Physica E 11 (2001) 186–189
www.elsevier.com/locate/physe
A method of measuring the contribution of the image potential
energy of the shallow impurity ground state in a quantum well
Tzong-Jer Yang
a; ∗, Shih-Ying Hsu
baDepartment of Electrophysics, National Chiao-Tung University, 1001 TA Hsueh Rd., Hsinchu, Taiwan 30050, ROC bDepartment of Electronic Design Engineering, UMCScienti(c Park, Hsinchu, Taiwan, ROC
Abstract
We show that the binding energy of the shallow impurity (at the center of the quantum well) ground state is strongly reduced by the presence of a metallic mirror at a few e.ective Bohr radii from the quantum well. For a given depth of the quantum well, we 1nd that the absolute value of the image potential energy without the metallic mirror is equal to that with a 1xed metallic mirror while a certain width of the quantum well is met. Hence, it is proposed that the binding energy of the shallow impurity ground state in the quantum well with and without the metallic mirror can be separately measured by the variation of the width of quantum well. The contribution of the image potential energy to the binding energy of the shallow impurity ground state may then be deduced. c 2001 Elsevier Science B.V. All rights reserved.
PACS: 73:40:L
Keywords: Metallic mirror; Image potential energy; Shallow impurity; Quantum well
1. Introduction
The role of image charge e.ect on the binding en-ergy of hydrogenic shallow impurity in a quantum well and the properties of exciton in quantum barrier has attracted much attention [1–4] in recent years. In a very recent work, Benoit ?a la Guillaume et al. [4] studied the quantum well exciton changes induced by
This work is partially supported by National Science Council
of Republic of China.
∗Corresponding author. Tel.: +886-3-5712121; fax: +886-3-5725230.
E-mail address: yangtj@cc.nctu.edu.tw (Tzong-Jer Yang).
the presence of a metal located at a distance from it. They 1nd that the exciton binding energy is strongly reduced by the presence of a metallic mir-ror at a few exciton Bohr radii from the quantum well. Their idea enables rise to us to ask about the e.ect of the metallic mirror on the binding energy of hydrogenic shallow impurity in the quantum well. The result would be similiar to the case of exciton. But when we investigate the issue deeply, we 1nd a new result which can be used to determine the contribution of binding energy due to image poten-tial. This is what we want to present in this short paper.
1386-9477/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
T.-J. Yang, S.-Y. Hsu / Physica E 11 (2001) 186–189 187 In this paper, we calculate the binding energy of
the ground state of hydrogenic impurity seated at the center of quantum well GaAs–Ga1−xAlxAs with the
inclusion of the e.ect due to the image charges (ex-cluding the self-energy image potential). In Section 2, the calculational method is presented. In Section 3, results and discussion are presented. Finally, we give our conclusion.
2. Formalism
The energy level of the ground state for on-center shallow donor in GaAs–Ga1−xAlxAs quantum well
with and without a metallic mirror at a few e.ective Bohr radius from the quantum well is studied within the framework of the e.ective mass approximation. The variational method is used to calculate the ground state energy of on-center shallow impurity (image po-tential not included) in a single quantum well. Then, we treat image potential (excluded self-energy image potential) to be a perturbed potential and use the per-turbation method to calculate the correction energy for both cases, without and with the metallic mirror on one side of the potential barrier.
The hydrogenic impurity is seated at the center of the quantum well with width L. The barrier height of the well is Vo. The Hamiltonian of this system is
H1=−˜ 2 2m∗ 1∇ 2+ V 1(r); |z| ¡L2; H2=−˜ 2 2m∗ 2∇ 2+ V 2(r) + V0; z ¡ −L2; H3=−˜ 2 2m∗ 2∇ 2+ V 3(r) + V0; z ¿L2; where m∗
1; m∗2 is the e.ective mass of,
respec-tively, quantum well (say GaAs) and barrier (say Ga1−xAlxAs). The image charge potential arises
from the di.erence of the dielectric constant be-tween GaAs( 1) and Ga1−xAlxAs( 2). V1(r); V2(r)
and V3(r) represent the sum of the Coulomb
inter-action due to electron-impurity ion, electron-image ion. Let us de1ne the dielectric mismatch between GaAs and Ga1−xAlxAs as p = ( 1− 2)=( 1+ 2) and
p= 2 1=( 1+ 2). The position of the image of the
impurity ion on the z-axis is z+ 0(n) = 2 n − n + 1 2 L 2 + zi + n + 1 2 L 2 − zi + zi; z− 0 (n) = −2 n + 1 2 L 2 + zi + n − n + 1 2 L 2 − zi + zi;
where ziis the position of the hydrogenic impurity in
the quantum well, [x] is the integer part of x and n is the index of the nth image charge of the impurity ion. Let =x2+ y2; r =2+ (z − zi)2. Then V1(r) =−e 2 1 1 r + V1++ V1−; where V+ 1 (r) =e 2 1 n=1p n 2 +[z − z0+(n)]2 −1=2
and V1− is the same expression as V+
1 except z+0
is replaced by z−
0(n); V2(r) = 1= 2pV1+(r) and
V3(r) = 1= 2pV1−(r) .
