M Tomak

S Balasulbramanian

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IC/85/202
INTERNATIONAL CENTRE FOR
THEORETICAL PHYSICS
en
RESONANT IMPUBITY STATES IN A QUANTIM WELL
INTERNATIONAL
ATOMIC ENERGY
AGENCY
UNITED NATIONS
EDUCATIONAL,
SCIENTIFIC
AND CULTURAL
ORGANIZATION
S. Balasulbramanian
and
M. Tomak
1985MIRAMARE-TRIESTE

IC '85 '2K: 1. INTRODUCTION
International Atomic Energy Agency
and
United Nations Educational Scientific and Cultural Organization
ItfTERHATIONAL CEHTRE FOR THEORETICAL PHYSICS
HESONAHT IMPURITY STATES IN A QUANTUM WELL
S. Balasubramanian * and H. Tomak **
International Centre for Theoretical Physics, Trieste, Italy.
ABSTRACT
The position of a resonant state relative to a localized
do'jor state and the vidth of the resonant state are calculated
for a hydrogenic donor in a GaAs-Ga Al As quantum well as a
function of the width of the well. The results are compared
with experiments and earlier theoretical estimates.
MIEAMARE - TRIESTE
August 1985
Hydrogenic donors in a semiconductor quantum well have been studied
extensively during the past few years With highly controllable growth
techniques, experimental studies on the properties of the shallow donors in
semiconductor quantum well structures have now become possible. For example,
2)
Jarosik et al. have reported the far-infrared magnetospectroscopic measurements
on shallow donors in GaAs-Al Ga As multiple quantum veil structures.
3) x
Perry et al. have carried out resonant Raman scattering experiments on
GaAs(Si doped)-Al Ga As quantum well structures and observed narrow, resonant
h)
donor states. Earlier, Priester et al. have calculated the position and
width of resonant states assuming an infinitely deep quantum well. For a
quantum well of finite depth, it is known, at least for the discrete donor
states, that the spreading of the donor wave function outside the well for small
well widths considerably alters the results of the infinite well case. The
samples used in the experimental work of Ref.3 are sufficiently thick (238 A
and h60 A) and the experimental resonant energy positions could he compared
with the calculations of Priester et al. It will he of interest to calculate
the resonant states as a function of well width for a finite well depth.
In the present work, we report our results of variational calculations
of the position and width of resonant states in a GaAs quantum well of finite
depth for different well widths.
II. THEORY
We consider a donor atom at the centre of a symmetrical quantum well
described by the potential energy function
'0 i
1*1 < T .
L.
2.
(l)
t To be submitted for publication.
* Permanent address: School of Physics, Hadurai KamaraJ University,
Madura!-625021, India.
•* Permanent address: Department of Physics, Middle East Technical
University, Ankara, Turkey.
The Hamiltonian for the donor electron Is given as
-2-
H • H (2)
where
(3)
m* ill Eq.(3) is the electron effective mass and € in Eq.(2) is the
static dielectric constant of CaAs. The eigenstates of H are
(It)
where S is the layer surface area and
Cos lEl < t (5)
for even parity states. The continuity of f^fz) and its derivative at
z = ± L/2 and the normalization condition determines the constants A, B and C
and the allowed values for a. B in Eq.(5) is related to a and the depth
of the well VQ. For odd parity states A is replaced by -A and coscca
by sinaz. We shall use the effective Bohr radius SL = €,<riS/m*e2 as the
unit of length and the effective Rydberg R = m*eh/2£2h2 as the unit of energy.
The eigenvalues Of E- are
E = (6)
where a is obtained as solutions of the equation
-3-
•*.t » . * !
(even parity)
(T)
**) Parity)
For the even parity states the argument of cosine is either in the first or third
quadrant. Similarly, for the odd parity states CoL/2) should be in the second
or fourth quadrant.
With the Impurity potential present, we take as a trial function for
the donor ground state
(8)
where the variational parameter a is obtained by minimizing the expectation
value of H. One thus obtains the ground state of hydrogenic donors associated
with any subband.
We wish to calculate a state above the first subband (in the continuum
of the first subband) and is resonant with the ground state of a donor associated
with the third subband. This resonant state is conveniently obtained if one
gets the energy around which the partial density of states has a peak. The
