- Email: [email protected]
The Stark effect on a bound polaron in parabolic quantum wires
14 August 1995 PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 204 (1995) 155-161
The Stark effect on a bound polaron in parabolic quantum wires T.C. Au-Yeung a School
of Electrical
and Electronic
h International ’ Applied Physics Department
a, C.C. Jong a, S.W. Gu b,c, Eddie M.C. Wong Engineering,
Nanyang
Technological
Centre for Material Physics, Academia
and Institute of Condensed
Unioersity, Singapore
Sinica, Shenyang
Matter Physics, Shanghai
110012,
a
2263, Singapore
China
Jiao Tong University, Shanghai
2000.10.
China
Received 21 March 1995; revised manuscript received 7 June 1995; accepted for publication 16June 1995 Communicated by J. Flouquet
Abstract We investigated the effect of electric fields on a bound polaron in parabolic quantum wires. The LLP-H variational method is used to calculate the ground state and first-excited state energies. We apply our calculation to GaAs.
1. Introduction The effects of electric fields on low-dimensional quantum systems have attracted a lot of interest in the past few years [l-7]. Bastard et al. [l] carried out variational calculations of the energy levels in quantum wells subjected to weak electric fields. Miller et al. [2] measured and explained the quantum-confined Stark effect on excitons in quantum wells. Brum et al. [8] studied the electric-field dependence of the impurity binding energy in quantum-well structures. Yoo et al. [9] reported the experimental observation of large electric fields on the electronic states of shallow impurities confined in semiconductor quantum wells, and Li and Gu [lo] studied the polaron effects on the quantum-confined Stark effect of bound polarons in finite quantum wells using the perturbation-variation technique. Also, Zhu and Gu [ll] investigated theoretically the electric-field effects on shallow donor impurity states in a harmonic quantum dot within the strong confinement regime. In this paper, we use the modified Lee-LowPines variational method (LLP-H) as proposed by Huybrechts [12] to calculate the ground state and first excited state energy levels of a bound polaron in a parabolic quantum wire, under the influence of an external electric field. The effects of the electric field and the parabolic potential on the ground state and the first excited state energies were studied. We applied our results to the case of GaAs.
2. Theory The Hamiltonian of the three-dimensional polaron constant electric field E = (Ed, c2, EJ and a parabolic
bound to a Coulomb impurity, in the presence of a potential irnwfx* + ~mdy* + +mw:z2 (where m is
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved 037S-9601(95)00451-3
SSDI
1.56
T.C. Au-Yeung et al. /Physics Letters A 204 (19951 155-161
the electron band mass), can be written as
(1) where H,
=
~(Vqeiq”aq
+ vq*e-iq’rU~),
(3a)
e2 (q2)1’2($_$j.
(3b)
ff = 81rfiWL0
In the above expressions, r = (x, y, z), wLo is the LO-phonon frequency, V is the volume of the polar crystal, and 8, and e0 are the optical and static dielectric constants, respectively. The LLP-H method consists of two successive transformations U, and U,, -iaCq
* raia,
(4a)
4
(4b) ’
4
where a is a variational parameter, and fq is assumed to be a function of 1q 1only. The total Hamiltonian H is transformed to p = U; ’ U,- ‘HU, U,. The ground state energy of fi (or H 1 is then calculated variationally by minimizing the expectation value of H, <4(x,
z)Iamlw~(x,
y,
y, z)>,
(5)
where 4(x, y, z) is the electronic function and IO) is the unperturbed expectation value (5) with respect to fq and f,* we find that y, z)I~Mml~(x,
(4(X,
=(4(x,
2l(4(x,
4
y, z)I~‘(‘-~)~‘~I~(x, Y, z)>l” fi%O
+
the trial wavefunction
=3
y,
A;(x-x0)=+A;(y-yo)2+h;(z-zo)2]). & into (61, we have
z)I~OI~IO)I4*(~~
(6)
a2h”q2/2m
of the displaced-harmonic-oscillator
y, z)=(*)“‘exp(-f[
E,=(+“(x,
the
y, z)
We use the ground state wavefunction
Substituting
state. By minimizing
Y, 2)1&P”-
-Clv,l
4,(x,
zero-phonon
y, 2))
WLO (vT+~t+~f)+h~~~(e,S,+e,6~+e,6,)
as a trial function,
(7)
157
T.C. Au-Yeung et al. /Physics Letters A -704(19951 155-161
sin% cos”+ i
X
exp
(
11;
i
co?
