solution (30). The result is further simplified by introducing the distribution function X X Gðk0 ; k1 ; k j sÞPðk1 ; k; k j sÞ f ðk; k j sÞ ¼ s k0

¼s

X

k1

Mðk0 ; k; k j sÞ

ð34Þ

k0

and by considering that X X vz ðk7Aðirk ÞÞf ðk j sÞ ¼ vz ðkÞ f ðk j sÞ k

ð35Þ

k

with f ðk j sÞ ¼ f ðk; k ¼ 0 j sÞ [22]. The final expression for the drift velocity X veff vz ¼ z ðkÞ f ðkÞ; k

veff z ðkÞ

¼ vz ðkÞ  i

X k

W1 ðk; k0 Þ

ð36Þ

0

is composed of two quite different contributions [22]. The first one, which is proportional to vz ðkÞ; has been widely used in the study of coherent carrier transport. The second one, which is proportional to the derivative of the scattering probability W1 ¼ @W =@kz jkz ¼0 ; is related to hopping transport in systems, whose Hamiltonian does not commute with the dipole operator (e.g., small polarons) [22]. For the SL system, we treat in this paper, the second contribution in veff z vanishes.

An equation for the distribution function f ðk j sÞ is easily derived from its definition in Eq. (34) and from Eq. (27). We obtain X # Gðk0 ; k j sÞ ½s þ IðkÞf ðk j sÞ ¼ s k0

X

þ

k

f ðk0 j sÞW ðk0 ; k j sÞ

ð37Þ

0

# ¼ Iðk; # k ¼ 0Þ calculated from with an operator IðkÞ Eq. (26). The first term on the right-hand side of Eq. (37) vanishes in the limit s- þ 0 as G is a regular quantity. From the final Eqs. (36) and (37), we can derive an expression for the drift velocity, which is particularly useful for the treatment of the WS transport regime, where the carrier motion is due to inelastic scattering and negative differential conductivity P is expected to appear. Taking into account k0 W1 ðk; k0 Þ ¼ 0 (valid for the treated SL model) and performing integration by parts in Eq. (36), we obtain from the kinetic equation (37) (for s- þ 0) 1 X vz ¼  ½Eðkz Þ  Eðkz0 Þf ðk0 ÞW ðk0 ; kÞ; ð38Þ eEz k;k0 P 0 where the sum rule k W ðk ; kÞ ¼ 0 has been taken into account. Ez denotes the z-component of the electric field. According to Eq. (38) only inelastic scattering leads to a finite drift velocity, which is proportional to the coupling constant of the respective scattering mechanism. This is the hopping transport regime based on the WS ladder. The diffusion coefficient is calculated from Eq. (8), the Wigner transformed version of which has the form

s 2 X @2 Dnn0 ðsÞ ¼  Mðk; k0 ; k j sÞ

: ð39Þ 2 @k @k 0 k;k0

n

n

k¼0

From this expression a term has to be subtracted, which diverges in the limit s- þ 0: The twoparticle function M is expressed by the vertex function G and the diffusion propagator P according to Eq. (22). In addition to the vertex function G and its derivative G1 ; the second derivative G2 appears, which is obtained from P2 in Eq. (29). Without taking into account the contribution BsG2 ; which disappears in the limit

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

s- þ 0; we obtain

solution of Eq. (42) in the form

s2 X X fGðk; k1 j sÞP2 ðk1 ; k0 j sÞ 2 k;k0 k

Dzz ðsÞ ¼ 

f1 ðk j sÞ ¼ jðk j sÞ þ

1

þ 2G1 ðk; k1 j sÞP1 ðk1 ; k0 j sÞg:

ð40Þ

This result is further simplified by taking into account the formal solutions (30), (31), and the sum rule (28) together with definition (34). The final result X Dzz ðsÞ ¼ vz ðkÞ f1 ðk j sÞ; k

f1 ðk j sÞ ¼ i

@ f ðk; k j sÞ

@kz k¼0

ð41Þ

has the same form as Eq. (36) for the drift velocity. The main difference consists in the replacement of the distribution function by a new function f1 ðk j sÞ; which solves the inhomogeneous integral equation X # ½s þ IðkÞf f1 ðk0 j sÞW ðk0 ; k j sÞ 1 ðk j sÞ  k0

1 ¼ ½vz ðk þ Aðirk ÞÞ þ vz ðk  Aðirk ÞÞf ðk j sÞ 2 X  is G1 ðk0 ; k j sÞ k0

 i

X k

f ðk0 j sÞW1 ðk0 ; k j sÞ:

