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The impurity energy levels in a cylindrical quantum well wire at weak magnetic field
Solid State Communications, Vol. 89, No. 1, pp. 13-16, 1994. Printed in Great Britain.
0038-1098/94 $6.00 + .00 Pergamon Press Ltd
THE IMPURITY ENERGY LEVELS IN A CYLINDRICAL QUANTUM WELL WIRE AT WEAK MAGNETIC FIELD Dong Bin and Wang You-Tong Department of Physics and Institute of Condensed Matter Physics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China
(Received 18 June 1993 by P. Bur&t) The variational and perturbation method is developed to solve the impurity levels problems of quasi-one-dimensional quantum well wires. The results given by this method are in good agreement with those derived by other methods.
IN THE PAST two decades many new quantum-well structures with dimensions comparable to the electronic de Broglie wavelength have been grown with the development of experimental techniques such as chemical vapor deposition, liquid-phase epitaxy and molecular-beam epitaxy. Due to their small sizes, quasi-one-dimensional quantum well wires and quasi-zero-dimensional quantum dots structures present some physical properties, such as optical and electronic transport characteristics, which are quite different from those of the threedimensional bulk semiconductor structures, quasitwo-dimensional quantum wells and superlattices which have been extensively studied. An understanding of the nature of the impurity states in semiconductor structure is one of the important problems in semiconductor physics. In the past few years, several theoretical studies [2-5] on impurity levels, as well as impurity binding energies in quantum-well wires have been performed since the calculations were first done by Bryant [1]. He studied the binding energies for the bound states of a hydrogenic impurity placed on the axis of a cylindrical quantum-well wire composed of GaAs and surrounded by GaAIAs. Weber, Schulz and Oliveira [2] calculated the binding energies as functions of the impurity position and the density of impurity states in GaAs-(Ga, AI)As QWWs with different rectangular cross sections and for infinite well depths. Brown and Spector [3] calculated the impurity binding energies of both axial and off-axis impurities for infinite and finite cylindrical confining potential QWWs. Using their results, Montenegro et al. [4] obtained the density of impurity state of cylindrical QWWs. However, to our knowledge, the properties of impurity in QWWs in the presence of external magnetic field have been less considered yet
although the similar problems [6-8] and magnetoexciton problems [9-13] in quasi-two-dimensional quantum wells have been studied very well. In this work, we calculate results of impurity energies in a static external magnetic field for cylindrical GaAs-(Ga, AI)As QWWs by using a novel variational and perturbation method. In previous work, to solve the QWWs impurity problems the variation method has been usually used because it is extremely difficult to deal with the non-separable 3D Coulomb potential. However, using the variational and perturbation method applied in the pioneer work by Gerlack, Richter and Pollmann [14], several other more detailed investigations have been performed to solve the magnetoexciton problems. Specially, several recent theoreucal studies [12, 13] in quantum wells have been completed by using this method, in which the real 3D Coulomb potential 1/r was replaced by the 2D potential Alp with the ansatz that the expectation two-dimensional Coulomb potential is equal to that of the threedimensional one in the first-order approximation
