CHIN.PHYS.LETT.
Vol. 21, No. 12 (2004) 2500
E ects of Crossed Electric and Magnetic Fields on Shallow Donor Impurity Binding Energy in a Parabolic Quantum Well 1
E. Kasapoglu1 , H. Sari1 , I. Sokmen2
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey 2 _ Department of Physics, Dokuz Eylul University, Izmir, Turkey
(Received 29 March 2004) We have calculated variationally the ground state binding energy of a hydrogenic donor impurity in a parabolic quantum well in the presence of crossed electric and magnetic elds. These homogeneous crossed elds are such that the magnetic eld is parallel to the heterostructure layers and the electric eld is applied perpendicular to the magnetic eld. The dependence of the donor impurity binding energy to the well width and the strength of the electric and magnetic elds are discussed. We hope that the obtained results will provide important improvements in device applications, especially for a suitable choice of both elds in the narrow well widths. PACS:
71. 55. Eq, 71. 55.
i, 73. 20. Dx
Recent years, many theoretical and experimental investigations have been performed on the issue of the hydrogenic binding of an electron to a donor impurity which is con ned within low-dimensional heterostructures.[1 14] The understanding of the electronic and optical properties of impurities in such a system is important because the optical and transport properties of device made from these materials are strongly a ected by the presence of shallow impurities. Parabolic structures are well known in designing infrared detectors with low leakage currents and low electric- eld sensitivity.[15;16] The e ect of an applied electric eld on the physical properties of such structures is of considerable interest.[17;18] For instance, the application of an electric eld in the crystal growth direction causes a polarization of the carrier distribution and shifts the quantum energy states, which can be used to control and modulate the intensity output of the device.[19] Also, magnetic and electric elds are e ective tools for studying the properties of impurities in heterostructures. A very interesting situation arises when a parabolic quantum well (PQW) is submitted to crossed magnetic and electric elds, because there is a particular eld regime value that the e ective potential turns into a double quantum well.[20] Thus, the use of the combined e ects of an electric and magnetic eld on impurity states is also interest. In this Letter, we report a calculation, with the use of a variational approximation, of the ground state binding energy of a hydrogenic donor impurity in a parabolic quantum well (PQW) in the presence of crossed electric and magnetic elds. Electric eld is applied parallel to the growth direction (z-direction), magnetic eld is applied perpendicular to the growth direction or electric eld. Parabolic compositional pro les are generated by alternate deposition of thin undoped layers of GaAs and Ga1 x Alx As of varying
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c 2004 Chinese Physical Society and IOP Publishing Ltd
thickness. Computer control is employed in the deposition. The relative thicknesses of the Ga1 x Alx As layers increase dramatically with distance from the well centres while those of the GaAs layers decrease. In our calculations, we obtain the explicit dependencies of the impurity binding energy upon the well width, the strengths of the electric and magnetic elds. We de ne the z-axis to be along the growth axis, and take the electric eld to be applied in the growth direction, and the magnetic eld to be applied in the x-axis, i.e. B = (B; 0; 0). We choose a vector potential A in the form A = (0; Bz; 0) to describe the applied magnetic eld. Within the framework of an e ective-mass approximation, the Hamiltonian of a hydrogenic donor impurity in a PQW in the presence of crossed electric and magnetic elds can be written as H
h i2 = 2m1 pe + ec A(re ) + V (ze ) e
+ eF ze
e2
"0
jre ri j ;
(1)
where me is the e ective mass, e is the elementary charge, pe is the momentum, "0 is the dielectric constant, r(= re ri ) is the distance between the carrier and the donor impurity site and V (ze ) is the con nement potential pro le for the electron in the zdirection. The PQW potential pro le and amplitude of normalized subband wave function of electron-j (~z )j2 versus the normalized position z~ = z=L are given in Figs. 1(a) and 1(b) for F = 0, B = 0 and F 6= 0, B 6= 0, respectively (vertical lines indicate the edges of barriers). The functional form of the con nement
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E. Kasapoglu et al.
potential is given by
8 V0 ; > > >
4V0 ze2 ; > > L2
V ze
:
2
ze
L= :
e
e
e2
; (3) + (zz zi )2 p where the term (= x2 + y2 ) is the distance between the electron and impurity in the x{y plane. Equation eB (3) does not contain the term m c ze py because the e expectation value of this term is identically zero for the chosen trial wave function. "0
p
2
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potential well with the Lb width. These bases are formed as r 2 cos h n z Æ i; (5) n (z ) = n L L b
where Æn = 0 if n is odd, and Æn = =2 if n is even. The solutions in the z-direction are described by (z ) =
PQW potential pro le and amplitude of normalized subband wave function of electron, (~z ) 2 , versus the normalized position z~ = z=L: (a) for case F = 0, B = 0 and (b) for case F = 0 and B = 0.