ð42Þ

0

The solution of this equation does not have the physical meaning of a distribution function and satisfies the sum rule X X s f1 ðk j sÞ ¼ vz ðkÞf ðk j sÞ: ð43Þ k

419

k

Up to this stage, the derivation of basic equations for the drift velocity and the diffusion coefficient is general and applies for an arbitrary orientation of the magnetic field. Now, we will focus on the special case, when the magnetic field is aligned parallel to the electric field. In addition, we assume that the dispersion relation has the form EðkÞ ¼ Eðk> Þ þ Eðkz Þ: In this case, the regular part of the diffusion coefficient with respect to the limit s- þ 0 (which determines the long-time behavior of the SL system) is easily identified by searching the

BðsÞ f ðk j sÞ: s

ð44Þ

In this ansatz the function j fulfils the sum rule P k jðk j sÞ ¼ 0: According to Eq. (43), we have X BðsÞ ¼ vz ðkÞ f ðk j sÞ  vz ðsÞ; ð45Þ k

which is used together with Eqs. (42) and (44) to derive an equation for the unknown function j: We obtain   eEz @ sþ jðk j sÞ _ @kz X jðk0 j sÞW ðk0 ; k j sÞ þ vz ðkÞ f ðk j sÞ ¼ k0

 vz ðsÞ  i

X

X

Gðk0 ; k j sÞ  is

X

k0

G1 ðk0 ; k j sÞ

k0 0

0

f ðk j sÞW1 ðk ; k j sÞ:

ð46Þ

k0

A solution of this equation requires the specification of initial correlations within the carrier system characterized by the vertex function G: We will treat a system that has reached the stationary state after both the external fields and the scattering processes were switched on at an early instant. Inserting the respective vertex function Gðk0 ; k; k j sÞ ¼ dk0 ;k into Eq. (46), we consider the limit s- þ 0 and identify the regular part of the diffusion coefficient by subtracting the respective diverging term. The result X X Dzz ¼ vz ðkÞjðkÞ; jðkÞ ¼ 0 ð47Þ k

k

is valid for arbitrary electric and magnetic field strengths. The function jðkÞ; which has not the character of a carrier distribution function, satisfies the inhomogeneous integro-differential equation X eEz @ jðkÞ ¼ jðk0 ÞW ðk0 ; kÞ þ vz ðkÞ f ðkÞ _ @kz 0 k X f ðk0 ÞW1 ðk0 ; kÞ ð48Þ  vz  i k0

with P vz ¼ vz ðs ¼ 0Þ: As a consequence of the sum rule k0 W1 ðk; k0 Þ ¼ 0 (which is satisfied when the Hamiltonian commutes with the dipole operator), the W1 contribution disappears in expression (36)

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

420

for the drift velocity. However, the same conclusion does not apply to the function j; from which the diffusion coefficient is calculated. The k derivative of the scattering probability ðW1 Þ never disappears in Eq. (48). This underlines the fact that the function jðkÞ has another physical meaning than the distribution function f ðkÞ: An equivalent exact expression for the diffusion coefficient, which is more suitable for WS localized eigenstates, is obtained by introducing a new function FðkÞ 1 FðkÞ ¼ jðkÞ  E*ðkz Þf ðkÞ; eEz X Eðkz Þ: ð49Þ E*ðkz Þ ¼ Eðkz Þ  kz

Expressing the diffusion coefficient by the quantity FðkÞ; we obtain X 1 Dzz ¼ ð*Eðkz Þ  E*ðkz0 ÞÞ2 f ðk0 ÞW ðk0 ; kÞ 2 2ðeEz Þ kk0 i X þ ð*Eðkz Þ  E* ðkz0 ÞÞf ðk0 ÞW1 ðk0 ; kÞ eEz kk0 1 X þ ð*Eðkz Þ  E* ðkz0 ÞÞFðk0 ÞW ðk0 ; kÞ; ð50Þ eEz kk0 which is exact for arbitrary field strengths, provided the higher-order subbands remain unoccupied. The first contribution on the right-hand side of Eq. (50) is relevant in the quasi-classical regime of high electric fields, when WS localization and intra-collisional field effects are not effective. A description of electric-field-induced quantum effects is possible by treating both the first and the second contribution in Eq. (50). The respective quantum-diffusion picture is most suitably expressed by switching to the WS representation. Taking into account the sum rule P 0 . W ðk ; k; k j sÞ ¼ 0 (valid for the Fr ohlich Hak miltonian), we obtain [23] X 1 ð*Eðkz Þ  E*ðkz0 ÞÞ2 W ðk0 ; kÞ 2 2ðeEz Þ kz0 k i X þ ð*Eðkz Þ  E*ðkz0 ÞÞW1 ðk0 ; kÞ eEz k0 k z