sin*0 sin’ 4
0
7
+--
+
-t’
Xexp
u;
v.;
I
q-
1
- + vf( r sin 0 cos f#~- 6,)’ + V~(Y sin 0 cos (f, - a?)’ [
v_f(r cos 0 - Cs,)‘] 1.
(8)
where 6, =xg/rP.
6, = y,/r,
v, = firpA,, l-a t= a
vl-_-J- 2rphz,
6, = zO/rp,
(9a)
vj = firpA.l,
(9b) (c)C)
'
e,=e~,r,/fiwL,, Y
7
=wq.o~
e, = e.qr,/hwLo, w,=
WJWLO,
es = e~3r,/~~Lo~
(9d)
w,=%/~Lo~
(9e)
is the Bohr radius In the above expressions R, = me4/32r2qfh’ is the Rydberg constant, a, = 4Tr&0h2/me’ and rp = (?i/2mwLo )‘I2 is the polaron radius. From (8) the ground state energy E,, involves SI c physical parameters, namely, e,, e2, e3, W,, Wz, and W,. For each set of values of the six parameters, we minimize E,, with respect to the seven variables t, u,, v2, v3, S,, 6, and 6, to obtain the ground state energy of H. Next, we use the first excited state wavefunction of the displaced-harmonic-oscillator as a trial function for the first excited state energy of H, 44 x, Y, 2) = A,
From (6), the first excited state energy E, is given by 7
E,=hbl&P’-
Similar to E,,, E, is a function of t, vl, u2, v3, 6,, 6, and a,, and it involves the six physical parameters e2. e3, W,, W, and W,. For each set of values of the parameters e,, ez, e3, WI, W, and W,, we minimize with respect to t, v,, v,,- vl,. 6,, Sz and S, to obtain the first excited state energy of H.
e,, E,
3. Results and discussions The minimization of E,, and E, was carried out by using a MATLAB minimization function. which is b::sed on the downhill simplex method. The results obtained were verified by using a minimization function in IMSL. We apply our method to the case of GaAs, where m = O.O66m, Cm, is the electron bare mass), cy = 0.1167,
1.58
T.C. Au-Y&q
0.1
-0.12 4.14 4.16 0.1.3
I 4
et al./Physics
Letters A 204 (1995) 155-161
4
:L-0
0.1
02
0.3
: : : : : :
0.4
0.5
00
0.7
OS
09
l.3 1
El
Fig. 1. Dependence of E, and E, on e, (a) at e, = e2 = 0, W, = W, = 0.1, W, = 0 and (b) at e, = e2 = 0, W, = W, = 4, W, = 0. All the energies are in units of fiw,,.
Fig. 2. Dependence of AE, on e3 (a) at e, = ez = 0, W, = W, = 0.1, W, = 0 and (b) at e, = e2 = 0, W, = W, = 4, W, = 0. All the energies are in units of h qo.