where ( )z,p means average value with respect to the product wave function of z-direction wave function Z(z) and o-plane one ~/,(p, 0). It is conceivable that we can generalize this method in a quasi-one-dimensional quantum well wire with the ansatz
It has to be noticed that an electron bound to an 13
14
E N E R G Y LEVELS IN A C Y L I N D R I C A L Q U A N T U M W E L L WIRE
impurity at the center of a QWWs never encounters the boundary and behaves as a three-dimensional electron bound to an impurity in bulk material when the radius of wire is very large. On the other hand, just as Bryant [1] has discussed earlier, for very small wires, infinitely bound states do not exist. Because the confinement potential is not great enough to confine electrons, the electrons leak out of the well and behave as three-dimensional electrons. Only for a wire with an intermediate radius, electrons behave as quasi-one-dimensional electrons. Thus, only in this case, the ansatz will provide rational approximation to the impurity problems inside QWWs. We assume that the effective mass approximation is valid and the effective mass of electron is isotropic and that a single hydrogenic impurity is fixed on a QWWs axis, z direction. Then the Hamiltonian of the system in the presence of a homogeneous magnetic field B along the QWWs axis is given by
H = ~2 + ~2z ( ~ + ½(e/c)Bp)2 ---z 2m ~ + 2m*
e2 er
v(p), (3)
where the vector potential is taken as:
AC>=½BP,
A t, = A z = O ,
for p > R forp
(4)
= #2p12m* +
2
2
(6)
2
(ll)
with m = 0, 4-1, + 2 , . . . ,
a = ½- ~---~lwl+ ½1ml + ½m, b = Iml + 1, where M(a,b,x) is the confluent hypergeometric function, a~ is the effective Bohr radius of the wire material (h2E/m*e2), a c = Vrh-c/eB and wc = eB/m*c is the Landau length and the cyclotron frequency of the electron at the external magnetic field, respectively. A is the normalized constant. The boundary condition ~b = 0 for p = R leads to the eigenvalue equation (12)
For given R/ac and given azimuthal quantum number m (i.e. given b), equation (12) determines the eigenvalues Ell since it determines a. Therefore, the eigenvalue of H is •
(5)
4
~)~2. 2e h
(13)
In order to simplify numerical evaluation, we shall discuss the weak magnetic field limit. Since in this case B ~ 0, [a[ is to become very large, we can use the following limit expression of the confluent hypergeometric function: lim M(a, b, x) = I'(b) eX/2(½bx - ax)l/4-b/27r -1/2
where
HII
2
M(a,b,p / 2 a c ) e x p ( - p /4ac), (10)
Z(z) = x/2(A/a;)3/2lz I exp (-Alzl/aS),
E = Ell -
where R is the QWWs radius. We introduce the 1D Coulomb potential with the parameter A and rewrite H as H = HII + H=,
2 Iml/2
M(a, b, R2/2~) = 0.
m* is the electronic effective mass, ~ the static dielectric constant of the wire material, and V(p) the confining potential. For the sake of simplicity, we assume that the potential barrier is infinite. (oo V(p)= 0
2
X(P) = (P /2ac)
Vol. 89, No. 1
a --.* - - 0 0
(P++ ½(elc)Bpl212m *,
Hz = p2/2m* - AeE/~z,
(7)
(8)
and A will be obtained by using the equation (2). The eigenfunctions of HII have already been derived in [15], and the eigenfunction of Hz is taken as the ground state wavefunction of the exact onedimensional hydrogen atom. Now the total eigenfunctions may be written as:
~b(p, ~, z) = AX(p) exp(imc~)Z(z),
(9)
(
x cos x/2bx - 4ax - - ~ +
4)
.
(14) The GaAs-(Ga, Al)As structure is taken as an example of numerical computations. The QWWs is made of GaAs and surrounded by (Ga, AI)As. This system has been already studied in detail by Bryant [1]. From his results, it can be seen that the wire may be regarded as a quasion-dimensional system when the wire radius is between 0.5a~ and 10-a~.
