multiquantum well structures

Superlattices

and Microstructures,

421

Vol. 5, No, 3, 1989

Influence of the Wavefunctions in Bopping Conduction in Superlattice/Uultiquantu Well Structures Yang

Kevin

Wang

and F. Brennan

School of Electrical Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0250 (Received 8 August 1988)

We present calculations of the polar optical phonon assisted hopping conduction in a two-well multiquantum well structure. The influence of different choices of the approximate wavefunctions commonly used in hopping conduction calculations are investigated. Specifically, we determine the transition probability using finite square well, triangular well, and Airy function wavefunctions. It is found that the choice of the wavefunctions dramatically influences the magnitude of the hopping conduction current.

Introduction Optimization of resonant tunneling diodes requires maximization of the peak-to-valley current ratio particularly for devices operated at 300 K. Recent experimental work [ll has yielded a record peak-to-valley current ratio of 14 at room temperature using an In0 53Ga0.47A~/ AlAs resonant tunneling barrier diode. Most theoretical studies of tunneling resonant devices to date have focused only on the nature of the resonant Lunneling channel [2,3]. These studies can only predict the negative differential resistance onset voltage. Further improvement in the design of resonant tunneling devices more theoretical requires elaborate studies which fully account for all of the present in devices, current processes these namely tunneling, assisted resonant phonon hopping conduct ion, and thermionic emission. The inclusion of non-resonant current processes, such as phono” assisted hopping and thermionic emission, enables the accurate determination of the off-resonance current and, hence, the peakto-valley currenL ratio. Of these mechanisms, polar optical phonon assisted tunneling is the most important. Doehler et al. [41 first that realized conduction in a superlattice can proceed via phonon assisted hopping of from electrons localized states between adjacent wells if the ptjtential drop over the superlattice period exceeds the miniband width. Alternatively, hopping conduct ion can dominate resonant tunneling processes when the potential barriers are sufficiently high and wide such that the electronic wavefunctions are localized within each well. In either case, the conduction is

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defined due to transitions between well localized spatial quantization states through the absorption or emission of phonons. Several approaches to this problem have recently been formulated [5-71, which illustrate the basic physics of the process as well as its relative import ante . However, in each of these calculations, different forms of the wavefunctions have been chosen making it difficult to ascertain the relative significance of hopping conduction. In this paper, we analyze the importance of the choice of wavefunctions in polar optical phonon assisted hopping conduction problems. It is found that the calculated results are very different depending upon the choice of the electronic wavefunctions.

Node1 Description I” polar semiconductors, dominant the electron-phonon interaction at room temperature is that of polar optical phonons , if the concentration of impurities is sufficiently low and resonant elastic tunneling can be neglected [81. The probability per unit time of a transition from initial state Ik,o> to final state \k’ ,u’> with a phonon emission or absorption is found using Fermi’s golden rule as [9],

W*(k,o;k’,o’;q)

= %

2 etw d

(Nq

+ $ i +)

P (1) *

/
Tiqr q2

Ik,al

2 6(Ek,

,

,,,

0 1989

- Ek (J f

3

Academic

~&‘u

Press Limited

422

Superlattices

The total transition rate from the ith subband to the jth subband of the neighboring well can be written as the sum of t$e abs_orption and emission processes as, J.. = J.. + J.. Jl J1 Jl’ where -Ifi

=

Z

~

Q q,

W*(k,o;k’,o’;Q,qz)fi(k)

Z

~

k,o

k’o’

[l Equation

(2)

can

-

fj(k’)]

(2)

be

rewritten

as

(Nq

+ + f +)

wavefunctions quantum wells, below.

are

Finite

Well

Square

=

s

$

d2k &

Q

determine the transition probafinite square well wavefunctions The wavefunctions are given as

b,(z)

=

6N

CN sin

(kN,(z-al)

CN sin

bN exp

exp[xN(=.l)I + hN) [-xN(z-z2)]

z<7.