E, = 10.9, &a = 12.83 and hw,, = 36.63 meV. The numerical results of our variational calculations are shown in the following diagrams. In Fig. la, the ground state energy E, and the first excited state energy E, are plotted against e3, with e, = e2 = 0, W, = W, = 0.1 and W, = 0. According to the curves, both E, and E, decrease with increasing e3 (or es). Fig. lb is similar, but with e, = e2 = 0, WI = W, = 4 and W, = 0, that is, we use larger parabolic frequencies WI and W,. The figure also shows that E, and E, decrease with increasing Ed. From Fig. 1, we see that the larger WI and W,, the larger E, and E,, if e3 is fixed. Our variational approach shows that when eg is sufficiently large ’ there is no bound state for our physical system. The change A E, in the ground state energy due to the presence of the electric field is plotted both in Fig. 2a (with e, = e2 = 0, W, = W, = 0.1 and W, = 0) and Fig. 2b (with e, = e2 = 0, WI = W, = 4 and W, = 0). Fig. 2 shows that AE, decreases with increasing e3, and that AE, increases with increasing W, and W, when e3 is fixed. Next, we plot E,, and E, against WI in Fig. 3 with e, = e2 = 0, e3 = 0.001, W, = 0.1 and W, = 0 (Fig. 3a1, and e, = e, = 0, e3 = 0.001, W, = 4 and W, = 0 (Fig. 3b). We see that both E, and E, increase with increasing WI, and the curves become linear when WI is sufficiently large, that is, both E, and E, vary linearly with large W,. Fig. 3 also shows that E, and E, increase with W, when W, is fixed. AE, is plotted against W, in Fig. 4, with similar conditions as in Fig. 3.
’ In Fig. la, e3 > 0.018 for E, and e3 > 0.19 for E,, and in Fig. lb, e3 > 0.16 for E0 and ~3 > 1.0 for E,.
T.C. Au-Yeung et al. /Physics
1.59
Letters A 204 (19951 155-161
Fig. 3. Dependence of E, and E, on W, (a) at el = ez = 0, e3 = 0.001, W, = 0.1, W, = 0 and (b) at e, = e, = 0. eT = 0.001, W, = 0. All the energies are in units of h wLo.
W, = 4,
at e, = e, = 0, e3 = 0.001, W, = 4, W, = 0. All
Fig. 4. Dependence of AE, on W, (a) at e, = e, = 0, e3 = 0.001, W, = 0.1, W, = 0 and(b) the energies are in units of fi wLo.
Figs. 5 and 6 are similar to Figs. 3 and 4, but with e, = e, = 0, e3 = 0.02, W, = 0.1 and 4 and W, = 0. That is, we use a larger e3. Fig. 5 also shows that E, and E, increase with W,, and that they become linearly related to W, when W, is sufficiently large. From Figs. 3 and 5 we once again see that E, and E, decrease with increasing e3 when all the other parameters are fixed. Figs. 4 and 6 show that AE, also decreases with increasing e3 when all the other parameters are fixed. We would like to point out that in Figs. 3-6 the limit W, = 0 corresponds to the quantum well (2D) situation. Finally, we consider the case e, = e2 = 0 and W, = W, = W,, which describes a quantum box. E, and E, are plotted against e3 in Fig. 7a with the condition W, = W, = W, = 0.1, and similarly in Fig. 7b with larger
-2
P
2
4
6
0
: 2
-WI 4
5
*
$0
Fig. 5. Dependence of E, and E, on W, (a) at e, = ez = 0, e3 = 0.02, W, = 0.1, W, = 0 and (b) at e, = e2 = 0, e3 = 0.02, W? = 4. W, = 0. All the energies are in units of fi wLo.
160
T.C. Au-Yeung et al. /Physics
ma T”
Letters A 204 (1995) 155-161
-b
AEa WI
0.01 -0.008 .0.m
.-
0.m
--
0.002 ..
0.009
1
-000(1
1
Fig. 6. Dependence of A&, on W, (a) at e, = eZ = 0, e3 = 0.02, W, = 0.1, W, = 0 and (b) at ej = e2 = 0, e3 = 0.02, Wz = 4, W, = 0. All the energies are in units of fi wLo.
a
0.2-
El
O0.
03
I
a
0.20
0.4
0.0
0.M
10-b
,.m
El
0.’ (I .7 -. 0 .. 5 .-
-& . -3 ..
0.8 ..