6
6
j j
We propose the following variational trial wave function for the electron bound to impurity (r) = (z)'(; ): (4) To solve the Schrodinger equation in the z-direction, we take the base of the eigenfunctions of the in nite
1 X
n=1
()
cn n z :
(6)
In calculating the wavefunction (z), we ensure that the eigenvalues are independent of the chosen in nite potential well width Lb and that the wavefunctions are localized in the well region and the wave function in the x{y plane, '(; ) is chosen to be the wave function of the ground state of a two-dimensional hydrogen-like atom. We have also used these functions (Eqs. (5) and (7)) in our previous studies,[21;22] 1 2 1=2 e = ; (7) '(; ) =
where is a variational parameter. The ground state impurity binding energy is obtained to be Eb
Fig. 1.
b
= Ez min h jH j i;
(8)
where Ez represents the ground-state energies of electron obtained from Schrodinger equation in the zdirection The values of the physical parameters used in our calculations are me = 0:0665m0 (m0 is the free electron mass), we have not discussed the e ect of a spatially dependent e ective mass (SDEM) on the electron in a nite PQW. Because the e ect of an SDEM on the general feature of the impurity binding energy is not signi cant, it is known that the e ect of an SDEM vanishes as the value of the well width and the magnetic elds increase.[23] The static dielectric constant e0 is assumed to be same GaAs and GaAlAs, e0 = 12:58, and V0 = 228 meV. These parameters are suitable in GaAs/Ga1 x Alx As heterostructures with an Al concentration of x = 0:3. Figure 2 shows the binding energies of a hydrogenic donor impurity located at the centre of the PQW under the crossed electric and magnetic elds as a function of the well width for di erent electric and magnetic eld values. As seen in this gure, when the dimension of quantum well increases, the impurity binding energy increases until it reaches a maximum value, and then decreases. This behaviour is related to the change of the electron con nement in the quantum well. When the dimension of well decreases, the con nement of electrons is strengthened, and therefore the impurity binding energy increases.
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When the dimension of quantum well is reduced to a small limited value, most of the electronic wave function begins to leak out of the well and therefore the impurity binding energy decreases. As the electric eld increases, for case F 6= 0, the coulombic interaction between the electron and a donor impurity located at the centre of the well decreases since the electron shifts to the left side of the well, and for F = 75 kV/cm the impurity binding energy becomes weaker than that at F = 0. Furthermore, the e ects of external elds are more evident especially for large well widths, since for large well widths, the geometric con nement becomes weaker and the con nement of external elds becomes predominant. The results obtained at present for B = 0 are the same as the results of the PQW in Ref. [22].
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localizes in the barrier which forms into a parabolic well by the e ect of the magnetic eld. In this case, the behaviour of electron is very similar to Landau-like states. (c) For B = 30 T, by increasing the magnetic eld value, we can increase the probability of nding the electron in the PQW again. Thus, one particle which delocalizes in the narrow wells can be localized again by increasing the magnetic eld value. Furthermore, in this case the potential pro le also behaves as an asymmetric double parabolic well. These results are valid in the case of crossed elds. One cannot obtained such behaviour in the case of non-crossed elds. This crossed con guration of the elds is very important for transport properties of semiconductor. Thus, we hope that these results provide important improvements in device applications, especially for narrow well widths and for a suitable choice of both elds.
Variation of the ground state binding energy of a hydrogenic donor impurity located in the centre of PQW under the crossed electric and magnetic eld as a function of the well width for di erent electric and magnetic eld values.
Fig. 2.
To see the combined e ects of the electric and magnetic elds in the narrow wells, the variation of the PQW potential pro le and amplitude of normalized subband wave function of electron, j (~z)j2 , versus the normalized position z~ = z=L, for L = 20 A, F = 150 kV/cm and di erent magnetic eld values are given in Figs. 3(a), 3(b), and 3(c). The e ects of the crossed electric and magnetic elds in the narrow wells leads to the interesting results: (a) For B = 0, electron is in the delocalization regime due to the well width and electric eld value. Electron localizes in the triangle well rather than in the PQW and thus, the Coulombic interaction between the electron and donor impurity located at the centre of the PQW decreases. (b) For B = 20 T, the new shape of the potential pro le becomes very interesting with the combined e ects of the electric and magnetic elds in the narrow wells. As seen in this gure, electron penetrates to the left side of the PQW by the e ect of the electric eld and
Variation of the PQW potential pro le and amplitude of normalized subband wave function of electron, (~z ) 2 , versus the normalized position z~ = z=L (horizontal line indicates the ground state energy in units of meV of the electron) for L = 20 A, F = 150 kV/cm, for (a) B = 0, (b) B = 20 T, (c) B = 30 T. Fig. 3.
j j
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As a result, we have studied theoretically the impurity binding energy in the PQW under the crossed electric and magnetic elds with the use of a variational approximation. Electric eld is applied parallel to the growth direction (z-direction), magnetic eld is applied perpendicular to the growth direction. We expect that the present study will be of great help for theoretical studies of the physical properties of PQWs.
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