Self-consistent modelling of resonant tunnelling structures

surface science

Surface Science 267 (1992) 392-395 North-Holland

Self-consistent modelling of resonant tunnelling structures T h o m a s Fiig Physics Department, Riso National Laboratory, DK-4000 Roskilde, Denmark

ard

A.P. J a u h o Physics Laboratory, Unirersity of Copenhagen, H.C. Orsted Institute, Universitetsparken 5, DK-2100 Copenhagen O, Denmark Received 16 June 1991; accepted for publication 9 July 1991

We report a comprehensive study of the effects of self-consistency on the /-V-characteristics of resonant tunnelling structures. The calculational method is based on a simultaneous solution of the effective-mass Schr6dinger equation and the Poisson equation, and the current is evaluated with the Tsu-Esaki formula. We consider the formation of the accumulation layer in the emitter contact layer in a number of different approximation schemes, and we introduce a novel way to account for the energy relaxation of continuum states to the two-dimensional quasi-botmd states appearing for certain applied voltag,.s and carrier densities at the emitter-barrier interface. We include the two-dimensional accumulation layer charge and the quantum well charge in our self-consistent scheme. We discuss the evaluation of the current contribution originating from the two-dimensional accumulation layer charges, and our qualitative estimates seem consistent with recent experimental studies. The intrinsic bistability of resonant tunnelling diodes is analyzed within several different approximation schemes.

Theoretical investigations of the self-consistent carrier density in resonant tunnelling structures have attracted considerable interest during recent years [1]. These studies hc.ve gained additional importance in efforts to establish theoretical criteria whether the bistability, observed under certain circumstances [2], is of intrinsic or extrinsic character. Many theoretical models have been used [3], and the general upshot seems to be that when the charge in the quantum well is included in the analysis, a bi- or multistable /-V-curve can result. A subtle point in the suggested computational schemes concerns the treatment of the accumulation layer at the emitter-barrier interface: many model "?nore this layer entirely, or trcat it within simple semiclassical models, which may lead to unphysical cusps [4] in the resulting carrier density. A proper treatment of the accumulation layer would require a nonlinear quantum-mechanical transport theory capable of treat-

ing both strong spatial inhomogeneity (size quantization) and carrier-phonon and carrier-carrier interactions, which are responsible for the relaxation towards the conduction band bottom. No such theory is available at present, and a~l analysis of some of the candidate theories can be found in ref. [5]. The Wigner function approach [6] may have the potential of becoming a firstprinciples tool for analyzing heterostructures under strongly nonequilibrium conditions. These calculations, however, invariably require large computer resources, which most likely will prevent their wide-spread use. Keeping the above complications in mind, the following considerations have motivated the present study. We want to develop a model, which satisfies the following criteria, based or.. physical plausibility and practicality: (i) the computational effort must not exceed (essentially) the one required in conventional semiclassical models (e.g.,

0039-6028/92/$05.00 ~:~ 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All rights reserved

Th. Fiig,A.P. Jauho / Self-consistentmodellingof resonanttunnellingstructures Ohnishi et al. in ref. [1] or Mains et al. in ref. [3]); (ii) the solution found for the charge density should be free of unphysical features: no cusps at interfaces, charge neutrality in the device should be guaranteed (i.e., no net charge transfer from the emitter to the collector), and no charge density oscillations may o c : , r far from the barriers; and (iii) a description of size quantization in the accumulation layer must be included in the model. A minimal model that satisfies the above criteria can be stated as follows. Consider first the semiclassical description of the contact layers, within which the charge density nc(X) is evaluated from

nc( x) = NcF,/2( I~ - Ec( x) ) kT

(1)

'

where Fj is the jth order F e r m i - D i r a c integral, and E¢(x) = E°(x)-eck(x) is the local conduction band edge, where E°(x) is the heterostructure potential, and ~b(x) is determined from the Poisson equation. Eq. (1) cannot be correct close to the e m i t t e r - b a r r i e r interface: due to size quantization, energies arbitrarily close to the local conduction band edge are not allowed. A simple way te account for this feature is to use the semiclassical charge density expression for the continuum states, and use a quantum mechanical expression for the states in the 'notch' formed at the interface. In this new model the charge density in the emitter layer is calculated from

iI I

kT

6

7 E r.. 'O

II

O

4 "-

X

c~

-1

-3 500

700

1

900 11 oo x (Oo)