1

z1

< z < z (3, 2 > z2

-h2k2/2mkT

e

- Bi

j,k*Q

isolated presented

Wavefunctions

CN sin

($ 6(E

of

161

i

*

to be those calculations

with

2 * f

Vol. 5, No. 3, 1989

[5,9]

1 d2Q + 4vl

P

taken the

in

We first bility assuming without bias.

fij’Q, J;;

and Microstructures,

EN)1’2

a + 2 Tan-‘(&)

= Nn

(3)

k * ~0)

,

XN = [5

(W-E,)]“2

where fij(Q)

= 7 pi

(z)e -‘=dz

(4)

(z+z’)+2(z+z’)dz’

(5)

C;

0

= [;

+ $

(k;

+ xi)]-’

and *

simple derivation, Using a obtain an expression for rate as J.. = C[ (Nq+l)I;; 31 with

the

one can finally total transition

+ NqISi]

m

C-Eph2J

B.. J1

2nkT

=

5

(Ej

- Ei

f

hue)

and

Iii

= 7 y

exp[&

(Q - +)2]dQ

0

N

9

1

=

huo/kT

where a is the well.

the

well

kN

width

= (2m

and W is

EN/k2)

the

1’2

depth

of

(6)

2

e hu

kN a + 2 6N = Nn

_ 1

e Results To obtain the total transition rate, one has to calculate the overlap integral, f. . , first. The most commmonly used wavefunctikas describing electrons in a multiquantum device are finite square wells [61, triangular wells and Airy functions [ll]. We use these [lOI, wavefunctions assume a two well, three and barrier, GaAs/Al,, 45Ga0 55A~ structure in which decoupled; the the wells are assumed’ to be

The calculated results of the transition rate are shown in Table 1. Both J12, which arises from transitions from the n=l to the n=2 level, arises from transitions and J21, which from the n=2 to the n=l quantum level, are considered. Inspect ion of Table 1 reveals that the l-2 transition rate decreases with increasing electric field strength while the 2-l transition rate increases with increasing electric field. The effect of the bias is included by shifting the quantum wells by an amount equal to the voltage drop. Hence, from if the transition is from 1 to 2, the Eq. (6), exponent, B.. , has a much larger magnitude than if the tranittion is from 2 to 1. It is interesting to note that the transition rates are asymmetrical with respect to the applied bias. This is due to the assumption of fixed subband energies within the wells implicit Therefore, in the finite square well model. since a positive voltage acts to lower the the level separation is quite electron energies, different in going from the 1~2 to the n=l level as opposed to going from the n=l to the n=2 since the sublevel under bias. In addition, band energy difference is much greater than the

Superlattices

and Microstructures, Table

Field

(kv/cm)

+

J12 J12(total)

using

finite

-10

wavefunctions

10

0.0

30

20

1.80x10’

1.21x107

7.89x106

5.22~10~

3.39x106

2.45~10~

1.57x108

1.01x108

6.48~10’

4.18~10’

2.71~10’

1.76~10’

2.88~10’

1.85x108

1.19x108

7.69~10’

4.97x107

3.23~10’

2.10x107

2:

Transition

rates

using

triangular

O2

wavefunctions

+

+

cl

~11

J12

512

J12

J21

J21

J21

10

0.0849

0.150

1.22~10~’

9.31~10~’

1.05~10~~

8.25~10~~

8.11~10~~

9.06~10~~

20

0.135

0.238

2.23~10~

1.39~10~’

1.51~10~~

4.43~10~~

6.13~10~~

5.04~10~~

30

0.177

0.312

5.58~10~

3.17x109

3.73x109

3.21~10~~

5.09~10~~

3.72~10~~

voltage drop the transition field.

subband

across rate

energy;

~~ = Second

subband

the structure chosen here, changes slowly with the

energy

B = (bl+b2)/6. and bl

(b;/2)

$,(z)

=

b,(z)

= A( 2/b;)

A=

z

exp

1 /2

z (l-Bz)

