* Corresponding author. Tel.: #886-3-5712121-56105; fax:
#886-3-5725230.
E-mail address: [email protected] (D.-S. Chuu).
E!ect of electron}phonon interaction on the impurity binding
energy in a quantum wire
Der-San Chuu*, Yueh-Nan Chen, Yuh-Kae Lin
Department of Electrophysics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 30050, Taiwan
Received 25 May 1999; received in revised form 20 December 1999
Abstract
The e!ect of electron}optical phonon interaction on the hydrogenic impurity binding energy in a cylindrical quantum wire is studied. By using Landau and Pekar variational method, the Hamiltonian is separated into two parts which contain phonon variable and electron variable, respectively. A perturbative-variational technique is then employed to construct the trial wave function for the electron part. The e!ect of con"ned electron}optical phonon interaction on the binding energies of the ground state and an excited state are calculated as a function of wire radius. Both the electron-bulk optical phonon and electron-surface optical phonon coupling are considered. It is found that the energy corrections of the polaron e!ects on the impurity binding energies increase rapidly as the wire radius is shrunk, and the bulk-type optical phonon plays the dominant role for the polaron e!ects. 2000 Elsevier Science B.V. All rights reserved.
PACS: 71.38#i; 73.20.Dx; 63.20.Kr
Keywords: Polaron; Quantum wire; Perturbative-variational technique
1. Introduction
During the past decades the development of the epitaxial crystal growth techniques such as mole-cular-beam epitaxy and metal}organic chemical vapor deposition has made the growth of the quasi-two-dimensional (quantum well) or quasi-one-di-mensional (quantum wire) [1}4] systems possible with controllable well thickness or wire radius. These quantum structures have been applied to
many semiconductor devices, such as high-elec-tron-mobility transistors. Recent progresses in growth and fabrication techniques have been able to fabricate the quantum wires with radii less than 100 As . Theoretically, the electronic properties of a hydrogenic impurity in the quantum well [5}8] and the quantum wire [9}16] have been studied by many authors. The impurity binding energies of a quantum wire with in"nite or "nite potential barrier [9] and with di!erent shapes of the cross-section [10}12] have been discussed. The e!ect of location [10}12] of impurities with respect to the wire axis was also studied previously. The emission line for quantum wires was observed [17] to be two to three times broader than that of quantum wells
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 5 1 - 8
and with 6}10 meV higher binding energy. It is expected that the same properties in quantum wells were further improved by the reduction of dimen-sionality to quasi-one-dimensional quantum wires. The physics of impurity states in quantum wire is very interesting because speci"c properties can be easily achieved by varying the wire radius. An elec-tron bound to an impurity on the axis of the quan-tum wire behaves like a bounded three-dimensional electron when the boundary is far away. However, as the wire radius is reduced, the electron con"ne-ment due to the potential barrier becomes very important. Especially in the quantum wire with in"nitely high potential wall, the total energy of the electron may change from negative to positive at a certain radius and "nally diverges to in"nity as the radius approaches zero. Furthermore, it is well known that the reduction of dimensionality in-creases the e!ective strength of the Coulomb inter-action. The binding energy E of the ground state of a hydrogenic impurity in N-dimension is given by E"[2/(N!1)]RH
W, where RHW"ke/2e is the e!ective Rydberg. Hence the dramatic change in the binding energy may serve as a clear signal for variation in the e!ective dimension of the quantum wire.
It is known that an electron weakly bound to a hydrogen impurity in a polar semiconductor will interact with the phonons of the host semiconduc-tor. In the past decade, many authors have studied the polaron e!ect on the binding energy of impurity or exciton in quantum well [18}26]. Recently, the electron}phonon e!ect on the binding energy of the donor impurity in a quantum wire with rectangular cross-section was reported [27}29]. It was found the polaron e!ect on the binding energy becomes sizeable as the electron gets more deeply bound. The polaron shifts in donor energy levels are found to be of the order of 10% in a weakly polar system. In studying the polaron e!ect on the impurity bind-ing energy, most of the previous works considered the interaction of the electron and bulk optical (BO) phonon only. However, in ionic crystal, the motion of an electron near the surface may be a!ected very much by the surface longitudinal op-tical (SO) phonon [30]. An electron may be trap-ped at the surface by the electron}SO phonon interaction. Besides, the electron}phonon
interac-tion Hamiltonian in the previous works was valid only for the bulk. Therefore, we will choose the Hamiltonian derived by Li and Chen [31], who considered the con"ned phonon modes in the cylin-drical quantum dot.
