Superlattices and Microstructures, Vol. 23, No. 2, 1998

Superlattice vertical transport with high-lying minibands

X. L. LeiState Key Laboratory of Functional Material for Informatics, Shanghai Institute of Metallurgy,

Chinese Academy of Sciences, 865 Changning Road,

Shanghai 200050, People’s Republic of China

I. C. da Cunha LimaFaculdade de Engenharia, Universidade Sao Francisco, Campus de Itatiba,

13.251-900 Itatiba, SP, Brazil and Departamento de Electronica Quantica, Instituto de Fıscia,

Universidade do Estado do Rio de Janeiro, Rua Sao Francisco Xavier 524,

20550-013 Rio de Janeiro, RJ, Brazil

A. TroperCentro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud 150, 22290 Rio de Janeiro, RJ, Brazil

(Received 15 July 1996)

We examine the role of high-lying minibands in superlattice vertical transport using anonparabolic balance-equation approach. We find that the inclusion of high-lying minibandsresults in a decrease of the electron temperature, a reduction of the peak drift velocity anda slow-down of the velocity-drop rate in the negative differential mobility (NDM) regime,in comparison with those predicted by a single miniband model. These effects becomesignificant when the strength of the electric field gets close to or falls in the NDM regime.

c© 1998 Academic Press LimitedKey words: vertical transport, minibands, superlattices.

1. Introduction

Bragg-diffraction-induced negative differential mobility (NDM) in superlattice vertical transport [1] hasattracted much attention in the literature for the past few years [2–10]. Since the early model suggested byEsaki and Tsu [1], many calculations have been carried out using Monte Carlo simulation [11–13], Boltzmannequation [7–9] and balance-equation methods [6]. The majority of these theoretical studies, however, werebased on the assumption that carriers are moving within a single miniband. Although the basic physicalfeature of these Bragg-diffraction-related phenomena is included in most single-miniband models, the carrierpopulation of high-lying minibands is not negligible for steady-state transport when the electric field is closeto or falls in the NDM regime, where the electron temperatureTe (equivalent to the energy) can be as highas, or even higher than, the energy distance between the bottoms of the first and second minibands of thesuperlattice. On the other hand, hot-electron transport in high-lying minibands has also been demonstratedexperimentally by injecting carriers of arbitrary energy into the semiconductor superlattice [14]. It is thusdesirable to pursue a theoretical study on superlattice NDM beyond the lowest-miniband model. In this paper

0749–6036/98/020243 + 06 $25.00/0 sm960299 c© 1998 Academic Press Limited

244 Superlattices and Microstructures, Vol. 23, No. 2, 1998

we report a theoretical analysis of miniband transport for a planar superlattice, with the role of high-lyingminibands included.

Consider a GaAs-based planar superlattice in which electrons move freely in the transverse(xy) plane andare subject to periodic potential wells and barriers of finite height along the growth axis (z-direction). Theelectron-energy dispersionεα(k) can be written as the sum of a transverse energyεk‖ = k2

‖/2m (m being theband mass of the carrier in the bulk semiconductor), and a tight-binding-type miniband energyεα(kz) relatedto the longitudinal motion:

εα(kz) = εα0+ 1α

2[1+ (−1)α cos(kzd)], (1)

whereα = 1,2, . . . is the miniband index,k = (k‖, kz) represents the three-dimensional wavevector andk‖ = (kx, ky) the in-plane wavevector(−∞ < kx, ky, <∞ and−π/d < kz ≤ π/d, d being the superlatticeperiod along thez-direction),εα0 is the bottom position and1α is the energy width of theαth miniband.Choosing the first miniband bottom as the energy zero(ε10 = 0), and denoting1 ≡ 11, we can write thefirst miniband energy spectrum as

ε1(kz) = 1

2[1− cos(kzd)] (2)

(with bottom atkz = 0), the second miniband energy spectrum as

ε2(kz) = ε20+ 12

2[1+ cos(kzd)] (3)

(with bottom atkz = π/d), and so on. These parameters of the miniband structure are easily calculated basedon the Kronig–Penny model once the superlattice periodd, well width a and barrier heightVb are given.

