Superlattice vertical transport with high-lying minibands

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

Superlattice vertical transport with high-lying minibands X. L. Lei State 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 Lima Faculdade de Engenharia, Universidade S˜ao Francisco, Campus de Itatiba, 13.251-900 Itatiba, SP, Brazil and Departamento de Electrˆonica Quˆantica, Instituto de F´ıscia, Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524, 20550-013 Rio de Janeiro, RJ, Brazil

A. Troper Centro Brasileiro de Pesquisas F´ısicas, 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 a nonparabolic balance-equation approach. We find that the inclusion of high-lying minibands results in a decrease of the electron temperature, a reduction of the peak drift velocity and a 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 become significant when the strength of the electric field gets close to or falls in the NDM regime. c 1998 Academic Press Limited

Key words: vertical transport, minibands, superlattices.

1. Introduction Bragg-diffraction-induced negative differential mobility (NDM) in superlattice vertical transport [1] has attracted much attention in the literature for the past few years [2–10]. Since the early model suggested by Esaki and Tsu [1], many calculations have been carried out using Monte Carlo simulation [11–13], Boltzmann equation [7–9] and balance-equation methods [6]. The majority of these theoretical studies, however, were based on the assumption that carriers are moving within a single miniband. Although the basic physical feature of these Bragg-diffraction-related phenomena is included in most single-miniband models, the carrier population of high-lying minibands is not negligible for steady-state transport when the electric field is close to or falls in the NDM regime, where the electron temperature Te (equivalent to the energy) can be as high as, or even higher than, the energy distance between the bottoms of the first and second minibands of the superlattice. On the other hand, hot-electron transport in high-lying minibands has also been demonstrated experimentally by injecting carriers of arbitrary energy into the semiconductor superlattice [14]. It is thus desirable to pursue a theoretical study on superlattice NDM beyond the lowest-miniband model. In this paper 0749–6036/98/020243 + 06 $25.00/0

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we report a theoretical analysis of miniband transport for a planar superlattice, with the role of high-lying minibands included. Consider a GaAs-based planar superlattice in which electrons move freely in the transverse (x y) plane and are subject to periodic potential wells and barriers of finite height along the growth axis (z-direction). The electron-energy dispersion εα (k) can be written as the sum of a transverse energy εkk = kk2 /2m (m being the band mass of the carrier in the bulk semiconductor), and a tight-binding-type miniband energy εα (k z ) related to the longitudinal motion: 1α (1) [1 + (−1)α cos(k z d)], εα (k z ) = εα0 + 2 where α = 1, 2, . . . is the miniband index, k = (kk , k z ) represents the three-dimensional wavevector and kk = (k x , k y ) the in-plane wavevector (−∞ < k x , k y , < ∞ and −π/d < k z ≤ π/d, d being the superlattice period along the z-direction), εα0 is the bottom position and 1α is the energy width of the αth miniband. Choosing the first miniband bottom as the energy zero (ε10 = 0), and denoting 1 ≡ 11 , we can write the first miniband energy spectrum as 1 (2) ε1 (k z ) = [1 − cos(k z d)] 2 (with bottom at k z = 0), the second miniband energy spectrum as ε2 (k z ) = ε20 +

12 [1 + cos(k z d)] 2

(3)

(with bottom at k z = π/d), and so on. These parameters of the miniband structure are easily calculated based on the Kronig–Penny model once the superlattice period d, well width a and barrier height Vb are given. As an example, we consider only the first and second miniband occupation, and assume that there are N electrons residing in this two-miniband system. When a uniform electric field E = (0, 0, E) is applied along the superlattice growth axis, electrons are accelerated by the field and scattered by impurities, phonons and among themselves. By symmetry, the average drift velocity and the frictional acceleration for each miniband are in the z-direction. We consider that intraband and interband Coulomb couplings are strong to give a unique electron temperature within the whole electron system, but the mutual drag and the particle exchange between two minibands due to interband electron–electron scattering are less important in comparison with those due to electron–phonon and electron–impurity scatterings and are neglected. In the framework of the balance-equation approach [15, 16] a transport state of the system can be described by the average lattice momentum (in the z-direction) p1 = (0, 0, p1 ) and p2 = (0, 0, p2 ), and the chemical potential µ1 and µ2 for each miniband, together with the electron temperature Te for the whole carrier system. In the steady transport state we have equations for the carrier population and the effective-force balance in each miniband and the energy balance for the whole system: d d N1 = − N2 = X ei12 + X e12p , dt dt N1 eE d 12 N1 v1 = + A1ei + A1e p + A12 ei + Ae p , dt m ∗1z N2 eE d 21 N2 v2 = + A2ei + A2e p + A21 ei + Ae p , dt m ∗2z d inter E = N eE · vd − Weintra p − We p . dt

(4) (5) (6) (7)

Here E = N1 ε1 + N1 ε2 is the average total energy of the system, N1 and N2 are the average numbers, v1 and v2 are the average velocities, and ε1 and ε1 are the average energies (per carrier) of the carriers populating the

Superlattices and Microstructures, Vol. 23, No. 2, 1998 first and second minibands (α = 1, 2) respectively: X f ([εα (k) − µα ]/Tα ), Nα = 2

245

(8)

k

2 X vα (k z ) f ([¯εα (k) − µα ]/Tα ), Nα k 2 X εα (k) f ([¯εα (k) − µα ]/Tα ), εα = Nα k

vα =

(9) (10)

and 1/m ∗αz is the zz-component of the ensemble- averaged inverse effective mass tensor of the αth miniband: 1/m ∗αz =

2 X d 2 εα (k z ) f ([¯εα (k) − µα ]/Tα ). Nα k dk z2

(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 v d = r n 1 v1 + r n 2 v2 ,

(13)

where r n 1 ≡ N1 /N and r n 2 ≡ N2 /N are the fractions of carriers populating the first and second minibands respectively. In the above equations, f (x) ≡ 1/[1 + exp(x)] is the Fermi distribution function, vα (k z ) ≡ dεα (k z )/dk z is the z-direction velocity function of carriers in the αth miniband, and ε¯ α (k) ≡ εkk + εα0 + εα (k z − pα ).