Most of the previous approaches concentrating on the polaron e!ect on the ground state of an impurity in a quasi-one-dimensional wire employ the variational method or perturbation method. Since the construction of variational trial wave functions is entirely based on physical intuition, the estimation of the accuracy of the result obtained from variational approach is very di$cult. Further-more, the perturbation method is only a good ac-cess to those systems with very small perturbation in most cases. Therefore, it would be most desirable to have an alternative approach which is not only simple but also e$cient to the quantum wire prob-lem. In this work, we employ a simple approxima-tion treatment which combines the spirit of both variational principle and perturbational approach to study the e!ect of electron}phonon interaction on the ground-state binding energy of a hydrogenic impurity located inside a quantum wire.
2. Theory
Consider now a hydrogenic impurity located on the axis of a rigid wall cylindrical quantum wire with a radius d. The Hamiltonian of the impurity electron interacting with the phonon can be ex-pressed as
H"H#H#H}#H#H}, (1)
where H is the electronic part of the Hamiltonian
H"!2k
R Rx# R Ry! 2k R Rz! e er#<(o). (2) <(o) is the con"ning potential which is assumed as <(o)"0 for o)d,R for o'd
(3) ande and k are the dielectric constants of the well and the e!ective mass of the electron. Recently, Li and Chen [31] have derived the con"ned longi-tudinal-optical phonon and surface phonon modes
of a free-standing cylindrical quantum dot of radius
d and height 2D. We will follow their Hamiltonian
and let D approach in"nity, such that the dot sys-tem can become a quantum wire. Therefore, H is the bulk phonon Hamiltonian which can be ex-pressed as
H"
L Ju*-aRLJaLJ,
(4) whereu*- is the dispersionless bulk optical (BO) phonon energy, aRLJ(aLJ) is the creation (annihilation) operator for BO phonon. H} is the interaction
between the electron and BO phonon which can be expressed as H}" L J
sL do J 2<LJ cos lp 2Dz ;(aLJ#aRLJ)# J 2<LJ sin lp 2Dz(aLJ#aRLJ) (5) with <LJ"<1 4 peu*-[(sL/d)J(sL)#(lp/2D)J(sL)] ;1 e! 1 e, (6)where JK is the mth-order Bessel function, sL is the
nth-root of J, and <"2pdD(DPR) is the
crys-tal volume. H is the surface optical phonon (SO) phonon Hamiltonian which can be expressed as
H"
L uBRLBL,
(7) whereu is the surface optical (SO) phonon en-ergy, BRL(BL) is the creation (annihilation) operator for SO phonon. H} is the interaction between
electron and SO phonon:
H}" L 2CLI
np 2Docos np 2Dz(BRL#BL) (8) with CL"1S 4peu DkL[I(kLd)!I(kLd)I(kLd)] ; 1 e(u)!e! 1 e(u)!e, (9) u"1# e!e e!e(u), (10)e(u)"!K(kLd)I(kLd)I(kLd)K(kLd), (11) where kL"np/2D, and S"pd. IK and kK are, respectively, the mth-order modi"ed Bessel func-tion of the "rst and second kind.
Following Landau and Pekar's variational ap-proach [32], the trial wave function can be written as
"W2"U(r);;Q"02, (12) where U(r) depends only on the electron coordi-nate, and"02 is the phonon vacuum state de"ned by
bO"02"0, aO"02"0, and ; is a unitary
trans-formation given by ;"exp
LJ (aRLJ fLJ!aLJ fHLJ), (13) ;"exp L (BRLgL!BLgHL). (14) where fLJ and gL are the variational function and the unitary operators ; and ; transform the bulk phonon and surface phonon operators as follows: ;RaRLJ;"aRLJ#f RLJ, (15) ;RaLJ;"aLJ#fLJ, (16) ;RBRL;"BRL#gRL, (17) ;RBL;"BL#gL. (18) The parameters fLJ, fH LJ, gL, gHL can be obtained byminimizing the 1"H"2 with respect to the para-meters fLJ, fH
1H2 turns out to be 1H2"1U(r)"H"U(r)2
!