2 --1 ..
1 .. 07
‘E,
-,.2 -
,
20
0.0
40
6.0
8.0
e3
to.0
Fig. 7. Dependence of E,, and E, on ej (a) at e, = e2 = 0, W, = W, = W, = 0.1 and (b) at e, = e, = 0, W, = Wz = W, = 4. All the energies are in units of fiwLo.
parabolic frequencies, namely, W, = IV, = W, = 4. Fig. 7 shows that both E, and E, decrease with e3 (or ~~1, and that they increase with the parabolic frequencies when e3 is fixed Al?,, is plotted against e3 in Fig. 8 with similar conditions as in Fig. 7. We see from Fig. 8 that AE, decreases with increasing e3.
a *a 0. 0.x)
&
+b
(4
0.12
-01 .-
0.0,
4.4 .-
0.02 -~
4.6 --
-0.m ..
-0.8 -.
4.04 ..
-1 -.
0.05 ..
-,.2-
Fig. 8. Dependence units of fi qo.
Jxe
..
-
of A& on e3 (a) at e, = ez = 0, WI = W, = W, = 0.1 and (b) at e, = e2 = 0, WI = W, = W, = 4. All the energies are in
T.C. Au-Yeung et al. /Physics Letters A 204 (1995) 155-161
I61
4. Conclusion We studied the electric-field effect on the bound polaron in parabolic quantum wires. The modified Lee-Low-Pines variational method (LLP-H) is used to deal with the electron-LO-phonon interaction. The displaced-harmonic-oscillator wavefunctions are USLL as trial functions for calculation of E, and E,. We considered two interesting cases, namely (a) E, = E: = w1 = 0 and (b) F, = c2 = 0 and w, = w, = o3 (quantum box), and we applied our results to the case of GaAs. In general, we found that the effects of the electric field on E,,. E, are small, and E,, E, and A& decrease with increasing Ed, when all the other parameters (E,, +, wl, w2 and w,) are fixed. Furthermore, in case (a) our results showed that there is no bound state of the physical system when c3 is sufficiently large, and, given that w2 and s3 are fixed, E, and E, increase with increasing o1 and become linearly dependent on w, when w, is sufficiently large. The energy reference is probably the GaAs conduction minimum, so that the unbound (to the impurity) states are quantum wire states with positive energy. Our result of no state bound to the impurity implies that E,, is positive. The reason why bound states disappear, although their energy decreases, is that the energy of the quantum wire increases with electric field. As for the case of the quantum box, we found that both E, and E, increase with the parabolic frequencies when ej is fixed.
References [l] G. Bastard, E.E. Mendez, L.L. Chang and L. Esaki, Phys. Rev. B 28 (1983) 3241. [2] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus. Phys. Rev. Lett. 53 (1984) 2173; Phys. Rev. B 32 (1985) 1043. [3] Y.J. Chen, Emil S. Koteles, B.S. Elman and C.A. Armientu, Phys. Rev. B 36 (1987) 4562. [4] D.P. Barrio, M.L. Glasser, V.R. Velasco and F. Garcia-Moliner, .I. Phys. Condens. Matter 1 (1989) 4339. [s] F. Borondo and J. Sanches-Dehesa, Phys. Rev. B 33 (1986) 8758. [6] D. Ahn and S.L. Chuang, Phys. Rev. B 3.5 (1987) 4149. [7] E.J. Austin and M. Jaros. Phys. Rev. B 38 (1988) 6326. [8] J.A. Brum, C. Priester and G. Allen, Phys. Rev. B 32 (1985) 2378. [9] B. Yoo. B.D. McCombe and W. Schaff, Phys. Rev. B 44 11991) 13152. [lo] Y.C. Li and SW. Gu, Phys. Rev. B 45 (1992) 12102. [ll] K.D. Zhu and S.W. Gu, Phys. Lett. A 181 (1993) 465. [12] W. Huybrechts. J. Phys. C IO (1977) 3761.