As an example, we consider only the first and second miniband occupation, and assume that there areNelectrons residing in this two-miniband system. When a uniform electric fieldE = (0,0, E) is applied alongthe superlattice growth axis, electrons are accelerated by the field and scattered by impurities, phonons andamong themselves. By symmetry, the average drift velocity and the frictional acceleration for each minibandare in thez-direction. We consider that intraband and interband Coulomb couplings are strong to give aunique electron temperature within the whole electron system, but the mutual drag and the particle exchangebetween two minibands due to interband electron–electron scattering are less important in comparison withthose due to electron–phonon and electron–impurity scatterings and are neglected. In the framework of thebalance-equation approach [15, 16] a transport state of the system can be described by the average latticemomentum (in thez-direction)p1 = (0,0, p1) andp2 = (0,0, p2), and the chemical potentialµ1 andµ2 foreach miniband, together with the electron temperatureTe for the whole carrier system. In the steady transportstate we have equations for the carrier population and the effective-force balance in each miniband and theenergy balance for the whole system:

d

dtN1 = − d

dtN2 = X12

ei + X12ep, (4)

d

dtN1v1 = N1eE

m∗1z

+ A1ei + A1

ep + A12ei + A12

ep, (5)

d

dtN2v2 = N2eE

m∗2z

+ A2ei + A2

ep + A21ei + A21

ep, (6)

d

dtE = NeE · vd −Wintra

ep −Winterep . (7)

HereE = N1ε1+ N1ε2 is the average total energy of the system,N1 andN2 are the average numbers,v1 andv2 are the average velocities, andε1 andε1 are the average energies (per carrier) of the carriers populating the

Superlattices and Microstructures, Vol. 23, No. 2, 1998 245

first and second minibands(α = 1,2) respectively:

Nα = 2∑

k

f ([εα(k)− µα]/Tα), (8)

vα = 2

Nα

∑k

vα(kz) f ([εα(k)− µα]/Tα), (9)

εα = 2

Nα

∑k

εα(k) f ([εα(k)− µα]/Tα), (10)

and 1/m∗αz is thezz-component of the ensemble- averaged inverse effective mass tensor of theαth miniband:

1/m∗αz =2

Nα

∑k

d2εα(kz)

dk2z

f ([εα(k)− µα]/Tα). (11)

The total number of carriers is

N = N1+ N2, (12)

and the average drift velocity of the carriers in the system is given by

vd = rn1v1+ rn2v2, (13)

wherern1 ≡ N1/N andrn2 ≡ N2/N are the fractions of carriers populating the first and second minibandsrespectively. In the above equations,f (x) ≡ 1/[1 + exp(x)] is the Fermi distribution function,vα(kz) ≡dεα(kz)/dkz is thez-direction velocity function of carriers in theαth miniband, and

εα(k) ≡ εk‖ + εα0+ εα(kz− pα). (14)

Equation (4) states that the rate of change of the carrier-population in the first miniband is due to interbandelectron–impurity scattering,X12

ei , and interband electron–phonon scattering,X12ep. Equations (5) and (6) state

that the rate of change of the average velocity of theαth (α = 1,2) miniband is due to the electric-fieldacceleration and the frictional acceleration which consists of contributions from intraband electron–impurityscattering,Aαei , the intraband electron–phonon scattering,Aαep, the interband electron–impurity scattering,

Aαβei , and the interband electron–phonon scattering,Aαβep . Equation (7) indicates that the rate of change ofthe total electron energy equals the energy supplied by the electric field minus the energy-loss to the latticethrough intraband electron–phonon coupling,Wintra

ep , and interband electron–phonon coupling,Winterep . The

relevant quantities are expressed in terms of the electron–impurity potential, the electron–phonon matrixelement, the electron–electron Coulomb potential, and the intraband and interband form factors related to thewavefunctions of the first and second minibands. Note that we have relationsX12

ei = −X21ei , andX12

ep = −X21ep.