LJ
<LJ
u*-
1U(r)"JsLdocos np 2Dz"U(r)2
! L CL u
1U(r)"I(kLo)cos np 2Dz"U(r)2
. (19) The axis of the wire is assumed to be along the
z direction. To solve the electronic part, one can
employ the perturbative-variational approach. Two variational parametersa and b are introduced by adding and subtracting two terms ae/eo and (b/2k)z into the original Hamiltonian H and then regroup H into three groups
H"H(b)#H(a)#H (a,b), (20) where H(b)"!2k R Rz# b 2k z, (21) H(a)"!2k
R Rx# R Ry! ae eo#<(o) (22) H (a, b)"ae eo! b 2k z! e er. (23)In the above equations, H (a, b) is treated as a per-turbation, and a and b are treated as variational parameters which can be determined by requiring the perturbation term to be as small as possible. Decomposing H into two terms H and H is equivalent to dividing the space into a two-dimen-sional (in xy plane) and a one-dimentwo-dimen-sional (in
z-axis) subspace. The unperturbed part of the
Hamiltonian H contains two terms, i.e. H and
H, where H represents the one-dimensional
harmonic oscillator, and H represents a two-di-mensional hydrogen atom located inside a quan-tum disk [16]. Both can be solved exactly. For illustration, the ground-state energy and wave function of the unperturbed part can be expressed as
E
(a, b)"E (b)#E (a), (24)
W (r,a, b)"u
(z;b)u (x, y;a), (25)
respectively, where u
(z;b) is the ground-state wave function of the 1D harmonic oscillator, and
u (x, y;a) is the ground-state wave function of the 2D hydrogen atom located at the center of an in"nite circular well. The ground-state eigenvalue and eigenfunction of the 1D harmonic oscillator can be expressed as E (b)"b2k, (26) u (z;b)"
b p e\@ X. (27) The ground-state eigenvalue and eigenfunction of the 2D hydrogenic impurity located at the center of an in"nite circular well can be obtained as [16](1) For E(0,
u (x, y;a)"Ne\K m K >
U("m"
#1/2!j,2"m"#1,m), (28) where m"ao, a"!8kE/, j"2kae/ ea, U(a,b,x) is the con#uent hypergeometric function, and N is the normalization constant.
(2) For E'0,
u (x, y;a)"NmKUK\ (g,m), (29) where m"ao, a"!8kE/, g"!kae/ ea, UK\ (g,m) is the irregular Coulomb wave function, and N is the normalization con-stant.
(3) The turning point for energy changing from
E'0 to E(0 in the quantum circle system may
be determined by setting d\ J
8ke e d "0 for m"0 (30) and d\ J8ke e d L0 for m"1. (31) The requirement of the continuity of the wave func-tions and its "rst derivative at boundary yields(1) For E(0:
Fig. 1. The ground-state energy (solid line) and the binding energy (dotted line) of a hydrogenic impurity located at the axis of a cylindrical wire as a function of the radius of the wire. (RyH and aH are the e!ective Rydberg and the e!ective Bohr radius.)
(2) For E'0:
UK\ (g,ad)"0. (33) The eigenvalues are then given as
E
(a)"
! kae
2ej for E(0, kae
2eg for E'0.
(34) The "rst-order energy correction can thus be ob-tained as
*E (a, b)"1U
(r;a, b)"H (a, b)"U (r;a, b)2 "1u
(x, y:a)"aeeo"u (x, y;a)2 !1u (z;b)"bz2k "u (z;b)2 !1U (r;a, b)" e er"U (r;a, b)2. The second term of the above equation can be integrated analytically and the result is
1u (z;b)"bz
2k "u (z;b)'u (x, y;a)2" b
k . (35) Then, the total energy up to the "rst-order per-turbation correction can then be obtained as
E(a, b)"E (b)#E (b)#*E (a, b). (36) The variational parameters are then chosen by re-quiring the total energy E(a,b) to be minimized with respect to the variation of a and b. This is equivalent to requiring
RE
Ra"0, (37)
RE
Rb"0. (38)
For the excited states, the eigenvalues and eigen-functions can be treated in the same way.
3. Results and discussions
We have calculated the e!ect of the con"ned longitudinal-optical phonon and surface phonon
interactions on the hydrogenic impurity located in a quantum wire. And the well potential is con-sidered as in"nite. Fig. 1 shows the ground-state energy as a function of the wire radius. The binding energy E of the hydrogenic impurity is de"ned as the energy di!erence between the ground-state en-ergy of the cylindrical wire system with and without the impurity, i.e.
E"E!E, (39)
where E is the ground-state energy of the quan-tum wire system without the impurity, while E is the ground-state energy of the quantum wire system with the impurity located on the axis of the cylindrical wire. One can see from Fig. 1 that the energy of the 1s state becomes negative when the wire radius is larger than 1.65aH. It means that the con"ning energy is larger than the Coulomb energy as the wire radius is smaller than 1.65aH. And one can also note that as the radius of the quantum wire is decreased, the ground-state energy increases. As the wire radius d becomes smaller, the electron is pushed toward the axis of the cylindrical wire. This makes the electron get close to the nu-cleus. As the electron gets close to the nucleus, both the ground-state energy and the binding energy increase rapidly. This is because the Coulomb
Fig. 2. The binding energy of the 2s excited state as a function of the wire radius. (RyH and aH are the e!ective Rydberg and the e!ective Bohr radius.)