(14)

Equation (4) states that the rate of change of the carrier-population in the first miniband is due to interband electron–impurity scattering, X ei12 , and interband electron–phonon scattering, X e12p . Equations (5) and (6) state that the rate of change of the average velocity of the αth (α = 1, 2) miniband is due to the electric-field acceleration and the frictional acceleration which consists of contributions from intraband electron–impurity scattering, Aαei , the intraband electron–phonon scattering, Aαe p , the interband electron–impurity scattering, αβ αβ Aei , and the interband electron–phonon scattering, Ae p . Equation (7) indicates that the rate of change of the total electron energy equals the energy supplied by the electric field minus the energy-loss to the lattice inter through intraband electron–phonon coupling, Weintra p , and interband electron–phonon coupling, We p . The relevant quantities are expressed in terms of the electron–impurity potential, the electron–phonon matrix element, the electron–electron Coulomb potential, and the intraband and interband form factors related to the wavefunctions of the first and second minibands. Note that we have relations X ei12 = −X ei21 , and X e12p = −X e21p . Therefore, the equation for the rate of change of the electron number populating the second miniband is identical to eqn (4). There are five variables: p1 , p2 , µ1 , µ2 , and Te . In the steady transport state (d N1 /dt = 0, d(N1 v1 )/dt = 0, d(N2 v2 )/dt = 0, and dE/dt = 0), we have two independent equations for the effective-force balance, one equation for particle number balance and one equation for energy balance. These four equations, together with constrain (12), form a complete set of equations for the determination of the above five variables if the total number of carrier, N , the electric field E and the lattice temperature T are given. As an example, we consider a GaAs/Alx Ga1−x As-based superlattices with barrier height Vb = 0.258 eV, period d = 8 nm and well width a = 6 nm, having the following parameters for the first and second miniband: 1 = 650 K, 12 = 2100 K and ε20 = 1500 K. The above equations for dc steady-state vertical transport under the influence of a uniform electric field, were solved at lattice temperature T = 300 K. Effects of both intraband and interband scatterings from randomly distributed charged impurities, acoustic phonons (through the deformation potential and piezoelectric couplings with electrons) and polar optic phonons (through Fr¨ohlich

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

80

0

20 1µ

15 rn2

–100

10

rn2 (%)

1 = 650 K T = 300 K 12 = 2100 K ²20 = 1500 K

1µ (K)

246

5 0

5 10 15 E (kV cm–1)

0 20

0.5

v1

0.4

40

0.3 p1

v2

0.2

20 p2

p1 and p2 (π/d)

v1 and v2 (km s–1)

–200

60

0.1 0

0 5

0

10

15

20

Electric field E (kV cm–1) Fig. 1. The average lattice momenta p1 and p2 (in units of π/d) and the average velocities v1 and v2 for the first and second miniband as functions of the electric field E. The inset shows 1µ ≡ µ2 − µ1 , and r n 2 .

80

1 = 650 K

12 = 2100 K

²20 = 1500 K

T = 300 K

60 vd 1500 40 Two band One band 900

Te

20

Te(K)

Drift velocity vd (km s–1)

2100

d = 8.0 nm Ns = 2.0 × 1015 m–2 0

300 0

5

10

15

20

Electric field E (kV cm–1) Fig. 2. The average drift velocity vd and electron temperature Te , obtained from the present two-band model (solid curves) and from a one-miniband model (chain curves).

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

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coupling with electrons) are taken into account, using bulk phonon modes of GaAs for simplicity. All the material 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 be Ns = 2.0 × 1015 m−2 and the strengths of impurity scattering are such that low-temperature linear mobility µ(0) = 1.0 m2 (Vs)−1 in both cases. The calculated average lattice momenta p1 and p2 and the average velocities v1 and v2 for the first and second miniband, 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, r n 2 , are plotted in the inset of this figure. Both p1 and p2 are in fact the average lattice-momentum deviations in the presence of the electric field from their zero-field values (zero). As expected, both p1 and p2 grow with increasing electric field, but the growth rate of p2 is much slower than that of p1 and the magnitude of p1 is much larger than p2 at small to medium field strength. The velocity for the upper miniband increases monotonically with increasing E, while the velocity for the lower miniband exhibits a peak around E ∼ 4.3 kV cm−1 before it decreases with increasing field. We find that the chemical potential of the upper miniband is lower than that of the lower miniband, and this difference is enhanced with increasing field-strength. The total average velocity vd of the present two-miniband model, obtained from eqn (13), is shown in Fig. 2 as a function of the electric field E, together with the electron temperature Te . For comparison, in the same figure we also plot the theoretical prediction of vd and Te 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 the electron temperature Te , but also results in an appreciable decrease of the peak drift velocity and a slow-down of the descending rate of vd in the NDM regime. These effects are significant at room temperature when the strength 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 differential mobility (NDM) regime, in comparison with those predicted by a single miniband model. These effects become 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 National Commission of Science and Technology of China, the Shanghai Foundation for Research and Development of 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.

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