Resonant tunneling through AlxGa1−xAsGaAs heterostructures

Superlattices and Microstructures, Vol. 4, No. 2, 1988

RESONANT

TUNNELING

THROUGH

245

AlxGal_xAS-GaAs

HETEROSTRUCTURES

L u i z A. C u r y D e p a r t a m e n t o de F i s i c a e C i ~ n c i a dos M a t e r i a i s I n s t i t u t o de F [ s i c a e Q u l m i c a de S a o C a r l o s Universidade de S a o P a u l o - 1 3 5 6 0 - S a o C a r l o s - SP - B r a z i l

Departamento

Nelson Studart de F i s i c a , U n i v e r s i d a d e F e d e r a l 5 0 7 3 9 - R e c i f e - PE - B r a z i l (Received

August

16,

de P e r n a m b u c o

1987)

A n i t e r a t i o n m a t r i x f o r m a l i s m is d e v e l o p e d to c a l c u l a t e the t r a n s m i s s i o n probability through multibarrier semiconductor heterostructures. The quasib o u n d a n d v i r t u a l s t a t e s are d e t e r m i n e d b y t a k i n g i n t o a c c o u n t the e x a c t s o l u t i o n of the e f f e c t i v e - m a s s equation with appropriate boundary conditions in the b a r r i e r s a n d w e l l s . T h e c u r r e n t d e n s i t y in d o u b l e - b a r r i e r d e v i c e s is c o m p u t e d as a f u n c t i o n of the v o l t a g e a p p l i e d in t h e d i r e c t i o n p e r p e n d i c u l a r to the h e t e r o j u n c t i o n i n t e r f a c e s . It is s h o w n t h a t the c o r r e c t r e p r e s e n t a t i o n of t h e p o t e n t i a l p r o f i l e a n d a d i r e c t e v a l u a t i o n of the c u r r e n t d e n s i t y a r e q u i t e i m p o r t a n t for c o m p a r i s o n w i t h e x p e r i m e n t a l current-voltage characteristics. The c o n t r i b u t i o n of t h e v i r t u a l s t a t e s at h i g h e n e r g i e s is a l s o a n a l y s e d .

I.

INTRODUCTION

Over the l a s t f e w y e a r s a lot of w o r k h a v e b e e n d e d i c a t e d to s t u d y resonant t u n n e l i n g in m u l t i b a r r i e r heterostructures. The g r o w i n g i n t e r e s t in t h i s e f f e c t is the p o s s i b i l i t y , among others, of constructing practical quantum well oscillators as a s o u r c e of very high f r e q u e n c i e s I . On the o t h e r hand, it m a y be w o r t h w h i l e to emphasize that reson a n t t u n n e l i n g w a s the ke[ concept in the s u p e r l a t t i c e proposal . The most studied system c o n s i s t s of a l t e r n a t i n g A l x G a l x As b a r r i e r s and G a A s w e l l s s a n d wiched- by ohmic contacts which furnish electrons to tunnel through the multibarrier s t r u c t u r e . M o s t of the w o r k is c o n c e n t r a t e d on h i g h - q u a l i t y doubleb a r r i e r s a m p l e s g r o w n b y M B E or MOCVD 3 However, sequential resonant tunneling through multiquantum well superlattices h a s a l s o b e e n o b s e r v e d ~'s From a t h e o r e t i c a l p o i n t of view, Tsu a n d E s a k i 6 w e r e the f i r s t to i n v e s tigate transport properties along a finite superlattice. T h e y i d e n t i f y the energy positions of the transmission p e a k s w i t h the q u a s i - b o u n d l e v e l s of the quantum w e l l s a n d c a l c u l a t e the t u n n e l ing c u r r e n t as a f u n c t i o n of the a p p l i e d v o l t a g e for a d o u b l e - a n d t r i p l e b a r r i e r . The s u p e r l a t t i c e w a s m i m i c k e d b y o n e - d i mensional p o t e n t i a l b a r r i e r s and wells