3b;b;/4(

given bl

(-b1z/2)

exp

b;-blb2+b;)]

(-b2z/2)

1’2

(8)

and

Is

A = 0.866 is

Uell$ Wavefunctions

The multiquantum well transition rate can also be calculated by assuming that each single quantum well under bias can be approximated by a simple triangular well [lo]. We use the following trial wavefunctions [ill

where

*I1

2.83~10’

(kv/cm)

= First

Triangular

square

4.34x107

Table

El

rates

-20

-30

J12

Field

Transition

1:

423

Vol. 5, No. 3, 1989

and

b31, as

is and

subband

A

F is

the

that

B = b1/3.

= [ (7’amq2/4ch2)(Ns

the

[ll]

bl

The

= b2,

parameter

[ll]

energy (312

where

found

applied

+ g is

given

nqF)2’3

~~~~~~~ 1’3 as

[ll

(9)

I

CN + 314) 2’3

(10)

field.

Table 2 shows the results of the calculation. In this model, the results are symmetrical with respect to the applied bias. This is obvious since a triangular well appears the same from left-to-right as from right-toleft. Therefore, only the positive bias results are presented.

424

Superlattices

Table 3:

Field

El

(kv/cm)

Ranlrition

e2

Vol. 5, No. 3, 1989

rates using Airy function solution

+

Cl

and Microstructures,

+

J12

J21

J12

J21

J21

20

0.059

0.255

1.01~10~

5.31x107

6.34~10~

8.82~10”

1.67~101~

1.05~10~1

30

0.049

0.246

1.81~10~

9.60~10~

1.14x108

7.48~10~’

1.41~10~~

8.89~101’

40

0.035

0.235

2.98~10~

1.56~10~

1.86x108

4.76~10~’

9.05~10~

5.67~10~’

= First

subband

energy;

~~ = Second

subband

energy

wells are not fully isolated. Interestingly, the calculated transition rates presented here using the finite square well wavefunctions and the Airy wavefunctions are an order of magnitude smaller than those for either the triangular well model or that of two coupled wells.

Conclusions

/ 240

Figure

1:

Numerically calculated electronic wavefunctions for the n=2 and n=l levels as a function of distance in a single quantum well under bias.

Airy Function Solutions The solution of the Schroedinger equation for a particle in an electric field is given by Airy functions 1121. By solving the Schroedinger equation directly, one obtains the wavefunctions as shown in Figure 1 [13]. The numerically calculated transition rates are listed in Table 3. As in the case of the finite square well, the 2-l transition rate decreases while the l-2 transition with increased bias, rate increases with increased bias. Examination of the calculated results for the three different wavefunctions selected here wavefunction triangular shows that the well earlier the closely to calculations agree In the calculations of Weil and Vinter [51. calculations presented in reference 151, the wavefunctions are found by assuming that the two wells interact sufficiently such that the quantum levels are common to both wells; the

We have presented results of hopping conduction in a two-well quantum well structure using three different approximations for the wavefunctions. The calculations show that the 2-l transition rate is larger by one to two orders of magnitude if triangular well wavefunctions are assumed as opposed to finite square well or Airy function solutions. The physical interpretation of this result is quite If either the finite square well or simple. are chosen, it is Airy function solutions implicitly assumed that the wells are noninterthe electronic states can be treated as acting; since this separate, isolated quantum states just correponds to choice of wavefunctions the Therefore, the overlap integral, that. degree to which the electronic states interact, is necessarily small and the resulting transition probability is also small. Physically, well corresponds to the multiquantum this structure consists of very weakly regime ; the very little interacting quantum wells with leakage of the electronic wavefunctions from one well to the next. Under these conditions, the under transition probability is asymmetric applied bias for transitions between the n=2 and n=l quantum levels since the energy separation between the levels in adjacent wells changes with the bias direction. can consider hopping Alternatively, one conduction in the superlattice regime, that in which the wells interact sufficiently such that the electronic wavefunctions must be calculated In this common to all of the wells present [51. the overlap of the wavefunctions is, of case, much greater and the resulting transicourse, tion probability is much larger. Under these