1X X Z Z ¼ ðZl  Zl 0 Þ2 WZll00Zll ðk0> ; k> Þ; ð51Þ 2 Z ;Z 0 k l

l

>

where Zl ¼ ld denotes the z component of the lattice vector. This equation is used to express the diffusion coefficient in Eq. (50) in the compact final form Dzz ¼

d2 X X 2 0 l f ðk> ÞW0l0l ðk0> ; k> Þ; 2 l k k0

ð52Þ

> >

valid in the region of WS localization, where the transport has hopping-like character. f ðk> Þ is the lateral distribution function, which is the solution of the kinetic equation (37) for s- þ 0 and Ot > 1 (with O being the Bloch oscillation frequency and t an appropriate scattering time). A similar expression for the drift velocity was derived in Ref. [23] X X vz ¼ d lf ðk0> ÞW0l0l ðk0> ; k> Þ: ð53Þ l

k> k0>

Both Eqs. (52) and (53) reveal the hopping character of the carrier motion in the WS regime. The applicability of these results is restricted to the regime of sufficiently high electric fields ðOt > 1Þ; where NDC may appear and the electronic states are WS localized. By introducing an effective scattering probability, which is renormalized by the electric field [23], Eqs. (52) and (53) are generalized in such a way that they are applicable for all electric field strengths including the Ohmic regime.

3. Calculation of the diffusion coefficient In this section, expressions for the drift velocity vz and the diffusion coefficient Dzz are derived, which are used for the numerical study of the stability criterion in Eq. (5). The calculation of the drift velocity proceeds in the same way as outlined in Ref. [24]. There is no need to repeat this calculation here. The treatment of the diffusion coefficient within the WS picture starts from our final result in Eq. (52). Scattering of electrons on polar-optical phonons is taken into account within the simple bulk-phonon model, which is assumed to demonstrate qualitative features of inelastic scattering in SLs. An additional complication is avoided here by assuming that the phonons remain

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

in equilibrium. The scattering probability written in the WS representation 0Zl 0 W0Z ðk> ; k> j sÞ l X Z N ¼ Re dt estþilOt Fql ðtÞ ql

0

2

X



Yðkz  qz ; kz ; 0; Zl Þ

k z  q q

P> k0> þ > ; k>  > ; q> j t ð54Þ 2 2 is composed of the magnetic-field dependent lateral part P> ðk0> ; k> ; q> j tÞ  h i q ¼ exp  E k0> þ > þ Aðirk0> Þ _ 2  i q  E k0>  >  Aðirk0> Þ dk0> ;k> ; 2

ð55Þ

the electron–phonon coupling contribution 2 Fql ðtÞ ¼ 2 j Mql j2 ½ðNql þ 1Þeioql t þ Nql eioql t  ð56Þ _ and a term, which describes the WS localization along the direction of the electric field Yðkz ; kz0 ; Zl ; Zl 0 Þ ¼ expfikz Zl  ikz0 Zl 0 þ iwðkz Þ  iwðkz0 Þg:

ð57Þ

In Eq. (56), Mql denotes the electron–phonon coupling matrix element for phonons of wave vector q in branch l: Nql and oql are the equilibrium phonon distribution function and the energy of polar-optical phonons, respectively. In Eq. (57), Zl ¼ ld denotes the SL sites and the periodic function wðkz Þ solves the differential equation d wðkz Þ ¼ E*ðkz Þ: ð58Þ eEz dkz The dispersion relation of the SL results from the effective-mass approximation with respect to k> and from the simple tight-binding description _2 k2> D þ ð1  cos kz dÞ: ð59Þ 2 2mn D denotes the miniband width and mn the effective mass of the lateral carrier dispersion. As in our previous approach, the lateral distribution function f ðk> Þ in Eq. (52) is replaced by the magneticEðkÞ ¼

421

field dependent Boltzmann distribution given by Eq. (30) in Ref. [24]. Taking into account the analytical result