0749 6036/88/020245 + 06 S02.00/0

of c o n s t a n t h e i g h t a n d d e p t h , e v e n u n d e r bias voltage, w h o s e o n l y e f f e c t w a s to decrease in a u n i f o r m w a y the bottom of the c o n d u c t i o n b a n d ( s t a i r c a s e p o t e n t i a l ) . The o b v i o u s a d v a n t a g e of t h i s approximation was that they could write the w a v e f u n c t i o n s as i n h o m o g e n e o u s p l a ne w a v e s . E f f e c t s c o m i n g f r o m the v a r i a t i o n of the e f f e c t i v e mass from layer to layer, depletion charges and nonparabolicity of the b a n d s t r u c t u r e w e r e all neglected. Subsequent works have tried to r e f o r m u l a t e the original Tsu and E s a k i m o d e l by t a k i n g s o m e of t h e s e effects into account or have merely simplified basic assumptions in o r d e r to turn the c a l c u l a t i o n s e a s i e r (for instance, W K B a p p r o x i m a t i o n , or 0-function p o t e n t i a l w e l l s ) 7'8 A g e n e r a l f o r m a l i s m of t r a n s m i s s i o n through multibarriers was briefly outl i n e d in a p r e v i o u s p a p e r ~. It is b a s e d on an i t e r a t i o n m a t r i x s c h e m e and t a k e s i n t o a c c o u n t the e x a c t s o l u t i o n s of the S c h r ~ d i n g e r e q u a t i o n in the b a r r i e r s and wells. In t h i s a p p r o a c h , we e m p l o y the envelope-function approximation where the rapidly varying Bloch component of the wave f u n c t i o n in i g n o r e d and the problem reduces to the familiar effective-mass equation. The relevant p a r a m e t e r s w h i c h e n t e r i n t o the c a l c u l a t i o n s are the b a n d o f f s e t s and e f f e c t i v e m a s s e s . The e f f e c t i v e - m a s s approximation

© 1988 Academic Press Limited

Superlattices and Microstructures, Vol. 4, No. 2, 1988

246 in tunneling phenomena could be questioned since it c o m p l e t e l y neglects mixing effects of the F and X valleys that should be important at large aluminum c o n c e n t r a t i o n where the F and X points have the same energy. However, a calculation using an empirical tightbinding approach has shown that the transmission probability is quite similar to that c a l c u l a t e d in the simple e n v e l o p e - f u n c t i o n a p p r o x i m a t i o n *° In this paper we use the iteration matrix formalism to calculate the transmission p r o b a b i l i t y as a function of the energy of the incident electron and hence the tunneling current versus the applied voltage. A minor modification in the previous work is made in order to include the massd ependent conditions on the d e r i v a t i v e of the wave function at the interfaces. We focus our attention in d o u b l e - b a r r i e r h e t e r o s t r u c t u r e s and compare the results with recent t u n n e l i n g - c u r r e n t measurements 11 . We find that a correct representation of the form of the potential

where n=2,3...2N, n even (odd) means the regions of barriers (wells) and V o is the uniform barrier height. The solution of the SchrSdinger equation can be exactly written in terms of linear combinations of the Airy functions as Tn (z) =CnAi (-Z//n-~n) +DnBi (-Z//n-6n) where even

~n =(E-Vo

)/eFln

(=E/eFln)

for

(2) n

(odd), I n = ( ~ 2 / 2 m ~ e F ) i/3 is a charac-

teristic length, and m R is the effective masses in the barriers and wells. The iteration matrix M. defined as 3 Cj = S. (Cj +i) (3) Dj 3 Dj+I can

be determined by using the boundary

conditions ~j = ~j+l and (m*j + i / m j*) ~ j'~ j +' l with j=2,3...2N-I in order to conserve the q u a n t u m flux at each interface. The result is

I Ai(~j+l)Bi'(~j) - ojAi'(ej+l)Bi(ej)

Bi(~j+l)Bi'(~j) - ~jBi'(~j+l)Bi(~j)

~jAi'(ej+l)Ai(ej) -Ai(Uj+l)Ai'(u j)

~jBi'(ej+l)Ai(~j) - Bi(~j+l)Ai'(~j)

(4)

3

and a correct c a l c u l a t i o n of the current with integration over all energies, are both n e c e s s a r y for a d e s c r i p t i o n of the c u r r e n t - v o l t a g e curves at large voltages.

where

the

prime means

differentiation

and ~ j = - z j / / j - 6 j , ~ j + l = - Z j / / j + l - 6 j + l

~j=m~lj/m~+llj+ 1 , z2j=JLb+ (j-l) L w II.