Superlattices

and Microstructures,

425

Vol. 5, No. 3, 1989

the energy level separation in the conditions, This structure is symmetric with applied bias. when the levels are common to is obvious since, the the symmetry of entire structure, the structure cannot be broken by applying a bias The use of triangular well wavefunctions [51. somewhat approximates this condition since these wavefunctions have a much greater spatial extent than either set of finite square well wavefuncapplicable to a multiwell and they are t ions structure which is symmetric with bias.

3.

4.

5.

found that the hopping It is further using rate derived transition conduction closely matches triangular well wavefunctions Hence, the simpler that for two coupled wells. used to roughly triangular well model can be current the conduction determine hopping in a coupled multiquantum well system, that of a Alternatively, finite square well superlattice. wavefunctions provide a more appropriate model determining hopping conduction for phonon between two isolated quantum wells, that of a multiquantum well device.

6.

7.

8.

9. - The authors would like to Acknowledgements Knight and D. Fouts of the Georgia thank P. technical their for Institute of Technology this preparation of the assistance in This work was supported in part by manuscript. Foundation President ial Nat ional Science the Young Investigator Program.

10.

1 References

1.

2.

T. InaLa, S. Muto, Y. Nakata, S. Sasa, T. Fujii, and S. Hiyamisu, “A pseudomorphic In0 53GaO 47As/AlAs resonant tunneling bartier with a peak-to-valley current ratio of 14 at room temperature,” Japanese Journal of Applied Physics, vol. 26, 1987. PP. L1332-L1334, 8. Ricco and M. Ya. Azbel, “Physics of resonant tunneling. The one-dimensional

13.

double-barrier case,” Physical Review R, vol. 29, pp. 1970-1981, 1984. “Tunneling in a finite R. Tsu and L. Esaki, superlattice,” Applied Physics Letters, vol. 22, pp. 562-564, 1973. G. H. Doehler, R. Tsu, and L. Esaki, “A new negative mechanism for differential conductivity in superlattices,” Solid-State Communications, vol. 17, pp. 317-320, 1975. Neil and r). Vinter, “Calculation of T. phonon-assisted tunnel ing between two Journal of Applied Physics, quantum wells,” vol. 60, pp. 3227-3231, 1986. J. F. Palmier, and A. Chomette, D. Calecki, “Hopping conduct ion in multiquantum well Sol id state structures,” J. Phys. C: Phys., vol. 17, pp. 5017-5030, 1984. Movaghar, J. and A. MacKinnon, R. Leo, “Electron transport in multiple-quantum well structures,” Semiconductor Science and Technology, vol. 3, pp. 397-410, 1988. R. Tsu and G. Doehler, “Hopping conduction in a superlattice,” Physical Review R, 1975. vol. 12, pp. 680-686, Price, P. J. “Two-dimensional electron transport in semiconductor layers. I. Phonon scattering,” Annals of Physics, vol. 133, pp. 217-239, 1981. K. S. Yoon, G. B. Stringfellow, and 4. J. Huber, “Monte Carlo calculation of characteristics velocity-field in GaInAs/InP and GaInAs/Al InAs single-well heterostructures,” Journal of Applied Physics, vol. 62, pp. 1931-1936, 1987. E. Yamaguchi, “Theory of defect scattering in two-dimensional multisubband electronic systems on 111-V compound semiconduct~>rs,” Journal of Applied Physics, vol. 56, pp. 1722-1727, 1984. and E. M. Lifshitz, Quantum L. D. Landau Third edition, pp.74-f5, Mechanics, Pergamon Press, 1977. K. F. Erennan and C. J. Summers, “Theory of tunneling in a variably spaced resonant Airy An well structure: multiquantum approach,” Journal of Applied function 1987. Physics, vol. 61, pp. 614-623,