2




X D qz d

2 sin Yðkz  qz ; kz ; 0; Zl Þ ¼ Jl ; ð60Þ

k _O 2 z

we arrive at an expression for the diffusion coefficient Dzz ¼

j Mql j2 d2 X _2 ql sinhð_oql =ð2kB TÞÞ
 N X D qz d 2 2 sin

l Jl _O 2 l¼N Z N

Re dt estþilOtþl_O=ð2kB TÞ cosðoql tÞ 0  q2> lB2

exp  2 sinhð_oc =ð2kB TÞÞ   _oc  cos oc t ; ð61Þ

cosh 2kB T

which has the same form as Eq. (40) in Ref. [24] for the current density. The main discrepancy consists in the appearance of l 2 in Eq. (61) instead of l in the expression for the current density. There are some problems in the calculation of the diffusion coefficient, which we encountered also in our transport study [24]. Without scattering, the energy spectrum of the quantum-box SL is completely discrete. In such a system, collisional broadening plays an essential role and caution has to be practiced in treating the limit s- þ 0 in Eq. (61). The correct treatment of this limit appears to be one in terms of a microscopic description of level broadening [16]. Such an approach is avoided here by the introduction of a phenomenological broadening parameter s-1=t: For a qualitative analysis of the onset of domain formation, this approximation seems to be justified. The high-field result in Eq. (61), which is valid in the NDC region (when O=sb1), simplifies further, when the q-dependence of the coupling matrix element is neglected (i.e., if j Mql j2 is replaced by Go20 ¼const., where o0 denotes the zone center polar-optical phonon frequency). In this case, all integrals in Eq. (61) can be calculated analytically

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

422

and we obtain N X




Dzz oc D ¼ l 2 Fl _O D0 sinhð_o0 =ð2kB TÞÞ l;m¼N
 _oc

exp  jmj 2kB T
 _O

cosh l 2kB T   s

s2 þ ðlO þ moc þ o0 Þ2

As expected, the diffusion coefficient is an even function of the electric field, in contrast to the drift velocity. A remarkable consequence of Eq. (65) is the reconstitution of the Einstein relation between the mobility and the diffusion coefficient ðm ¼ eDzz =kB TÞ for relatively high electric fields satisfying Do_OokB T:



ð62Þ

with field-independent reference values for the diffusion coefficient D0 ¼ v0 d=2 and the drift velocity v0 ¼ mn o20 Gd=p_3 : oc denotes the cyclotron frequency. In Eq. (62), we introduced the function

 Z D 1 p D sin x ; ð63Þ Fl dx Jl2 ¼ _O p 0 _O the values of which rapidly decrease with increasing j l j: For the sake of completeness, we add the final expression for the drift velocity
 N X vz oc D ¼ lFl _O v0 sinhð_o0 =ð2kB TÞÞ l;m¼N
 _oc jmj

exp  2kB T
 _O

sinh l 2kB T   s

; ð64Þ s2 þ ðlO þ moc þ o0 Þ2 which was derived and discussed in our previous work (Eq. (42) in Ref. [24]). Both Eq. (62) for the diffusion coefficient and Eq. (64) for the drift velocity exhibit resonances, whenever lO þ moc þ o0 ¼ 0 is satisfied (with l; m being any integers). These cyclotron-Stark-phonon resonances are superimposed on a background, which decreases with increasing electric field. From the Eqs. (62) and (64) we obtain a simple relationship between the drift velocity and the diffusion coefficient, when the inequality D=_Oo1 is satisfied. In this case, only the contributions for l ¼ 71 are relevant and we obtain vz d _O Dzz ¼ coth : ð65Þ 2 2kB T

4. Numerical results and discussion Two features are conspicuous in the equations for the drift velocity and the diffusion coefficient. This is on the one hand the appearance of NDC and on the other hand the occurrence of combined cyclotron-Stark-phonon resonances. The latter peculiarity results from the WS and Landau quantization. It is a pure quantum effect, which is associated with the electric- and magnetic-field dependence of the scattering, i.e. intracollisional field effects. In Fig. 1, the relative drift velocity (dashed line) and the relative diffusion coefficient (solid line) are shown as a function of the electric field for B ¼ 10 T: Both curves exhibit cyclotron-Stark-phonon resonances at O ¼ moc þ o0 ðm ¼ 0; 1; y; 9Þ indicated by thin vertical lines. These resonances are increasingly washed out with increasing lifetime

Fig. 1. The relative diffusion coefficient Dzz =D0 (solid line) and drift velocity vz =v0 (dashed line) are shown as a function of the electric field for D=_o0 ¼ 1; B ¼ 10 T; n ¼ 1018 =cm3 ; T ¼ 300 K; and d ¼ 50 nm: Collisional broadening is treated by a phenomenological parameter s=o0 ¼ 0:1: The positions of the cyclotron-Stark-phonon resonances located at O ¼ moc þ o0 are marked by thin vertical lines.