Therefore, the equation for the rate of change of the electron number populating the second miniband isidentical to eqn (4).

There are five variables:p1, p2, µ1, µ2, andTe. In the steady transport state(d N1/dt = 0,d(N1v1)/dt =0,d(N2v2)/dt = 0, anddE/dt = 0), we have two independent equations for the effective-force balance, oneequation for particle number balance and one equation for energy balance. These four equations, togetherwith constrain (12), form a complete set of equations for the determination of the above five variables if thetotal number of carrier,N, the electric fieldE and the lattice temperatureT are given.

As an example, we consider a GaAs/Al xGa1−xAs-based superlattices with barrier heightVb = 0.258 eV,periodd = 8 nm and well widtha = 6 nm, having the following parameters for the first and second miniband:1 = 650 K,12 = 2100 K andε20 = 1500 K. The above equations for dc steady-state vertical transport underthe influence of a uniform electric field, were solved at lattice temperatureT = 300 K. Effects of both intra-band and interband scatterings from randomly distributed charged impurities, acoustic phonons (through thedeformation potential and piezoelectric couplings with electrons) and polar optic phonons (through Fr¨ohlich

246 Superlattices and Microstructures, Vol. 23, No. 2, 1998

00

20

40

v 1 an

d v 2

(km

s–1

)

p 1 an

d p 2

(π/d

)

60

80

5 10

Electric field E (kV cm–1)

15 200

0.1

0.2

0.3

0.4

0.5

1 = 650 K T = 300 K

12 = 2100 K ε20 = 1500 K

v1

p1 v2

p2

0–200

–100

1µ

(K

)

0

5 10E (kV cm–1)

15 200

5

10

rn2 (

%)15

20

1µ

rn2

Fig. 1. The average lattice momentap1 andp2 (in units ofπ/d) and the average velocitiesv1 andv2 for the first and second minibandas functions of the electric fieldE. The inset shows1µ ≡ µ2 − µ1, andrn2.

00

20

40

Dri

ft v

eloc

ity v

d (k

m s

–1)

Te(

K)

60

80

5 10

Electric field E (kV cm–1)

15 20300

900

1500

2100

Two bandOne band

d = 8.0 nm Ns = 2.0 × 1015 m–2

Te

1 = 650 K 12 = 2100 K ε20 = 1500 K T = 300 K

vd

Fig. 2. The average drift velocityvd and electron temperatureTe, obtained from the present two-band model (solid curves) and froma one-miniband model (chain curves).

Superlattices and Microstructures, Vol. 23, No. 2, 1998 247

coupling with electrons) are taken into account, using bulk phonon modes of GaAs for simplicity. All thematerial parameters employed in the calculation are typical values of GaAs and are the same as those used in[6]. The carrier sheet density is assumed to beNs = 2.0× 1015 m−2 and the strengths of impurity scatteringare such that low-temperature linear mobilityµ(0) = 1.0 m2 (Vs)−1 in both cases.