The contribution from the image potential to the ground state energy of the hydrogenic impurity is expected to be small because p1 if x ≤ 0:4 in Ga1−xAlxAs. Hence, the Hamiltonian can be rewritten
as H = H0+ H; where H0=−˜ 2 2m ∇2− e2 1[2+ (z − zi)2]1=2 + V (z) and V (z) = 0 for |z| ¡L 2; V0 for |z| ¿L2; H= V+ 1 (r) + V1−(r) for |z| ¡L2; V2(r) for z ¡ −L2; V3(r) for z ¿L2:
188 T.-J. Yang, S.-Y. Hsu / Physica E 11 (2001) 186–189 Since the wave function of the electron is mostly
con-1ned in the quantum well, the electron’s e.ective mass and dielectric constant in the quantum well may be used in the expression of H0. The ground state wave
function of H0cannot be obtained analytically. Here,
we use the variational method to get the ground state energy of H0. The tried wave function is
0= f(z) exp
−12+ (z − zi)2
;
where f(z) is equal to A cos(k; z) for |z| ¡ L=2 and Be−k2|z|for |z| ¿ L=2.
f(z) is the ground state wave function in one-dimensional potential well without impurity and k1and k2are de1ned separately as k1=2m∗E0=˜ and
k2=2m∗(V0− E0)=˜.
E
0is the ground state energy in this one-dimensional
potential well and f and df=dz are forced to be con-tinuous at the boundary z =±L=2. Then, two equations containing E
0 are obtained and E0is determined.
The ground state energy of the hydrogenic impurity at the center of the quantum well, excluding the image potential contribution, can be determined by the vari-ation of 0|H0| 0 = E0 with respect to to have a
minimum value E0. Also, the binding energy of
elec-tron in the quantum well is equal to U = E
0−E0. Now
the correction due to the image potential LE is deter-mined by the perturbation method and the corrected binding energy is U= U − LE.
Next, let us consider a metallic mirror located away from the quantum well at a distance d. Between the metallic mirror and the quantum well, there is a quan-tum barrier (here it is Ga1−xAlxAs) (see Fig. 1). Now,
the image potential will be changed due to the pres-ence of the metallic mirror. Then, the correction of the binding energy due to this new image potential LE
would be di.erent from LE. It is interesting to make a comparison between LE and LE over the
varia-tion of the width of quantum well L (in e.ective Bohr radius a∗) for the impurity seated at the center of the
well.
3. Results and discussion
The quantum well is GaAs. The e.ective Bohr radius a∗= 103:4 MA and the e.ective Rydberg Ry∗=
5:29 meV. In the case of no metallic mirror, the
Fig. 1. Schematic diagram of metal |Ga1−xAlxAs|GaAs|Ga1−x
AlxAs.
correction LE due to the image potential is negative because p is positive. LE becomes smaller while the width of the quantum well increases. The bar-rier height V0 increases, the dielectric mismatch also
becomes bigger, which will give rise to an increase in LE. Now, in the presence of the metallic mirror, the correction energy LE is positive, because the
metallic mirror will give rise to a stronger Coulomb interaction between the negative image charge and electron. But LEwill become smaller as V
0increases
for a given width of the quantum well. This is due to the electron wave function being more localized in the quantum well; the e.ect of the metallic mirror will be less. For a given V0, LE will become larger
as the width of the quantum well decreases from 10a∗
until a certain width. Now, how about the e.ect on LE while we change the position of the metallic
mirror for a given V0 and the width of the quantum
well? We have carried out the calculation of LE
over the range of d from 5a∗ to 800a∗for a given
width L = a∗ and V
0= 10Ry∗. LE would approach
the value of LE as d → ∞. We 1nd the value of LE at d = 800a∗ to be quite consistent with LE as
expected. Here, we specially attach Fig. 2 to show the results of the absolute value of the correction due to image potential with and without the metallic mirror for a given V0= 50R∗y over the range of L = 0:9a∗to
1:3a∗. We 1nd that |LE| = |LE| = 1:57×10−2Ry∗at
L = 1:1665a∗. The binding energy of the ground state
of the donor shallow impurity (including image poten-tial) is U= U −LE. Thus, U
T.-J. Yang, S.-Y. Hsu / Physica E 11 (2001) 186–189 189
Fig. 2. The absolute value of the contribution of the image potential to the binding energy of the impurity at the center of the quantum well with (|LE|) and without (|LE|) the metallic mirror varies with the width of quantum well for V0= 50Ry∗.
the metallic mirror. U
m= U − |LE| with the metallic
mirror, therefore U
wm− Um = 2|LE| 0:166 meV.
Hence, the binding energy of the ground state of the impurity with and without the metallic mirror can be measured seperately. Then, the image potential con-tribution to the binding energy can be determined at a certain width of the quantum well.
4. Conclusion
We have demonstrated how to get the contribution of the image potential to the binding energy of the shallow impurity at the center of a quantum well. The method we propose here is a general one which may be applied to the case of the shallow impurity center not only at the center of the quantum well,
but also the dielectric constant of the barriers on the sides of quantum well may be set to di.erent values. The self-energy of the image potential can be eas-ily incorporated into our future work. We do believe that our idea to obtain the contribution of the ground state binding energy due to image potential cannot be changed without self-energy of the image potential. References
[1] C. Mailhiot, Y.C. Chang, T.C. McGill, Phys. Rev. B 26 (1982) 4449.
[2] M. Kumagai, T. Takagahara, Phys. Rev. B 40 (1989) 1239. [3] D.B. Tran Thoai, R. Zimmermann, M. Grundmann, D.
Bimberg, Phys. Rev. B 42 (1990) 5906, and references therein. [4] C. Benoit ?a la Guillaume, M. Combescot, O.