GENERAL

FORMALISM

z2j+l=j (Lb+Lw)

Let us review now the general procedure discussed in our earlier work. We consider the s u p e r l a t t i c e structure, c o nsisting of N barriers under an applied electric field F c h a r a c t e r i z e d by a potential given by

0

if z<0

Vo-eFz

if n even and 0
-eFz

if n odd and 0
-eFL

if z>L

. Here ib and Lw are bar-

rier and well width respectively. Using the matching conditions at the outside interfaces (z=O and z=L), we obtain the connection between the reflection and transmission amplitudes through a 2x2 matrix S defined as

where V n (z) =

and

(i)

S=P

R 2N-I H j=2

M..Q, ]

(6)

with

! 1

/

iki/bAi(8)

- AAi'(8)

iki/bBi(8)

- ABi' (8)

I

iki/bAi(8)

+ AAi' (8)

iki/bBi(8)

+ ABi' (8)

(7) P =

~

Superlattices and Microstructures, Vol. 4, No. 2, 1988

247

and

\ [Bi' (y) + ~ % B i ( y ) / A ] e x p ( i k L L )

[~;~

- EikLIAi (y)/A + Ai' (y)] exp (ikLh)

Here k i = ( 2 m w.E / ~ 2) ~12,kL=[2m~(IVLi +E) i~2]i/2

III.

APPLICATION

TO

(8)

)

(~)/A - ~ ' I~)]exp (-~u L)

DOUBLE

-

BARRIER

HETEROSTRUCTURES

8=(Vo-m)/emfb,

Y =-L/lb+ 6 and A =m~/m~.

Finally the transmission cient is obtained as

coeffi-

T*T =(I/S] i)*(i/Sl]) (kL/k i)

(9)

For comparison, it is useful to exhibit the iteration matrix in our formulation for the staircase model corrected for mass variation. In this approximation 2N S= K Mj, j=l where

(i0)

now + ~j exp(kj+l-kj)z j

M.

Numerical results of the transmission probability as a function of the energy were displayed in Ref.9 for double, triple, and quintuple barriers. We have also seen that the staircase model is a good a p p r o x i m a t i o n in the limit of low electric fields. Now, we are concerned with the w e l l - s t u d i e d AlxGal_ x A s -GaAs double-barrier devices. In particular, we want to compare our results with I-V characteristics measured by Mendez et al. II . In the present calculations we have used the following physical parameters: m~=0.0665m O , m~=0.1Olm O (m O is the free-electron mass), V o = 0 . 2 9 6 e V , Lw=60A and Lb=I00A.

~] exp-(kj+l+k j) zj (ii)

=

3 ~j exp(kj+l+kj)z j

with

~

The wave

l

=

(l±m~kj+i/m~+ikj)/2.

vectors

k

are defined

i[(2m~/~2) (E+IVjl] I~

k.= 3 I[(2m{/~2) (Vj-E)] I/2 , % where

6~ exp-(kj+l-kj)z j

0 as

if j odd

-7 (12)

if j even

Vj=Vo-(J-2)AV/2 (=-(j-I)AV/2)

~ ~4

for

j even (odd) and AV=eFL/N. The expression of the iteration matrix Mjis much simpler than that of Ref.6. From the transmission coefficient expression (Eq.9), we can compute the current density component along the superlattice as

-RI

-28

-55 0

em*kB@ J = ~2- - ~

I~

Fl+exp (EF-E)/kB@ l T*T log [l+exD (~_E_eV)/kBo] dE (i3)

o where @ is the temperature, k B is the Boltzmann constant and E F is the Fermi energy.