E N E R G Y LEVELS IN A C Y L I N D R I C A L Q U A N T U M W E L L WIRE
Vol. 89, No. 1
By using above asymptotic expansion [equation (14)], a straightforward calculation leads to
Enm=½mhwc+~ mr-~
2lml+3~rJ4
m*g 2
A,m(R)
(15)
where m is magnetic quantum number and n is the nth root of the eigenvalue equation [equation (12)]. In our evaluation, we choose four magnetic field strength 1, 10, 102, and 103G. But in the limit of weak field, the difference of energies due to different field strength is very small. So values only at B = 102G against r ( = R/a~) are depicted in Fig. 1 for four different impurity states. These values are obtained numerically from equation (15). As expected, energies increase monotonically as the radius decreases. When the radius is very small, the energies are enhanced dramatically. While for larger wire radii, the ground state energies may be negative because of the attractive effect of the Coulomb potential. This is consistent with [1]. In Fig. 2 we display the variation parameter A as a function of wire radii for four impurity states. It is shown that the parameter A is independent of magnetic field due to weak strength limit. It is quite clear that the parameter A increases monotonically with the decrease of radius. From the tendency of the curve, we can deduce that A should approach 1 as the radius R ~ 0. As discussed in [15], E°m corresponds to the subband energy at zero magnetic field and E °
15t 13 11 9 7 5 3 1 -1
1.00f 0.85
~2 h2
4
15
h. 0.70I 0"55I 0 400
10 r
Fig. 2. The parameter A as a function of the radius of the cylindrical well wire. The (n,m) sequence is {(0, 0), (0, +1)} for the sohd curves and {(1, 0), (1, +1)} for the dashed curves from top to bottom. corresponds to the ground-state energy. Comparing our results of the ground-state energies E°0 with those of [5] (Fig. 4), we found that they are in good agreement and the difference between them is less than 10% of the ground state energies. Therefore, we have reason to believe that the novel variational and perturbation method presents a good approximation for the energies of shallow impurity in QWWs with an appropriate radius. Finally, we would like to stress that the variational parameter A should be dependent on the magnetic field strength. It can be seen from the equation (2) which defines A values that the magnetic field dependence of the parameter A is introduced by the eigenfunction of the system. However, in the weak strength limit, the eigenfunctions are hardly affected by the magnetic field, and A approximately is independent of the magnetic field. Summing up, we have developed a novel variational and perturbation procedure to investigate the energies of shallow impurities in cylindrical G a A s (Ga, AI)As quantum well wire with infinite potential barrier and obtained the results which are consistent with those derived by other methods.
REFERENCES
2I
4I
6I
I
8
I
10
1. 2. 3.
Fig. 1. Lowest-order subband energies versus the radii of the cylindrical quantum well wire for an infinite potential at a magnetic field strength 102G, where r = R/a~. The subbands are identified by their quantum numbers (n, m). From bottom to top, these curves correspond to {(0, 0), (0, 1), (1, 0), (1, 1)}, respectively.
4. 5. 6.
G.W. Bryant, Phys. Rev. B29, 6632 (1984). G. Weber, P.A. Schulz & L.E. Oliveira, Phys. Rev. B38, 2179 (1988). J.W. Brown & H.N. Spector, Phys. Rev. B35, 3009 (1987). N.P. Montenegro, J. Lopez-Gondar & L.E. Oliveria, Phys. Rev. !i43, 1824 (1991). D.S. Chuu, C.M. Hsiao & W.N. Mei, Phys. Rev. B46, 3898 (1992). Chanduri & K.K. Bajaj, Solid State Commun. 52, 967 (1984).
16 7. 8. 9. 10. 11.
ENERGY LEVELS IN A CYLINDRICAL QUANTUM WELL WIRE R.L. Greene & K.K. Bajaj, Phys. Rev. B31,913 (1985). G. Brozak & D.M. Larsen, Phys. Rev. 1340, 1265 (1989). K.J. Nash, M.S. Skolnick, P.A. Claxton & J.S. Roberts, Phys. Rev. B39, 10943 (1989). Chanduri & K.K. Bajaj, Phys. Rev. B31, 6498 (1985). D.A. Broido et al., Phys. Rev. B37, 2725 (1988).
12. 13. 14. 15.
Vol. 89, No. 1
X.L. Zheng, D. Heiman & B. Lax, Phys. Rev. B40, 10523 (1989). Kyu-Soek Lee, Yoshinobu Aoyagi & Takuo Sugano Phys. Rev. B46, 10269 (1992). B. Gerlack, D. Richter & J. Pollmann, Z. Phys. B66, 419 (1987). N.C. Constantinou, M. Masale & D.R. Tilley, J. Phys. C4, 4499 (1992).
