Intrinsic bistability in the resonant tunneling diode

Superlattices

and Microstructures,

397

Vol. 5, No. 3, 1989

INTRINSIC BISTABILITY IN THE RESONANTTUNNELINGDIODE* N. Kluksdahl, A.M Kriman, and D. K. Ferry Center for Solid State Electronics Research Arizona State University Tempe, AZ 85282 C. Ringhofe? Department of Mathematics Arizona State University Tempe, AZ 85287 (Received

9 August,

1988)

Quantum transport in the resonant tunneling diode is modeled here with the Wigner formalism including self-consistent potentials for the The calculated I-V characteristics show an intrinsic first time. differential conductivity region of the bistability in the negative We show that intrinsic bistability results from charge storage curve. shifting of the internal potential of the device. the subsequent and The cathode spacer layers is investigated. The effect of undoped and quantization of depletion region of the RTD shows a strong layer is electrons in a deep triangular potential well if no spacer The potential drop in the cathode well reduces the barrier present. height to ballistic electron injected from the cathode, enhancing the A finite relaxation valley current and reducing the peak-valley ratio. time for- the electr-ons increases the negative resistance, reduces the to valley ratio of the current, and causes a ‘soft’ hysteresis in peak the bistable region. The spacer layer prevents the formation of a deep and the distribution does not at the cathode barrier, quantum well deplete as sharply as without the spacer layer.

I.

Introduction

technology has advanced Microfabrication with each advance giving a reduction rapidly, Devices fabricated in the sizes of features. with MOCVDand MBE have features as small as a On these spatial scales, nanometers. few evident. quantization effects are quite Theoretical models of these small devices, however, have not kept pace. The quantum structure which has been most recently is the resonant tunneling studied diode (RTD)le3. A thin AlGaAs barrier is grown Next, a GaAs on a GaAs substrate by MBE. is grown on the barrier, then a quantum well second AlGaAs barrier is added. Contacts are resistive GaAs layers. The two made through AlGaAs barriers and the GaAs well constitute a tunneling I-V resonant system. The characteristic of this two terminal device has

a strong region of negative differential conductance (NDC), which results from resonant tunneling in the device. The quantum model we shall employ uses the Wigner function. Though it has been around for quite some time, and its properties have been 4-7 well-investigated Wigner functions have only been recently’ applied to electronic 8-10 transport . The Wigner formalism offers many advantages for quantum modeling. Itis a phase-space description and scattering is a 11 local phenomenon . In this paper, we present the first results of utilizing a fully selfconsistent potential coupled to the Wigner function transport equation. This model shows the presence of intrinsic bistability. We examine the bistability, and the role played by undoped spacer layers adjacent to the AlGaAs barriers. II.

* Work supported in part Research and (+) the Scientific Research.

0749-6036/89/030397+05$02.00/0

by the Office of Naval Air Force Office of

The Wigner distribution

For a mixed state, 4-6 form

function

the Wigner function

has

the well-known

0 1989 Academic

Press Limited

398

Superlattices

fw(X,k)=

;

Jdx eikx

p(X+$,

X-$,.

(1)

It is easily seen that there is no requirement in the definition of the Wigner function (1) which requires it to be a positive quantity. The equation of motion of the Wigner function is derived from the quantum Liouville equation by using the Wigner-Weyl transform. The equation &

where

of

motion

fw(x,p,t)

1 H s dF B(x,y,F)

M(x,y,P)=

is 4’6

- i &

(p = Kk) fw(x,p,t)

fw(x,p+F,t),

= (2)

[ dy eiPY’“[V(x+$)-V(x-$)I.

(3)

The kineticterm is identical to the kinetic term of the Boltzmann equation. The potential term in however, is nonlocal in the (3), position of the potential and in the momentum distribution function. These of the nonlocalities give rise to the quantum 11 . corrections in the equation of motion It might seem that a correct quantum mechanical steady-state solution may be found by specifying the correct boundary conditions and solving (2) with the time derivative set to The fallacy in this procedure is subtle. zero. mechanical boundary The correct quantum conditions Presuppose knowledge of the state of which is a the system at the boundaries, the internal potential. Thus, function of the boundary conditions implies knowledge of knowledge of the solution without the Wigner function equation of motion. may be All orders of quantum effects the initial Wigner function incorporated into through the use of the adjoint equation to the Liouville equzr;:zrtt;,d numerically ’