' 0J

' OR

, 03

, 04

, 0.5

E(eV)

Fig.l - L o g a r i t h m of the transmission p r o b a b i l i t y versus the energy of the incident electron for a d o u b l e - b a r r i e r device (100A-60A-100A). The eletric field F=4xl0"V/cm. The physical parameters are given in the text.

l

EF

®

0

l

l

l

l

[

,

*

,

,

, 200

I

s

F:40

kV/cm

300

d i a g r a m for the same in F i g . l . In t h e G a A s s o l i d l i n e s d e n o t e the a n d the d a s h e d line corresponding to the in the transmission in F i g . l .

L(~)

*

11.2

18Z5

,

AIGoAs

.__2%4:?_

I00

l

Fig.2 - Energy heterostructure as q u a n t u m w e l l , the quasi-bound states is a v i r t u a l s t a t e first three peaks probability shown

-104

0

40

,.i

>

296

AIGo A s - G a A s -

o.

w

-32

-28

- 24

-20

-i6

-12

-8

i 0.4 V (Volts]

0,447 V

i 0.6

i 0.8

Fig.3 L o g a r i t h m of the transmission p r o b a b i l i t y v e r s u s the a p p l i e d voltage for the same double-barrier d e v i c e as in F i g . l . T h e e n e r g y of the i n c i d e n t elect r o n f r o m the l e f t G a A s e l e c t r o d e is 15 meV.

i 0.2

0,0965 V

i j.._ 1.0

h) 4::. CO

Superlattices and Microstructures, Vol. 4, No. 2, 1988 F i g o l shows a t y p i c a l plot of the transmission probability versus the energy for F = 4 x l 0 4 V / c m . We o b s e r v e two sharp peaks a s s o c i a t e d with the quasi-bound levels of the quantum well and broad peaks c o r r e s p o n d i n g to v i r t u a l states at high energies. In Fig.2 we plot the e n e r g y d i a g r a m c a l c u l a t e d very a c c u r a t e l y by i n s p e c t i o n of the peaks l o c a t i o n s of the transmission p r o b a b i l i t y shown in Fig.l. It is apparent from this figure that the virtual state at 254.9 meV must be sensitive to the d e t a i l e d form of the potential, in p r e s e n c e of the applied electric field, and it is not at all surprising that this e n e r g y region is not correctly d e s c r i b e d by the staircase potential. Fig.3 shows the t r a n s m i s s i o n probability as a f u n c t i o n of the applied voltage for incident electrons with energy E=lSmeV. A good quantitative a g r e e m e n t is found b e t w e e n the t h e o r e t i cal and e x p e r i m e n t a l data of q u a s i - b o u n d states of sample B in Ref.ll. However, we must point out that the experim e n t a l r e s u l t s can always be w e l l - f i t t e d in this way by only a d j u s t i n g the e n e r g y of the incident electron. In o r d e r to make a more r e a l i s t i c comparison one must c o m p u t e the t u n n e l i n g current. Fig.4 shows the c a l c u l a t e d J - V curves, w h i c h e x h i b i t features similar to the e x p e r i m e n t a l ones. We found a reasonable a g r e e m e n t b e t w e e n the p o s i t i o n s of the experimentally measured (indicated by arrows in Fig.4) and c a l c u l a t e d peaks a s s o c i a t e d w i t h the b o u n d states of the q u a n t u m well. The a b s o l u t e intensity of the peaks are d i f f i c u l t to compare b e c a u s e there are other sources of current and we c o n s i d e r here only the elastic resonant component. At high bias, the a g r e e m e n t is not so good. We have found other negative resistance regions w h i c h m a y be a t t r i b u t e d to high energy v i r t u a l states in the continuum a r i s i n g from q u a n t u m - m e c h a n i c a l reflections at the interfaces. Furthermore, virtual states have r e c e n t l y b e e n observed in transport measurements of heavily-doped q u a n t u m w e l l s 12 . On the other hand, M e n d e z et al. interpreted their data by a s s u m i n g an alternative mechanism. The o b s e r v e d f e a t u r e s at high bias w o u l d be a s c r i b e d to r e s o n a n t tunn e l i n g t h r o u g h X - s t a t e s of the B r i l l o u i n zone. Even though, we can not d i s c a r d this p o s s i b i l i t y , we can not also ignore the c o n t r i b u t i o n of v i r t u a l states in tunneling processes. The o b v i o u s conc l u s i o n is that we need a more complete theory of r e s o n a n t t u n n e l i n g in superlattices. In summary, we have used a simple a p p r o a c h to d e s c r i b e the perpendicular