0038-1098/94 $6.00 + .00 Pergamon Press Ltd
THE IMPURITY ENERGY LEVELS IN A CYLINDRICAL QUANTUM WELL WIRE AT WEAK MAGNETIC FIELD Dong Bin and Wang You-Tong Department of Physics and Institute of Condensed Matter Physics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China
(Received 18 June 1993 by P. Bur&t) The variational and perturbation method is developed to solve the impurity levels problems of quasi-one-dimensional quantum well wires. The results given by this method are in good agreement with those derived by other methods.
IN THE PAST two decades many new quantum-well structures with dimensions comparable to the electronic de Broglie wavelength have been grown with the development of experimental techniques such as chemical vapor deposition, liquid-phase epitaxy and molecular-beam epitaxy. Due to their small sizes, quasi-one-dimensional quantum well wires and quasi-zero-dimensional quantum dots structures present some physical properties, such as optical and electronic transport characteristics, which are quite different from those of the threedimensional bulk semiconductor structures, quasitwo-dimensional quantum wells and superlattices which have been extensively studied. An understanding of the nature of the impurity states in semiconductor structure is one of the important problems in semiconductor physics. In the past few years, several theoretical studies [2-5] on impurity levels, as well as impurity binding energies in quantum-well wires have been performed since the calculations were first done by Bryant [1]. He studied the binding energies for the bound states of a hydrogenic impurity placed on the axis of a cylindrical quantum-well wire composed of GaAs and surrounded by GaAIAs. Weber, Schulz and Oliveira [2] calculated the binding energies as functions of the impurity position and the density of impurity states in GaAs-(Ga, AI)As QWWs with different rectangular cross sections and for infinite well depths. Brown and Spector [3] calculated the impurity binding energies of both axial and off-axis impurities for infinite and finite cylindrical confining potential QWWs. Using their results, Montenegro et al. [4] obtained the density of impurity state of cylindrical QWWs. However, to our knowledge, the properties of impurity in QWWs in the presence of external magnetic field have been less considered yet
although the similar problems [6-8] and magnetoexciton problems [9-13] in quasi-two-dimensional quantum wells have been studied very well. In this work, we calculate results of impurity energies in a static external magnetic field for cylindrical GaAs-(Ga, AI)As QWWs by using a novel variational and perturbation method. In previous work, to solve the QWWs impurity problems the variation method has been usually used because it is extremely difficult to deal with the non-separable 3D Coulomb potential. However, using the variational and perturbation method applied in the pioneer work by Gerlack, Richter and Pollmann [14], several other more detailed investigations have been performed to solve the magnetoexciton problems. Specially, several recent theoreucal studies [12, 13] in quantum wells have been completed by using this method, in which the real 3D Coulomb potential 1/r was replaced by the 2D potential Alp with the ansatz that the expectation two-dimensional Coulomb potential is equal to that of the threedimensional one in the first-order approximation
where ( )z,p means average value with respect to the product wave function of z-direction wave function Z(z) and o-plane one ~/,(p, 0). It is conceivable that we can generalize this method in a quasi-one-dimensional quantum well wire with the ansatz