Equation (2) may then be to find steady-state

and Microstructures,

solutions which include all orders of quantum corrections. We use a scattering-state basis to find the density matrix p(x,x’), which is then transformed into the initial Wigner function13’14. Calculation of the density matrix is done self-consistently. the equilibrium Wigner Figure 1 shows distribution function at 300K for the structure of Fig. 2. The Wigner function calculated by scattering states is characterized by a thermal distribution far from the barriers. Oscillations in the distribution near the resulting from quantum repulsion from barrier, 8,13,14 the barrier are evident

The Wigner function equation of motion is the position modeled on a discretized grid in The modeled and momentum variables x and p. region, from x = 0 to x = L, is divided into a with a mesh size Ax chosen such spatial mesh, that features of interest such as potential are adequately represented. The barriers, periodic in discrete Wigner function is momentum,

with

in momentum is number of

a period

1.

of g,

discretized

into

meshes ranging

from &

and this a

period

convenient

to &.

The discretized equation is solved Lax-Wendroff explicit time differencing, the retains the second-order terms in

using which Taylor

Stability of an of f(t+bt)? error in the scheme means that equation remains bounded. For a the CourantaX and a time step At, 16 gives a Friedrichs-Lewy stability criterion condition for and sufficient necessary stability:

expansion explicit discretized mesh size

-1 5E7 Fig.

Vol. 5, No. 3, 1989

Equilibrium Wigner distribution for the self-consistent resonant tunneling diode.

Momentum

Superlattices

and Microstructures,

the velocity where v is the solution. component of stability and convergence of

the fastest of Studies of the this approach will

17

. be presented elsewhere potentials are self-consistent Fully the introduced by adding Poisson’s equation to The self-consistent initial Wigner system. distribution includes the effects of background An iterative the ionized donors. charge from procedure is required to find the correct The initial Wigner distribution. initial function,

the background

399

Vol. 5, No. 3, 1989

charge

N:(x),

and

the

self-consistent potential V(x) are input to (2) Solution proceeds by and Poisson’s equation. the then adjusting forward, stepping (2) Steady-state potential via Poisson’s equation. conditions may be tested by examining the timeinternal variant fluctuations in both the potentials and the Wigner function. We model scattering with the relaxation-

in which undoped GaAs consider a structure layers of 5 nm thickness are adjacent to the AlGaAs barrier layers, as shown in Fig. 2b. The steady-state I-V curve of the device is calculated by applying an incremental bias potential to the cathode contact, then stepping and Poisson’s equation until steady-state (2) steady-state From the conditions are reached. the current in the device is distribution, bias potential is again calculated, and the After the bias has been increased incremented. decremented to its maximum, the potential is toward zero, with the current being calculated The resultant I-V curves, shown along the way. show a peak-valley ratio of in Fig. 3, respectively, at approximately 2:l and 3:1, 300K . Early models of the RTD assumed that all is dropped across the barrier-well potential

with 101* cmm3 donors. The barriers and the will also quantum well are undoped. We

The self-consistent internal structure8-10’18. the device, for an applied bias potential of near the valley of the I-V curve, shows that in the absence of the spacer layers only about one third of the potential is dropped across the the remaining potential, a barriers. Of the majority is dropped at the cathode end of Wigner the At the cathode, device. At high distribution shows heavy depletion. this region forms a triangular quantum bias, within a of electron well with quantization bound state. The deep triangular well at the cathode Because has implications for the I-V cur-ve. is at the well is deep, the top of the barrier the same energy as the cathode contact. nearly Electrons which are injected from the cathode the barrier ar-e able to and ballistically to This process travel over the quantum barriers. leads to a greatly enhanced valley current, and the device. reduces the peak-valley ratio of The potential distribution at O.$ V bias: is shown in Fig. 4a. We contrast this in Fig. 4b to that for the spacer layer structure. the A second difference results from of the spacer layers on the scattering. effect Without ionized donors, the electron mean-free

Fig.

Fig.

which lumps all dissipation time approximation, parameter. macroscopic processes into one other crude in comparison to Al though easily the resulting expression is formalisms, This approach has been shown to evaluated. effectively remove correlation and to shift the 18 IV characteristics of the RTD . III.

Application

to

the Diode

Resonant

Tunneling

The self-consistent has been applied to The structure (RTD). two AlGaAs 2a) has 0.3 eV high and 5 nm

Wigner function model a resonant tunneling diode under consideration (Fig. quantum barriers which are thick. These barriers are

by a 5 separated Outside the barriers,

nm GaAs quantum well’. the device is GaAs, doped

Posiiion

2a.