13oo

o 1500

Fig. 1. Self-consistent cha,'ge density and band-edge potential calculated with the Thomas-Fermi model, eq. (1). Full curve: increasing bias; broken curve: decreasing bias. calculations we only included the lowest energy state in the notch. Before turning to our numerical calculations we make the following comments concerning eq. (2). (i) While eq. (2) removes the cusp in charge density implied by eq. (1), it still leads to a charge density with a discontinuous first derivative at the interface (see figs. I and 2 for typical charge densities). (ii) the 2D-states are assumed to have the same chemical potential as the extended 3

["

0

i XlXi(x)l z

I

z

>

log 1 + exp

-

v

nc( x) = NcF1/2( l't - ~c( x) ) + ~2p2°kT

393

8

i

E cO

(2)

Here E c ( x ) = m a x [ E c ( - ~ ) , Ec(x)] (this choice ensures that quasibound states below the asymptotic c o n d u c t i o n band edge are treated q u a n t u m

mechanically, while all other states are treated semiclassically), p2D = m * / ~ h 2, and the sum runs over the notch states. The way we have identified the notch states is described elsewhere [7]. Here it suffices to note that in our actual numerical

4 ._..

x

x

1

-3 500

700

900 1100 x (ao)

1 , 3 0 0 1500

Fig. 2. Same as fig. 1, but for the new model, eq. (2).

394

Th. Fiig, A.P. Jauho / Self-consistent modelling of resonant tunnelling structures

states. Only a dynamical quantum kinetic calculation can confirm this choice, but since thermalization processes generally occur on a very fast time-scale, we believe that our choice is a reasonable one. Our numerical procedure is standard and has been described elsewhere [7]. The structural parameters for the double barrier system studied in this work are as follows: barrier and well thicko nesses 50 A, barrier height 0.23 eV, and in contact layers E F = 10 meV. Figs. 1 and 2 show the self-consistent charge densities and potentials obtained with eqs. (1) and (2), respectively, with the bias chosen to correspond to the maximum bistability in the / - V - c u r v e (Vappl = 0.24 V for fig. 1, and Vappn= 0.2 V for fig. 2). For the old method, our findings are quite similar to those of Mains et al, [3]. While the potentials are quite similar, the charge densities show some important differences. As remarked above, the new method (fig. 2) leads to a much smoother charge density than that resulting from eq. (1). In the old method, the (unphysically) large charge density at the interface is more effective in screening the applied field, and, therefore, the structure must be biased stronger to reach maximum current (see also figs. 3 and 4 below). In the new method, the accumulation layer charge is more sensitive to bistability, while the quantum well charges show rather similar behaviour in both methods. In the calculation of the current we have used the standard Tsu-Esaki formula [8]. We also examined corrections to this formula [9], but did not find significant differences. In our calculation of the current we only included the continuum states, because in our opinion, Tsu-Esaki formula is applicable only if one can associate a momentum to a given state, which is not the case for quasi-bound states in the a-'cumulation layer. Clearly, in a quantitative evaluation of the current, one should take these states into account, but, "-"~ . . .tulxateiy, . . . . . "" to Oily IKIIUWICUge, ' ...... ' - J- - there exists ulntvt almost no theoretical work addressing the problem of how to combine on equal footing currents originating from continuum states and 2D-states. An estimate for the current can be constructed by following the suggestion of Lassnig and Boxleiter [10]: one should consider the life-times ~- of the

20

15

C

E u

~<10 0

0 0.0

b

,

~

L

t

i

l

i

i

t

,

L

,

i

,

i

0.1

i

i

i

0.2

0.3

Vooo (v) Fig. 3. Current-voltage characteristics, calculated for the model defined by eq. (1). Full curve: increasing bias; broken curve: decreasing bias.