249

1500 -

1200 E 0

"~ < 900

B

i

-=~

600

0

0

,

0

8

VOLTAGE

2

, (V)

Fig.4 - Theoretical J-V characteristics at zero t e m p e r a t u r e for the same h e t e r o s t r u c t u r e as in p r e v i o u s figures. The a r r o w s i n d i c a t e the p o s i t i o n s of the negative-resistance r e g i o n s in the exp e r i m e n t a l c u r v e s (Ref. ii).

transport properties in semiconductor heterostructures. We have c a l c u l a t e d the energy of b o u n d and v i r t u a l states in q u a n t u m w e l l s w h i c h are in good a g r e e m e n t with experimental results of doubleb a r r i e r devices. The p r o b a b i l i t y transmisssion and the c u r r e n t d e n s i t y v e r s u s the bias v o l t a g e are also computed.

Acknowledgments - We wish to thank C.E.T. G o n ~ a l v e s da Silva and G i l m a r E. M a r q u e s for u s e f u l d i s c u s s i o n s . We t h a n k also Prof. B,I. Halperin for enlightening discussions on this and related topics and for a critical reading of the m a n u s c r i p t . One of us (N.S.) a c k n o w l e d g e s Universidade Federal de Sao Carlos for a leave of absence. This r e s e a r c h was s u p p o r t e d in part by F u n d a G a o de A m p a r o a P e s q u i s a do Estado de Sao Paulo (Fapesp) and Conselho N a c i o n a l de D e s e n v o l v i m e n t o Cient{rico e T e c n o l 6 g i c o (CNPq). This paper is b a s e d in part on the M.Sc. dissertation s u b m i t t e d by L.A.C. to the Instituto de F { s i c a e Q u f m i c a de Sao Carlos, U n i v e r s i d a d e de Sao Paulo.

Superlattices and Microstructures, VoL 4, No. 2, 1988

250 REFERENCES i.

2. 3.

4.

5.

6. 7.

T.C.L.G. Sollner, P.E. T a n n e n w a l d , D.D. Peck and W.D. Goodhue, Appl. Phys. Lett. 4, 1319 (1984). L. Esaki, J. Q u a n t u m Electron. QE22, 1611 (1986). A.R. Bonnefoi, T.C. McGill, R.D. Burnham and G.B. Anderson, Appl. Phys. Lett. 50, 344 (1987) and ref e r e n c e s therein. F. Capasso, K. M o h a m m e d and A.Y. Cho, Appl. Phys. Lett. 48, 478 (1986). T. Nakagawa, H. Imamoto, T. K o j i m a and K. Ohta, Appl. Phys. Lett. 49, 73 (1986). R. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562 (1973). M . O . V a s s e l , J . L e e and H.F. Lockwood, J. Appl. Phys. 54, 5206 (1983).

8.

B. R i c c o and M. Ya. Azbel, Phys. Rev. B 29, 1970 (1984) and 29, 4356 (1984). 9. L.A. Cury and N. Studart, Superl a t t i c e s and M i c r o s t r u c t u r e s , ~, 175 (1987). i0. T. Ando : " T h e o r y of S e m i c o n d u c t o r Heterostructures-Many-body ~d int e r f a c e Effects" in Proc. 3 Braz i l i a n School Phys. Semiconductors, C a m p i n a s (1987). ii. E.E. Mendez, E. Calleja, C.E.T. G o n g a l v e s da Silva, L.L. C h a n g and W.I. Wang, Phys. Rev. B33, 7368 (1986). 12. M. Heiblum, M.V. Fischetti, W.P. Dumke, D.J. Frank, I.M. Anderson, C.M. Knoedler and L. Osterling, Phys. Rev. Lett. 58, 816 (1987).