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

broadening due to interface roughness scattering, electron–phonon, and Coulomb interaction. Related cyclotron-Stark resonances, at which polaroptical phonons do not participate, have been experimentally identified [15]. Relation (65) between the drift velocity and the diffusion coefficient is well fulfilled for all electric field strengths treated in Fig. 1. When the inequalities D; kB To_O are satisfied, the diffusion coefficient exhibits the same electric-field dependence as the drift velocity. At these electric-field strengths, we have the simple relationship Dzz ¼ vz d=2: The asymptotic field dependence of both quantities depends on the character of the most relevant scattering mechanism. According to inequality (5), the sharp structures of the drift velocity shown in Fig. 1 determine in a complicated manner the onset of domain formation. There are alternating electric-field intervals, where the derivative of the drift velocity with respect to the electric field becomes positive so that according to Eq. (5) a homogeneous field distribution is stabilized. An example is shown in Fig. 2, in which the shaded areas indicate, at which field strengths Im o becomes positive so that a uniform electric field cannot exist throughout the whole SL. The main effect of a quantizing magnetic field on the onset of the domain formation consists in the creation of alternating stable branches, which

Fig. 2. Left-hand side of criterion (5) as a function of the electric field. The same set of parameters has been used as in Fig. 1. Within the shaded area, a uniform electric-field distribution becomes unstable.

423

Fig. 3. Ez –B plane, in which Im o > 0 and Im oo0 are marked by the shaded and white areas, respectively. According to criterion (5), a stable uniform electric field cannot exist in the gray region. These instability branches are separated by white stripes, for which a homogeneous electric field is stable. The Landau fan-like structure of this pattern is illustrated by two sets of straight lines calculated from O ¼ moc 7o0 : The same set of parameters has been used as in Fig. 1.

separate instability regions from each other. This is shown in more detail in Fig. 3 as a function of both the electric and magnetic field. Within the shaded areas of the Ez –B plane, Im o is positive and therefore a homogeneous electric field is unstable according to criterion (5). The instability area intersects branches, in which a homogeneous electric field exists. This is the result of our linear stability analysis. To which extend these islands of stability manifest themselves in experiment depends on the influence of higher-order corrections, which may give rise to multistability. The stabilizing effect of the magnetic field with respect to the formation of electric-field domains in SLs is due to the Landau quantization, which introduces additional energy levels. The fan-type character of the stripes is illustrated by two sets of straight lines, which are calculated from O ¼ moc 7o0 for various integers m: This is in accordance with experimental and other theoretical results [18,21], which claim that the appearance of additional stable branches in the intermediate magneticfield range is due to scattering-assisted interLandau-level transitions. With increasing lifetime

424

P. Kleinert, V.V. Bryksin / Physica B 334 (2003) 413–424

like structure of this pattern is due to interLandau-level scattering. Additional stable branches, which are due to a magnetic field of medium strength, have been experimentally identified [18]. To our knowledge, the close relationship between the stable regions in the electric- and magnetic-field plane and cyclotron-Stark-phonon resonances as stressed in this paper has not been studied from an experimental point of view. References

Fig. 4. The same as in Fig. 3 for a larger broadening parameter s=o0 ¼ 0:3: In the gray area of the Ez –B plane, it holds Im o > 0 so that according to criterion (5) a uniform electric field cannot exist.

broadening, the fan-type splitting is increasingly washed out, but does not disappear at all as shown in Fig. 4, where the collisional broadening is stronger. Magnetic-field induced islands of stable uniform electric-field distributions still exist, but at higher magnetic field strengths. The disappearance of extra stable branches at extremely high magnetic fields as reported in Ref. [18] is not reproduced by our density-matrix approach. Further studies seem to be necessary to elucidate the origin of this phenomenon.

5. Summary We have carried out a microscopic analysis of electric-field-domain formation under the influence of a quantizing magnetic field, which affects the carrier motion within the layers. Special results have been obtained for the NDC region, where the energy spectrum is completely discrete due to WS and Landau quantization. Both in the drift velocity and in the diffusion coefficient, combined cyclotron-Stark-phonon resonances appear. These structures give rise to a complex scenario of the domain formation. New branches appear in the Ez –B plane, for which both the density profile and the electric-field distribution are uniform. The fan-

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