states, extracted from the density of states, and evaluate the current from J = Q / ~ - . We have performed such an estimate [7], and in general find large currents, e.g., the ground state in the notch leads to a current comparable to the maximum current from the continuum states. The estimate depends critically on the value of the chemical potential for the notch states, which, as r e n m k e c above, can only be determined with a dynamical theory. One should also note that the simple argument of lining up the quantum-well state with the occupied states, which can be applied for continuum states, is no longer applicable: one should consider the notch and the quantum well as a strongly coupled double well system. Curiously, our estimate agrees well with the trend of several recent experimental investigations [11]: the 2D-states seem to lead currents comparable to, or exceeding the continuum states related current. The current densities obtained with the two methods are shown in fig. 3 ava. 4. The significant differenc," in addition to the shift toward lower voltages for the new method, as mentioned above. is the much larger bistable region. We hasten to point out that for decreasing biases our calculational scheme had some difficulty in converging, and it showed a tendency to overshoot the cur-

Th. Fiig, A.P. Jauho / Self-consistent modelling of resonant tunnelling structures 20

15 I

E

~<1o 0

0 0.0

j

I

0.1

0.2

r"

0.3

Vo.o (v)

Fig. 4. Same as fig. 3, but for the new model, eq. (2).

rent obtained for increasing currents. An extensive numerical study is needed to make quantitative predictions. We are, however, confident that the larger bistable region for the new method is a real effect. In summary, we present a simple method to evaluate the self-consistent /-V-curves for resonant tunnelling diodes, including the effects of 2D-accumulation layer states. As an application, we study the bistablity of these devices.

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T.C.L.G. Sollner, Phys. Rev. Lett. 59 (1987) 1622; T.J. Foster, M.I. Leadbeater, L. Eaves, M. Henini, O.H. Hughes, C.A. Payling, F.W. Sheard, P.E. Simmonds, G.A. Toombs, G. Hill and M.A. Pate, Phys. Rev. B 39 (1989) 6205; A. Zaslavsky, V.J. Goldman, D.C. Tsui and J,E. Cunningham, Appi. Phys. Lett. 53 (1988) 1408; J.G. Chen, C.H. Yang and R.A. Wilson, J. Appl. Phys. 69 (1991) 4132. [3] F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228; W. PStz, J. Appl. Phys. 66 (1989) 2458; R.K. Mains, J.P. Sun and G.I. Haddad, Appl. Phys. Lett. 55 (1989) 371; M. Rahman and J.H. Davies, 3cmicond. Sci. Technol. 5 (1990) 168; J.A. Sofo and C.A. Balseiro, Phys. Rev. B 42 (1990) 7292. [4] W. Frensley, Solid State Electron. 32 (1989) 1235; Rev. Mod. Phys. 62 (1990) 745. [5] A.P. Jauho, Solid State Electron. 32 (1989) 1265. [6] W.R. Frensley, Phys. Rev. B 36 (1987) 1570; N.C. Kluksdahl, A.M. Kriman, D.K. Ferry and C. Ringhofer, Phys. Rev. B 39 (1989) 7720; K.L. Jensen and F.A. Buot, Phys. Rev. Lett. 66 (1991) 1078. [7] T. Fiig and A.P. Jauho, Appl. Phys. Lett. 59 (1991) 2245. [8] R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. [9] The velocity, correction suggested by S. Collins, D. Lowe and J.R. Barker, J. Appl. Phys. 63 (1988) 142, led to a correction of less than one percent for the maximum current, while two-body effects, as discussed by D.D. Coon and H.C. Liu, Appl. Phys. Lett. 47 (1985) 172, gave slightly larger effects (although not as large as suggested by the original authors). [10] R. Lassnig and W. Boxleiter, Solid State Commun. 64 (1987) 979. [11] L. Eaves et al., Appl. Phys. Lett. 52 (1988) 212; M. Buchanan, H.C. Liu, T.G. Powell and Z.R. Wasilewski, J. Appl. Phys. 68 (1990) 4313; J.S. Wu, C.Y. Chang, C.P. Lee, K.H. Chang, D.G. Liu and D.C. Liou, Appl. Phys. Lett. 57 (1990) 2311; B. Jogai, C.I. Huang, E.T. Koenig and C.A. Bozada, J. Vac. Sci. Technol. B 9 (1991) 142.