ELSEVIER
LETTERS
A
Physics Letters A 204 (1995) 155-161
The Stark effect on a bound polaron in parabolic quantum wires T.C. Au-Yeung a School
of Electrical
and Electronic
h International ’ Applied Physics Department
a, C.C. Jong a, S.W. Gu b,c, Eddie M.C. Wong Engineering,
Nanyang
Technological
Centre for Material Physics, Academia
and Institute of Condensed
Unioersity, Singapore
Sinica, Shenyang
Matter Physics, Shanghai
110012,
a
2263, Singapore
China
Jiao Tong University, Shanghai
2000.10.
China
Received 21 March 1995; revised manuscript received 7 June 1995; accepted for publication 16June 1995 Communicated by J. Flouquet
Abstract We investigated the effect of electric fields on a bound polaron in parabolic quantum wires. The LLP-H variational method is used to calculate the ground state and first-excited state energies. We apply our calculation to GaAs.
1. Introduction The effects of electric fields on low-dimensional quantum systems have attracted a lot of interest in the past few years [l-7]. Bastard et al. [l] carried out variational calculations of the energy levels in quantum wells subjected to weak electric fields. Miller et al. [2] measured and explained the quantum-confined Stark effect on excitons in quantum wells. Brum et al. [8] studied the electric-field dependence of the impurity binding energy in quantum-well structures. Yoo et al. [9] reported the experimental observation of large electric fields on the electronic states of shallow impurities confined in semiconductor quantum wells, and Li and Gu [lo] studied the polaron effects on the quantum-confined Stark effect of bound polarons in finite quantum wells using the perturbation-variation technique. Also, Zhu and Gu [ll] investigated theoretically the electric-field effects on shallow donor impurity states in a harmonic quantum dot within the strong confinement regime. In this paper, we use the modified Lee-LowPines variational method (LLP-H) as proposed by Huybrechts [12] to calculate the ground state and first excited state energy levels of a bound polaron in a parabolic quantum wire, under the influence of an external electric field. The effects of the electric field and the parabolic potential on the ground state and the first excited state energies were studied. We applied our results to the case of GaAs.
2. Theory The Hamiltonian of the three-dimensional polaron constant electric field E = (Ed, c2, EJ and a parabolic
bound to a Coulomb impurity, in the presence of a potential irnwfx* + ~mdy* + +mw:z2 (where m is
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved 037S-9601(95)00451-3
SSDI
1.56
T.C. Au-Yeung et al. /Physics Letters A 204 (19951 155-161
the electron band mass), can be written as
(1) where H,
=
~(Vqeiq”aq
+ vq*e-iq’rU~),
(3a)
e2 (q2)1’2($_$j.
(3b)
ff = 81rfiWL0
In the above expressions, r = (x, y, z), wLo is the LO-phonon frequency, V is the volume of the polar crystal, and 8, and e0 are the optical and static dielectric constants, respectively. The LLP-H method consists of two successive transformations U, and U,, -iaCq
* raia,
(4a)
4
(4b) ’
4
where a is a variational parameter, and fq is assumed to be a function of 1q 1only. The total Hamiltonian H is transformed to p = U; ’ U,- ‘HU, U,. The ground state energy of fi (or H 1 is then calculated variationally by minimizing the expectation value of H, <4(x,
z)Iamlw~(x,
y,
y, z)>,
(5)
where 4(x, y, z) is the electronic function and IO) is the unperturbed expectation value (5) with respect to fq and f,* we find that y, z)I~Mml~(x,
(4(X,
=(4(x,
2l(4(x,
4
y, z)I~‘(‘-~)~‘~I~(x, Y, z)>l” fi%O
+
the trial wavefunction
=3
y,
A;(x-x0)=+A;(y-yo)2+h;(z-zo)2]). & into (61, we have
z)I~OI~IO)I4*(~~
(6)
a2h”q2/2m
of the displaced-harmonic-oscillator
y, z)=(*)“‘exp(-f[
E,=(+“(x,
the
y, z)
We use the ground state wavefunction
Substituting
state. By minimizing
Y, 2)1&P”-
-Clv,l
4,(x,
zero-phonon
y, 2))
WLO (vT+~t+~f)+h~~~(e,S,+e,6~+e,6,)
as a trial function,
(7)
157
T.C. Au-Yeung et al. /Physics Letters A -704(19951 155-161
sin% cos”+ i
X
exp
(
11;
i
co?
