Theory of resonant states of hydrogenic impurities in quantum wells

Theory of resonant states of hydrogenic impurities in quantum wells

S. T. Yen

Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

共Received 17 February 2002; revised manuscript received 6 May 2002; published 30 August 2002兲 The binding energy and the density-of-states spectrum of resonant impurity states in quantum well structure have been theoretically studied with variation of the impurity position taken into account, using the multisub-band model and the resolvent operator technique. Calculations for the 2 p0 resonant state in a

GaAs-Al0.2Ga0.8As quantum well have been performed. It has been found that there can be a considerable

resonant coupling in the 2 p0 state, causing a⬃0.1 ps capture or escape time of electrons between the 2p0

localized state and the first subband states. The maximum shift of the impurity energy is in general of the order of 0.1 meV, much smaller than the maximum binding energy of the 2 p0state.

DOI: 10.1103/PhysRevB.66.075340 PACS number共s兲: 73.20.Hb

I. INTRODUCTION

The study of the electronic states of a hydrogenic impu-rity in a semiconductor quantum-well 共QW兲 structure has been a subject of considerable interest for the last two decades.1–15 There have been numerous reports on calcula-tions of the impurity states in such a system.1–14 However, most of the studies are restricted to the calculation of the binding energies of the ground state or some low-lying ex-cited states attached to the first subband.1–10In general, the calculation relies on the variational technique accompanied with a proper trial function. For the ground state, a good trial function should keep the calculation not too complicated and give the expectation energy of the state as low as possible. Typical forms for the trial function are bound and expressed as a linear combination of Gaussian orbitals2,11 or as prod-ucts of the envelope function of the first subband edge and a single or a linear combination of Gaussian 共or exponential兲 orbitals.1,3–9Making use of the cylindrical symmetry of the systems, it is not difficult to obtain proper trial functions for the excited states below the first subband, such as the 2 p_⫾ states.2,3,6 –9,11 However, the trial function for the 2 p₀ state, which is cylindrically symmetric and has a node along the growth direction, has to be chosen with caution. When the 2 p₀ donor level lies below the lowest subband for a wide QW, it is a localized state and a bound trial function is rea-sonable. Such a state is orthogonal to other localized impu-rity states below the first subband and can therefore be ob-tained simultaneously with the ground state in a matrix diagonalization.2,11 It has been confirmed that as the QW width reduces, the on-center 2 p0 donor level moves with the second subband and then overlaps with the continuum of the first-subband states.3,6,8 This indicates that the 2 p0 state should be attached to the second subband rather than the first one. Consequently, a trial function has been proposed in the form of the envelope function of the second-subband edge multiplying a linear combination of Gaussian or exponential orbitals.8,12 However, such a trial function of the 2 p0 state, which when below the first subband has to stay orthogonal to the ground state, becomes inconvenient for an asymmetric system 共with an impurity in an asymmetric QW or an off-center impurity in a symmetric QW兲.

When the 2 p0state lies above the first subband, it may be

resonant with the first subband and no longer localized. The bound trial function becomes questionable for such a state which is composed of a bound part and an extending part. Priester et al., first pointed out the resonance behavior of impurity states attached to higher-order subbands.12 They proposed a type of trial function for the bound part and treated the resonant coupling with the technique of resolvent operators. Since then, the 2 p0 state has been treated with various trail functions.11–14 The investigations were, never-theless, restricted to symmetric systems where the 2 p0 state has an odd parity, leading to no coupling with the first sub-band. In this case the 2 p₀state is localized. This allows us to write down easily a proper trial function for the 2 p0 state. When the impurity is away from the QW center, the parity breaks down and the 2 p0 state is in resonance with the first subband. It is not trivial to determine a proper trial function for the bound part of the state since it is not a stationary state of the whole system and need not be orthogonal to other localized impurity states below the first subband.

The resonance of the 2 p0state has been observed in reso-nant Raman scattering experiments.15The resonant states can serve as hot-carrier traps for the mechanism leading to a negative differential conductance.16 They are also expected to play an important role in achievement of intraimpurity population inversion for terahertz stimulated emission.17It is therefore of scientific and technical importance to have a theoretical method for the investigation of the resonant im-purity states. However, until now there has not existed any report on a treatment of the resonant impurity states in an asymmetric system.18

In this paper, we will theoretically study resonant states of a shallow donor arbitrarily lying in a two-dimensional QW structure based on the multisubband model. The density-of-states共DOS兲 spectra of the resonant states are obtained using the resolvent operator technique. For illustration, we calcu-late the binding energy, the DOS spectrum width, and the resonance energy shift of the 2 p0state in a QW as a function of the impurity position. Considerable coupling is found. It causes a capture or escape time as short as ⬃0.1 ps. The paper is organized as follows. In the following section, we present a theory of resonant impurity states in a QW. The calculated results of the 2 p0state in GaAs-Al0.2Ga0.8As QW

are given in Sec. III. Finally, the conclusion is drawn in Sec. IV.