Fig. 3. The energies of the wire radius modi"ed by the con"ned BO and SO phonon. The solid line stands for the BO phonon e!ect, and the dashed line for the SO phonon e!ect. (RyH and
aH are the e!ective Rydberg and the e!ective Bohr radius.)
potential, which varies with &1/d (d is the wire radius), becomes more negative, while the kinetic energy of the electron, which varies with &1/d (by the uncertainty relation), increases more rapidly. As a result, the ground state energy is increased as the electron gets close to the nucleus. The binding energy de"ned in Eq. (39) is e!ectively the negative sign of the Coulomb interaction energy between the electron and the nucleus, i.e. &1/d, therefore, the binding energy of the electron is also increased as the electron gets near to the nucleus. Our results show that for small wire radius, the binding ener-gies are in good agreement with the previous results [12,16]. As the radius becomes very large, our re-sult approaches the correct limit 1RH while the previous work [14] can only yield a value of 0.22RH. The large discrepancy of the previous work may be due to the arti"cial dividing of the varia-tional trial wave function into a one-dimensional hydrogen atom and a two-dimensional hydrogen atom and thus forces the creation of an additional node of the wave function at z"0. In this work, the trial wave function is adopted to be in the form of 1D harmonic oscillator wave function instead of the one-dimensional hydrogen atom. This prevents our wave function from introducing any additional node at z"0. Fig. 2 presents the 2s excited state binding energies as the functions of wire radius. One can note from the "gure that as the wire radius increases, the binding energy approaches 0.25RH which gives correctly the limiting value of 3D hy-drogen atom.
Fig. 3 presents the con"ned BO phonon and SO phonon e!ects as functions of wire radius. With increasing the wire radius, the magnitude of the con"ned BO phonon e!ect decreases from large value and then approaches to the bulk value. When the wire radius is less than 1.5aH, the polaron e!ect increases rapidly. One might think as the radius becomes very small, the con"ned BO phonon e!ect should approach zero, like the case in quantum well [33]. In fact, similar results were obtained by Oshiro in a spherical quantum dot [34]. They found that the polaron energy shift is enhanced as the dot radius becomes small. This is due to the fact that the electron becomes completely localized (E approaches in"nity) in small wire (or dot) radius while the binding energy approaches 4RH in small
well width. In the case of quantum well, the con-"ned SO phonon e!ect plays the dominant role for small well width [33]. But in quantum wire, the con"ned SO phonon is less important, just like that in quantum dot system [34]. This is because the surface area of a quantum wire (or quantum dot) decreases with the radius. Thus, the number of vibration modes of con"ned SO phonon becomes fewer.
In Fig. 4, three curves are presented. The dotted curve represents the binding energy of the impurity
Fig. 4. The binding energy with/without phonon e!ect. The dotted line stands for the binding energy without the phonon e!ect. The dashed line stands for that only BO phonon e!ect on the binding energy, and the solid line for both the BO and SO phonons e!ects on the binding energy. (RyH and aH are the e!ective Rydberg and the e!ective Bohr radius.)
without considering the interactions between the electron and phonon. The dashed curve represents the binding energy of the impurity with only the con"ned BO phonon e!ect being taken into ac-count. While the solid curve is the binding energy of the impurity including both con"ned BO phonon and SO phonon e!ects in the calculation. Com-pared with the impurity binding energy, the con-"ned SO phonon is negligible in quantum wire. We then conclude that because of the similarity in geometry, the behavior of the polaron e!ect on the quantum wire system is like that on the quantum dot system.
4. Conclusion
In this work, analytical solutions for the e!ects of the electron}phonon interaction on the binding energies of an impurity located inside a quantum wire are obtained by a simple but e$cient perturbation-variation method. As the radius becomes very large, the correct limiting value can be obtained. We have also discussed both the con"ned BO and SO phonon e!ects. We found that the con"ned BO phonon e!ect is prominently for a quantum wire with a small radius. We have also found that the energy corrections of the
polar-on e!ects polar-on the impurity binding energies increase rapidly when the wire radius is less than 1.5aH. Acknowledgements
This work was supported partially under the Grant no. NSC 89-2112-M-009-038 by the Nati-onal Science Council, Taiwan.
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