The calculated average lattice momentap1 andp2 and the average velocitiesv1 andv2 for the first and secondminiband, are shown in Fig. 1. The chemical potential difference between these two minibands,1µ ≡ µ2−µ1,and the fraction of the number of carriers accommodating the second miniband,rn2, are plotted in the inset ofthis figure. Bothp1 andp2 are in fact the average lattice-momentum deviations in the presence of the electricfield from their zero-field values (zero). As expected, bothp1 and p2 grow with increasing electric field, butthe growth rate ofp2 is much slower than that ofp1 and the magnitude ofp1 is much larger thanp2 at smallto medium field strength. The velocity for the upper miniband increases monotonically with increasingE,while the velocity for the lower miniband exhibits a peak aroundE ∼ 4.3 kV cm−1 before it decreases withincreasing field. We find that the chemical potential of the upper miniband is lower than that of the lowerminiband, and this difference is enhanced with increasing field-strength. The total average velocityvd of thepresent two-miniband model, obtained from eqn (13), is shown in Fig. 2 as a function of the electric fieldE, together with the electron temperatureTe. For comparison, in the same figure we also plot the theoreticalprediction ofvd andTe obtained by assuming that only the lowest miniband is relevant (one-miniband model),which is the result of [6]. It turns out that the existence of a high-lying miniband not only greatly reduces theelectron temperatureTe, but also results in an appreciable decrease of the peak drift velocity and a slow-downof the descending rate ofvd in the NDM regime. These effects are significant at room temperature when thestrength of the electric field gets close to or falls in the NDM regime.

In summary, we find that the existence of high-lying minibands results in a decrease of electron temperature,a reduction of the peak drift velocity and a slow-down of the velocity-drop rate in the negative differentialmobility (NDM) regime, in comparison with those predicted by a single miniband model. These effectsbecome significant when the strength of the electric field gets close to or falls in the NDM regime.

Acknowledgements—The authors thank the National Natural Science Foundation of China, the NationalCommission of Science and Technology of China, the Shanghai Foundation for Research and Developmentof Applied Materials, the Brazilian Research Council (CNPq), the Foundation for Research in Rio de Janeiro(FAPERJ), and PROCORE from Universidade S˜ao Francisco for support of this work.

References

[1] }}L. Esaki and R. Tsu, IBM J. Res. Dev.14, 61 (1970).[2] }}A. Sibille, J. F. Palmier, H. Wang, and F. Mollot, Phys. Rev. Lett.64, 52 (1990).[3] }}F. Beltram, F. Capasso, D. L. Sivco, A. L. Hutchinson, and S. N. G. Chu, Phys. Rev. Lett.64, 3167 (1990).[4] }}H. T. Grahn, K. von Klitzing, K. Ploog, and G. H. D¨ohler, Phys. Rev.B43, 12094 (1991).[5] }}R. Tsu and L. Esaki, Phys. Rev.B43, 5204 (1991).[6] }}X. L. Lei, N. J. M. Horing, and H. L. Cui, Phys. Rev. Lett.66, 3277 (1991); J. Phys.: Condens. Matter.4,

9375 (1992).[7] }}A. A. Ignatov, E. P. Dodin, and V. I. Shashkin, Mod. Phys. Lett.B5, 1087 (1991).[8] }}J. F. Palmier, A. Sibille, G. Etemadi, A. Celeste, and J. C. Portal, Semicond. Sci. Technol.B7, 283 (1992).[9] }}R. R. Gerhardts, Phys. Rev.B48, 9178 (1993).

[10] }}H. J. Hutchinson, A. W. Higgs, D. C. Herbert, and G. W. Smith, J. Appl. Phys.75, 320 (1994).[11] }}P. J. Price, IBM J. Res. Dev.17, 39 (1973).[12] }}D. L. Andersen and E. J. Aas, J. Appl. Phys.44, 3721 (1973).[13] }}M. Artaki and K. Hess, Superlattices Microstruct.1, 489 (1985).[14] }}P. England, J. R. Hayes, E. Colas, and M. Helm, Solid State Electron.32, 1213 (1989).

248 Superlattices and Microstructures, Vol. 23, No. 2, 1998

[15] }}X. L. Lei and C. S. Ting, Phys. Rev.B30, 4809 (1984);32, 1112 (1985).[16] }}X. L. Lei, Phys. Stat. Sol. (b)170, 519 (1992).[17] }}X. L. Lei, D. Y. Xing, M. Liu, C. S. Ting, and J. L. Birman, Phys. Rev.B36, 9134 (1987).