sin*0 sin’ 4
0
7
+--
+
-t’
Xexp
u;
v.;
I
q-
1
- + vf( r sin 0 cos f#~- 6,)’ + V~(Y sin 0 cos (f, - a?)’ [
v_f(r cos 0 - Cs,)‘] 1.
(8)
where 6, =xg/rP.
6, = y,/r,
v, = firpA,, l-a t= a
vl-_-J- 2rphz,
6, = zO/rp,
(9a)
vj = firpA.l,
(9b) (c)C)
'
e,=e~,r,/fiwL,, Y
7
=wq.o~
e, = e.qr,/hwLo, w,=
WJWLO,
es = e~3r,/~~Lo~
(9d)
w,=%/~Lo~
(9e)
is the Bohr radius In the above expressions R, = me4/32r2qfh’ is the Rydberg constant, a, = 4Tr&0h2/me’ and rp = (?i/2mwLo )‘I2 is the polaron radius. From (8) the ground state energy E,, involves SI c physical parameters, namely, e,, e2, e3, W,, Wz, and W,. For each set of values of the six parameters, we minimize E,, with respect to the seven variables t, u,, v2, v3, S,, 6, and 6, to obtain the ground state energy of H. Next, we use the first excited state wavefunction of the displaced-harmonic-oscillator as a trial function for the first excited state energy of H, 44 x, Y, 2) = A,
From (6), the first excited state energy E, is given by 7
E,=hbl&P’-
Similar to E,,, E, is a function of t, vl, u2, v3, 6,, 6, and a,, and it involves the six physical parameters e2. e3, W,, W, and W,. For each set of values of the parameters e,, ez, e3, WI, W, and W,, we minimize with respect to t, v,, v,,- vl,. 6,, Sz and S, to obtain the first excited state energy of H.
e,, E,
3. Results and discussions The minimization of E,, and E, was carried out by using a MATLAB minimization function. which is b::sed on the downhill simplex method. The results obtained were verified by using a minimization function in IMSL. We apply our method to the case of GaAs, where m = O.O66m, Cm, is the electron bare mass), cy = 0.1167,
1.58
T.C. Au-Y&q
0.1
-0.12 4.14 4.16 0.1.3
I 4
et al./Physics
Letters A 204 (1995) 155-161
4
:L-0
0.1
02
0.3
: : : : : :
0.4
0.5
00
0.7
OS
09
l.3 1
El
Fig. 1. Dependence of E, and E, on e, (a) at e, = e2 = 0, W, = W, = 0.1, W, = 0 and (b) at e, = e2 = 0, W, = W, = 4, W, = 0. All the energies are in units of fiw,,.
Fig. 2. Dependence of AE, on e3 (a) at e, = ez = 0, W, = W, = 0.1, W, = 0 and (b) at e, = e2 = 0, W, = W, = 4, W, = 0. All the energies are in units of h qo.