It has to be noticed that an electron bound to an 13
14
E N E R G Y LEVELS IN A C Y L I N D R I C A L Q U A N T U M W E L L WIRE
impurity at the center of a QWWs never encounters the boundary and behaves as a three-dimensional electron bound to an impurity in bulk material when the radius of wire is very large. On the other hand, just as Bryant [1] has discussed earlier, for very small wires, infinitely bound states do not exist. Because the confinement potential is not great enough to confine electrons, the electrons leak out of the well and behave as three-dimensional electrons. Only for a wire with an intermediate radius, electrons behave as quasi-one-dimensional electrons. Thus, only in this case, the ansatz will provide rational approximation to the impurity problems inside QWWs. We assume that the effective mass approximation is valid and the effective mass of electron is isotropic and that a single hydrogenic impurity is fixed on a QWWs axis, z direction. Then the Hamiltonian of the system in the presence of a homogeneous magnetic field B along the QWWs axis is given by
H = ~2 + ~2z ( ~ + ½(e/c)Bp)2 ---z 2m ~ + 2m*
e2 er
v(p), (3)
where the vector potential is taken as:
AC>=½BP,
A t, = A z = O ,
for p > R forp
(4)
= #2p12m* +
2
2
(6)
2
(ll)
with m = 0, 4-1, + 2 , . . . ,
a = ½- ~---~lwl+ ½1ml + ½m, b = Iml + 1, where M(a,b,x) is the confluent hypergeometric function, a~ is the effective Bohr radius of the wire material (h2E/m*e2), a c = Vrh-c/eB and wc = eB/m*c is the Landau length and the cyclotron frequency of the electron at the external magnetic field, respectively. A is the normalized constant. The boundary condition ~b = 0 for p = R leads to the eigenvalue equation (12)
For given R/ac and given azimuthal quantum number m (i.e. given b), equation (12) determines the eigenvalues Ell since it determines a. Therefore, the eigenvalue of H is •
(5)
4
~)~2. 2e h
(13)
In order to simplify numerical evaluation, we shall discuss the weak magnetic field limit. Since in this case B ~ 0, [a[ is to become very large, we can use the following limit expression of the confluent hypergeometric function: lim M(a, b, x) = I'(b) eX/2(½bx - ax)l/4-b/27r -1/2
where
HII
2
M(a,b,p / 2 a c ) e x p ( - p /4ac), (10)
Z(z) = x/2(A/a;)3/2lz I exp (-Alzl/aS),
E = Ell -
where R is the QWWs radius. We introduce the 1D Coulomb potential with the parameter A and rewrite H as H = HII + H=,
2 Iml/2
M(a, b, R2/2~) = 0.
m* is the electronic effective mass, ~ the static dielectric constant of the wire material, and V(p) the confining potential. For the sake of simplicity, we assume that the potential barrier is infinite. (oo V(p)= 0
2
X(P) = (P /2ac)
Vol. 89, No. 1
a --.* - - 0 0
(P++ ½(elc)Bpl212m *,
Hz = p2/2m* - AeE/~z,
(7)
(8)
and A will be obtained by using the equation (2). The eigenfunctions of HII have already been derived in [15], and the eigenfunction of Hz is taken as the ground state wavefunction of the exact onedimensional hydrogen atom. Now the total eigenfunctions may be written as:
~b(p, ~, z) = AX(p) exp(imc~)Z(z),
(9)
(
x cos x/2bx - 4ax - - ~ +
4)
.
(14) The GaAs-(Ga, Al)As structure is taken as an example of numerical computations. The QWWs is made of GaAs and surrounded by (Ga, AI)As. This system has been already studied in detail by Bryant [1]. From his results, it can be seen that the wire may be regarded as a quasion-dimensional system when the wire radius is between 0.5a~ and 10-a~.