partial density of states is related to the imaginary part of Green's function
for the resolvent operator (£' - H ) ~ , where H is the Hamiltonian in Eq.(2).
If we write E. as the impurity ground state energy the diagonal part of Green's
function is given by
-1
(9)
where V.fc is the matrix element of the impurity potential between the localized
impurity ground state (wave function in Erj,(8)) and the band state (wave function
in Eli. C O ) . The position of the resonant level and its width are given,
respectively, by
-l i -
III. RSSULTSAND DISCUSSIONS
and
(10)
(11)
The svumnation may "be replaced "by an integration with respect to energy.
Suhstitut
equat ion
b ing for the matrix elements, E is obtained as the solution of the
H
E - F. - f
where
g(p) in Eq,(l3) is given by
h,
The binding energy of a donor ground state associated with the third
subband (even parity) as a function of well width is presented in Fig.l. This
figure also shows the separation of the resonant level from the impurity level,
again as a function of well width. As the well width decreases the separation
increases andthen starts decreasing. Thus qualitatively we observe similar
results for the resonant state and the localized ground state when a finite well
is considered. The "turning over" is however not sharp for the resonant states.
In order to make direct contact with the experiments we have presented
in Table 1 our results for the two wells of widths 238 A and lt60 A. Since
the Al content in the two experimental samples were different, this has also
been taken into accovuit in the present calculations. Transition energies are
in reasonable agreement with the experimental results.
Table 2 shows the variation of the width of the resonant state as a
function of well width. The widths are of the same order of magnitude as the
separation of the resonant state from the localized state. This feature is
sinnilar to the results in Eef.lt. The numerical values of Ref.'t, however, are
quite different from the results presented in our work.
One can repeat the calculations for the odd parity states. Experimentally
3)transitions involving the first and second subbands have been observed
expect qualitatively the same features for this case also with regard to
variations with well widths.
In summary, we have calculated the position and width of resonant
donor states involving the first and third subband for a GaAs quantum well
with finite depth.
We
The width of the resonant state 'becomes
16a'- F<v (15)
The numerical results are presented in the next section.
ACKNOWLEDGMENTS
The authors would like to thank Professor Ahdus Salad, the International
Atomic Energy Agency and UNESCO for hospitality at the International Centre for
Theoretical Physics, Trieste.
-6-
REFERENCES
TABLE 1
1) For a brief review, see:
R.L. Greene and K.K. BajaJ, Sol. State Com. 53, 1103 (1985).
2) H.C. Jarosik, B.D. McCombe, B.V. Shanahrook, J. Comas, J; Ralston and
G. Wicks, Phys. Rev. Lett. 5]*,, 1283 (1985).
3) T.A. Perry, R. Merlin, B.V. Shanabrook and J. Comas, Phys. Rev. Lett. 5^,
2623 (1985).
It) C. Prtester, G. Allan and M. Lannoo, Phys, Rev. B2£, 3l»o8 (198M.
5) R.C. Miller, D.A. Kleinman and A.C. Gossard, Phys. Rev. B29_, 7085 (l9Sit).
6) F. Dingle in FestKorper-Probleme (Advances In Sol. State Phys.) Vol.XV,
Ed. H,J. Quelaser (Pergamon Vieveg,Braunschweig, 1975).
Comparison of the calculated values with experimental results
2.!tl
U.66
0.23
0.2U
0.070
(0.06U)
0.026
(0.03UJ
Th.
70.37
(66.6U)
25.36
(2lt.6U)
Eicp.
23. h
Th,
63.31
(59-76)
20.23
(19.76)
Exp.
^ 58
SI .
Ej. is the Impurity ground state associated vlth the first subband and
Eg. , is the third subband energy. Quantities in brackets are estimates
based on the 'Miller rule'5' for getting Vn. The other values are
based on the 'Dingle rule' '.
-7-
TABLE 2
Variation of width of the resonant state (T)
vith well width (L) for V = £9.2
1.20
1.50
1.60
2. 00
2.20
2.1+1
3.50
It.66
r
(R)
0.001125
0.006058
0.006957
0.009335
0.010602
0.003>*39
O.01761t9
O.02ltlt60
E,(R)
08
X
Q015
0.010
0005
20 30
<«,!
4.0
0000
Fig.l Variation of donor hinding energy (E^) with well width (L)
for a composition x = 0.2^.
Variation of the separation of the resonant state from the
localized state (E - E,) as a function of the well width
(L) for a composition x = 0.2U.
-10-



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