Conduction The barriers thick.

band of the simulated RTD. and well are each 5 nm

2b.

(nm)

Modified RTD conduction band. Undoped spacer layers 5 nm thick are placed between the barriers and the resistive GaAs layers.

400

Superlattices

0.1

0.2

Applied

Fig.

3a.

4a.

,i pp

the slope

of

the NDC region

of

the curve19. Finally, the bistability of the NDC region of the I-V curve is found to be much sharper, again a consequence of the reduced Rn. A controversial aspect of the RTD is the source of observed bistability in the I-V curve in the NDC regime. Intrinsic bistability is thought to result from charge storage within the quantum well, which changes the potential and field near the positions of

3b.

the barriers3. the conduction

e d

This modifies band edges on

Fig.

4b,

r’uientiol

Self-consistently calculated I-V curve for the modified RTD. The peak-valley ratio is enhanced, and the negative differential conductivity shows reduced resistance.

Position

Self-consistent potential for the RTD with an applied bias of 0.4 V. A deep quantum well is formed between the cathode and the barrier, leading to a quantized state and an overall electron depletion.

increases

Fig.

(nm)

path is longer, and the relaxation time is correspondingly increased. More of the potential is dropped across the barriers, causing a higher barrier to the ballistic which reduces electrons, the valley current. The effect upon the valley current is clearly evident in the calculated I-V curve of Fig. 3b. Another major difference caused by the buffer layer and reduced scattering is the reduced Rn, which

Vol. 5, No. 3, 1989

0:5

(V)

Self-consistently calculated I-V curve of the RTD. An intrinsic bistability is present in the negative differential conductivity region.

Position

Fig.

0:4

0.3

Potential

and Microstructures,

(nm)

Self-consistent potential for the modified structure with an applied bias of 0.4 v. The spacer layer prevents formation of the deep quantum well and quantized state.

either side of the barriers and of the resonant level in the well. Since the current is strongly affected by the positions of these bands and the resonant level, subtle shifts in stored charge can greatly affect the current tunneling through the resonant level between the barriers. Extrinsic bistability, on the other hand, results from the external circuit and may either cause, or be a result of, the device oscillating at very high frequency about 19 . Such bistability is then a the bias point function of the external circuit rather than the quantum well structure. While experiments have observed in the operation of RTDs, there is bistability 3,19-21 . If all disagreement as to its source oscillations of the device are supressed, any observed bistability would have to be intrinsic. Because the negative resistance R n is low, however, debate centers around whether

Superlattices

and Microstructures,

in fact been suppressed, all oscillation has and extrinsic bistability observed. The calculation of the I-V curve does not effects. external circuit include Any which is found, therefore, must be bistability With fully selfnature. intrinsic in potentials, a soft bistability, consistent found similar to ferromagnetic hysteresis, is This model in the NDR region of the I-V curve. shows one region of bistability with a finite resulting strictly from differences in the R n’ the potentials and self-consistent distributions on either side of the bistable region. Both effects are related to the charge This is true for the quantum well. stored in both structures modeled, although the details of the I-V curves differ somewhat. the RTD is In an experimental circuit, loaded, which changes the slope of the NDC region. Experiments have shown two bistable regions of the I-V curve, each of which has a R observed very large What may be n’ the intrinsic bistability experimentally is being heavily modified by the external circuit parameters. VI.

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Vol. 5, No. 3, 1989

Summary and Conclusions

We have briefly addressed the issues of using the Wigner function to model quantum The discretized equation of transport devices. motion is found to be stable and convergent when the CFL conditions for stability are satisfied. The nonlocality of the equation of motion may be removed at the boundaries, which leads to realization of ’ideal’ contacts. These ideal contacts serve as a perfect source sink for of incoming electrons and a perfect 22,23 electrons which leave the device We have self-consistent added fully potentials to the Wigner function model of the resonant tunneling diode. The calculated I-V show clear curves bistability, which results from charge storage in the quantum well and also in a triangular potential well which forms at the interface between the cathode bulk GaAs and the AlGaAs barrier. Since the model does include not external circuit effects, this bistability must be intrinsic. Modification of the structure to include buffer layers next to the barriers changes the internal potentials. The negative resistance reduced is through the reduced scattering in the undoped regions, which sharpens the bistable region of the calculated I-V curve.

References 1. 2.

3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14.

15.

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19. 20.

21. 22. 23.

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