E, = 10.9, &a = 12.83 and hw,, = 36.63 meV. The numerical results of our variational calculations are shown in the following diagrams. In Fig. la, the ground state energy E, and the first excited state energy E, are plotted against e3, with e, = e2 = 0, W, = W, = 0.1 and W, = 0. According to the curves, both E, and E, decrease with increasing e3 (or es). Fig. lb is similar, but with e, = e2 = 0, WI = W, = 4 and W, = 0, that is, we use larger parabolic frequencies WI and W,. The figure also shows that E, and E, decrease with increasing Ed. From Fig. 1, we see that the larger WI and W,, the larger E, and E,, if e3 is fixed. Our variational approach shows that when eg is sufficiently large ’ there is no bound state for our physical system. The change A E, in the ground state energy due to the presence of the electric field is plotted both in Fig. 2a (with e, = e2 = 0, W, = W, = 0.1 and W, = 0) and Fig. 2b (with e, = e2 = 0, WI = W, = 4 and W, = 0). Fig. 2 shows that AE, decreases with increasing e3, and that AE, increases with increasing W, and W, when e3 is fixed. Next, we plot E,, and E, against WI in Fig. 3 with e, = e2 = 0, e3 = 0.001, W, = 0.1 and W, = 0 (Fig. 3a1, and e, = e, = 0, e3 = 0.001, W, = 4 and W, = 0 (Fig. 3b). We see that both E, and E, increase with increasing WI, and the curves become linear when WI is sufficiently large, that is, both E, and E, vary linearly with large W,. Fig. 3 also shows that E, and E, increase with W, when W, is fixed. AE, is plotted against W, in Fig. 4, with similar conditions as in Fig. 3.
’ In Fig. la, e3 > 0.018 for E, and e3 > 0.19 for E,, and in Fig. lb, e3 > 0.16 for E0 and ~3 > 1.0 for E,.
T.C. Au-Yeung et al. /Physics
1.59
Letters A 204 (19951 155-161
Fig. 3. Dependence of E, and E, on W, (a) at el = ez = 0, e3 = 0.001, W, = 0.1, W, = 0 and (b) at e, = e, = 0. eT = 0.001, W, = 0. All the energies are in units of h wLo.
W, = 4,
at e, = e, = 0, e3 = 0.001, W, = 4, W, = 0. All
Fig. 4. Dependence of AE, on W, (a) at e, = e, = 0, e3 = 0.001, W, = 0.1, W, = 0 and(b) the energies are in units of fi wLo.
Figs. 5 and 6 are similar to Figs. 3 and 4, but with e, = e, = 0, e3 = 0.02, W, = 0.1 and 4 and W, = 0. That is, we use a larger e3. Fig. 5 also shows that E, and E, increase with W,, and that they become linearly related to W, when W, is sufficiently large. From Figs. 3 and 5 we once again see that E, and E, decrease with increasing e3 when all the other parameters are fixed. Figs. 4 and 6 show that AE, also decreases with increasing e3 when all the other parameters are fixed. We would like to point out that in Figs. 3-6 the limit W, = 0 corresponds to the quantum well (2D) situation. Finally, we consider the case e, = e2 = 0 and W, = W, = W,, which describes a quantum box. E, and E, are plotted against e3 in Fig. 7a with the condition W, = W, = W, = 0.1, and similarly in Fig. 7b with larger
-2
P
2
4
6
0
: 2
-WI 4
5
*
$0
Fig. 5. Dependence of E, and E, on W, (a) at e, = ez = 0, e3 = 0.02, W, = 0.1, W, = 0 and (b) at e, = e2 = 0, e3 = 0.02, W? = 4. W, = 0. All the energies are in units of fi wLo.
160
T.C. Au-Yeung et al. /Physics
ma T”
Letters A 204 (1995) 155-161
-b
AEa WI
0.01 -0.008 .0.m
.-
0.m
--
0.002 ..
0.009
1
-000(1
1
Fig. 6. Dependence of A&, on W, (a) at e, = eZ = 0, e3 = 0.02, W, = 0.1, W, = 0 and (b) at ej = e2 = 0, e3 = 0.02, Wz = 4, W, = 0. All the energies are in units of fi wLo.
a
0.2-
El
O0.
03
I
a
0.20
0.4
0.0
0.M
10-b
,.m
El
0.’ (I .7 -. 0 .. 5 .-
-& . -3 ..
0.8 ..
2 --1 ..