E N E R G Y LEVELS IN A C Y L I N D R I C A L Q U A N T U M W E L L WIRE
Vol. 89, No. 1
By using above asymptotic expansion [equation (14)], a straightforward calculation leads to
Enm=½mhwc+~ mr-~
2lml+3~rJ4
m*g 2
A,m(R)
(15)
where m is magnetic quantum number and n is the nth root of the eigenvalue equation [equation (12)]. In our evaluation, we choose four magnetic field strength 1, 10, 102, and 103G. But in the limit of weak field, the difference of energies due to different field strength is very small. So values only at B = 102G against r ( = R/a~) are depicted in Fig. 1 for four different impurity states. These values are obtained numerically from equation (15). As expected, energies increase monotonically as the radius decreases. When the radius is very small, the energies are enhanced dramatically. While for larger wire radii, the ground state energies may be negative because of the attractive effect of the Coulomb potential. This is consistent with [1]. In Fig. 2 we display the variation parameter A as a function of wire radii for four impurity states. It is shown that the parameter A is independent of magnetic field due to weak strength limit. It is quite clear that the parameter A increases monotonically with the decrease of radius. From the tendency of the curve, we can deduce that A should approach 1 as the radius R ~ 0. As discussed in [15], E°m corresponds to the subband energy at zero magnetic field and E °
15t 13 11 9 7 5 3 1 -1
1.00f 0.85
~2 h2
4
15
h. 0.70I 0"55I 0 400
10 r
Fig. 2. The parameter A as a function of the radius of the cylindrical well wire. The (n,m) sequence is {(0, 0), (0, +1)} for the sohd curves and {(1, 0), (1, +1)} for the dashed curves from top to bottom. corresponds to the ground-state energy. Comparing our results of the ground-state energies E°0 with those of [5] (Fig. 4), we found that they are in good agreement and the difference between them is less than 10% of the ground state energies. Therefore, we have reason to believe that the novel variational and perturbation method presents a good approximation for the energies of shallow impurity in QWWs with an appropriate radius. Finally, we would like to stress that the variational parameter A should be dependent on the magnetic field strength. It can be seen from the equation (2) which defines A values that the magnetic field dependence of the parameter A is introduced by the eigenfunction of the system. However, in the weak strength limit, the eigenfunctions are hardly affected by the magnetic field, and A approximately is independent of the magnetic field. Summing up, we have developed a novel variational and perturbation procedure to investigate the energies of shallow impurities in cylindrical G a A s (Ga, AI)As quantum well wire with infinite potential barrier and obtained the results which are consistent with those derived by other methods.
REFERENCES
2I
4I
6I
I
8
I
10
1. 2. 3.
Fig. 1. Lowest-order subband energies versus the radii of the cylindrical quantum well wire for an infinite potential at a magnetic field strength 102G, where r = R/a~. The subbands are identified by their quantum numbers (n, m). From bottom to top, these curves correspond to {(0, 0), (0, 1), (1, 0), (1, 1)}, respectively.
4. 5. 6.
G.W. Bryant, Phys. Rev. B29, 6632 (1984). G. Weber, P.A. Schulz & L.E. Oliveira, Phys. Rev. B38, 2179 (1988). J.W. Brown & H.N. Spector, Phys. Rev. B35, 3009 (1987). N.P. Montenegro, J. Lopez-Gondar & L.E. Oliveria, Phys. Rev. !i43, 1824 (1991). D.S. Chuu, C.M. Hsiao & W.N. Mei, Phys. Rev. B46, 3898 (1992). Chanduri & K.K. Bajaj, Solid State Commun. 52, 967 (1984).
16 7. 8. 9. 10. 11.
ENERGY LEVELS IN A CYLINDRICAL QUANTUM WELL WIRE R.L. Greene & K.K. Bajaj, Phys. Rev. B31,913 (1985). G. Brozak & D.M. Larsen, Phys. Rev. 1340, 1265 (1989). K.J. Nash, M.S. Skolnick, P.A. Claxton & J.S. Roberts, Phys. Rev. B39, 10943 (1989). Chanduri & K.K. Bajaj, Phys. Rev. B31, 6498 (1985). D.A. Broido et al., Phys. Rev. B37, 2725 (1988).
12. 13. 14. 15.
Vol. 89, No. 1
X.L. Zheng, D. Heiman & B. Lax, Phys. Rev. B40, 10523 (1989). Kyu-Soek Lee, Yoshinobu Aoyagi & Takuo Sugano Phys. Rev. B46, 10269 (1992). B. Gerlack, D. Richter & J. Pollmann, Z. Phys. B66, 419 (1987). N.C. Constantinou, M. Masale & D.R. Tilley, J. Phys. C4, 4499 (1992).