1 .. 07
‘E,
-,.2 -
,
20
0.0
40
6.0
8.0
e3
to.0
Fig. 7. Dependence of E,, and E, on ej (a) at e, = e2 = 0, W, = W, = W, = 0.1 and (b) at e, = e, = 0, W, = Wz = W, = 4. All the energies are in units of fiwLo.
parabolic frequencies, namely, W, = IV, = W, = 4. Fig. 7 shows that both E, and E, decrease with e3 (or ~~1, and that they increase with the parabolic frequencies when e3 is fixed Al?,, is plotted against e3 in Fig. 8 with similar conditions as in Fig. 7. We see from Fig. 8 that AE, decreases with increasing e3.
a *a 0. 0.x)
&
+b
(4
0.12
-01 .-
0.0,
4.4 .-
0.02 -~
4.6 --
-0.m ..
-0.8 -.
4.04 ..
-1 -.
0.05 ..
-,.2-
Fig. 8. Dependence units of fi qo.
Jxe
..
-
of A& on e3 (a) at e, = ez = 0, WI = W, = W, = 0.1 and (b) at e, = e2 = 0, WI = W, = W, = 4. All the energies are in
T.C. Au-Yeung et al. /Physics Letters A 204 (1995) 155-161
I61
4. Conclusion We studied the electric-field effect on the bound polaron in parabolic quantum wires. The modified Lee-Low-Pines variational method (LLP-H) is used to deal with the electron-LO-phonon interaction. The displaced-harmonic-oscillator wavefunctions are USLL as trial functions for calculation of E, and E,. We considered two interesting cases, namely (a) E, = E: = w1 = 0 and (b) F, = c2 = 0 and w, = w, = o3 (quantum box), and we applied our results to the case of GaAs. In general, we found that the effects of the electric field on E,,. E, are small, and E,, E, and A& decrease with increasing Ed, when all the other parameters (E,, +, wl, w2 and w,) are fixed. Furthermore, in case (a) our results showed that there is no bound state of the physical system when c3 is sufficiently large, and, given that w2 and s3 are fixed, E, and E, increase with increasing o1 and become linearly dependent on w, when w, is sufficiently large. The energy reference is probably the GaAs conduction minimum, so that the unbound (to the impurity) states are quantum wire states with positive energy. Our result of no state bound to the impurity implies that E,, is positive. The reason why bound states disappear, although their energy decreases, is that the energy of the quantum wire increases with electric field. As for the case of the quantum box, we found that both E, and E, increase with the parabolic frequencies when ej is fixed.
References [l] G. Bastard, E.E. Mendez, L.L. Chang and L. Esaki, Phys. Rev. B 28 (1983) 3241. [2] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus. Phys. Rev. Lett. 53 (1984) 2173; Phys. Rev. B 32 (1985) 1043. [3] Y.J. Chen, Emil S. Koteles, B.S. Elman and C.A. Armientu, Phys. Rev. B 36 (1987) 4562. [4] D.P. Barrio, M.L. Glasser, V.R. Velasco and F. Garcia-Moliner, .I. Phys. Condens. Matter 1 (1989) 4339. [s] F. Borondo and J. Sanches-Dehesa, Phys. Rev. B 33 (1986) 8758. [6] D. Ahn and S.L. Chuang, Phys. Rev. B 3.5 (1987) 4149. [7] E.J. Austin and M. Jaros. Phys. Rev. B 38 (1988) 6326. [8] J.A. Brum, C. Priester and G. Allen, Phys. Rev. B 32 (1985) 2378. [9] B. Yoo. B.D. McCombe and W. Schaff, Phys. Rev. B 44 11991) 13152. [lo] Y.C. Li and SW. Gu, Phys. Rev. B 45 (1992) 12102. [ll] K.D. Zhu and S.W. Gu, Phys. Lett. A 181 (1993) 465. [12] W. Huybrechts. J. Phys. C IO (1977) 3761.