II. THEORY

The effective-mass Hamiltonian for an electron bound to a donor in a QW structure can be written as

H⫽H0⫹Vc共r兲, 共1兲

where H₀ is the impurity-free Hamiltonian and can be writ-ten as H0⫽⫺ ⳵ ⳵z 1 mr共z兲 ⳵ ⳵z⫺ⵜ兩兩 2_{⫹V0共z兲. 共2兲}

Vc is the Coulomb potential energy of the impurity, having the following expression:

Vc共r兲⫽⫺

2 ␬共z兲

冑

␳2_⫹共z⫺z

i兲2

. 共3兲

The Hamiltonian is written in the dimensionless form in which the energy and the length are in units of the effective Rydberg R* (R*⫽m0*e4/2⑀02ប2) and effective Bohr radius a* (a*⫽ប2_⑀0/m

0

*e2) of the well material, respectively. (m₀* and ⑀0 are the effective mass and the static dielectric constant, respectively, of the material making up the well.兲 The z axis is chosen to be along the growth direction of the layers. mr(z) is the relative effective mass, defined as the ratio of the effective mass at z to that of the well material, m₀*. Here V0(z) is the potential energy of the two-dimensional QW in the absence of the impurity.␬(z) is the relative dielectric constant at z with respect to that of the well material. The impurity is assumed to be at ␳⫽0 and z⫽zi.

The impurity-free Schro¨dinger equation

H0␺nk⫽Enk␺nk 共4兲 can be readily solved with the eigenfunctions expressed as

␺nk⫽ 1

冑

Ae ik•␳_f

n共z兲, 共5兲

where n is the subband index, k is the in-plane wave vector, and A is the area of the QW for normalization of the eigen-functions.

The wave functions of resonant states can be divided into two parts: a bound part and an extending part. Consider the energy EI of an impurity state of interest to lie between the bottoms of the (␯⫺1)th and the ␯th subbands: that is, E_{␯⫺1,k⫽0}⬍EI⬍E␯k⫽0. This impurity state is conventionally called as being attached to the ␯th subband. It may be in resonance with the mutually overlapped (␯⫺1) lowest sub-bands 共subbands 1,2, . . . ,␯⫺1). The wave function of the resonant state as ␳→⬁ can be written in a linear combina-tion of solucombina-tions of the Schro¨dinger equacombina-tion, which now can be regarded to be impurity free—that is,

␺(␯m)_⫽␺ B (␯m)_⫹␺ X (␯m)_{, 共6兲} where ␺B (␯m)_{共␳→⬁,␾,z兲⫽e}im␾

兺

n⭓␯ C_n(␯m)K_n关Im共k_n兲␳兴 f_n共z兲, 共7a兲 ␺X (␯m)_{共␳→⬁,␾} ,z兲⫽eim␾

兺

n⬍␯ Cn (␯m) Hn (1)_共k n␳兲fn共z兲, 共7b兲 with k_n⫽

冑

E_I⫺E_nk⫽0. Here K_nis the modified Bessel func-tion of the second kind of order n and Hn

(1)

is the Hankel function of the first kind of order n. The␾-dependent func-tion eim␾, where m is an integer, arises from the cylindrical symmetry of the system. C_n(␯m)’s are coefficients of the linear combination. It is clear that kn is imaginary for n⭓␯ (EI ⬍Enk⫽0) and real for n⬍␯ (EI⬎Enk⫽0). With the help of the fact that, as ␳→⬁,

Kn关Im共kn兲␳兴→ e⫺Im(kn)␳

冑

2␲Im共kn兲␳ 共8a兲 and H_n(1)共k_n␳兲→

冑

2 ␲kn␳ e⫺i(n/2⫹1/4)␲eikn␳_{, 共8b兲} we know that ␺_B(␯m) in Eq. 共7a兲 decays with ␳ and should correspond to the bound part of the resonant state attached to the␯th subband, while␺_X(␯m)in Eq.共7b兲 behaves divergently as outgoing waves and should be responsible for the extend-ing part. Since the plane waves eik•␳can serve as a complete set of basis functions for expressions of the␳-dependent en-velope functions multiplying by eim␾, the bound part ␺B

(␯m)_{(␳→⬁) in Eq. 共7a兲 can be thought of as a linear} com-bination of higher-subband states ␺_nk with n⭓␯ while the extending part ␺X

(␯m)

(␳→⬁) in Eq. 共7b兲 is in a linear com-bination of lower-subband states␺nkwith n⬍␯. It is there-fore easy to divide the impurity resonant state into the bound and extending parts if the wave function over the whole range is assumed to be in the form

␺(␯m)_⫽eim␾

兺

n⫽1 ⬁ Y_n(␯m)共␳兲f_n共z兲⫽␺_B(␯m)⫹␺_X(␯m), 共9兲 where ␺B (␯m)_⫽eim␾

兺

n⫽␯ ⬁ Y_n(␯m)共␳兲fn共z兲 共10a兲 and ␺X (␯m)_⫽eim␾

兺

n⫽1 ␯⫺1 Yn (␯m)_共␳兲f n共z兲. 共10b兲

Comparing Eq. 共10兲 with Eq. 共7兲, we find that the ␳-dependent functions Y_n(vm) should have the asymptotic be-havior that, as␳→⬁,

Yn (␯m)_→C n (␯m) Kn关Im共kn兲␳兴 共11a兲 for n⭓␯ and Y_n(␯m)→C_n(␯m)H_n(1)共kn␳兲 共11b兲 for n⬍␯. Since the set of plane waves eik•␳is complete for expressions of all eim␾Y_n(␯m)(␳), it is concluded that the bound part of the resonant state is attributed entirely to the subband states with n⭓␯, while the extending part is com-posed only of the subband states with n⬍␯. Due to the or-thogonality of the subband states, we have

具

␺nk兩␺B

(␯m)

_典

_{⫽0 共12a兲}

for n⬍␯ and thus

具

␺X (␯m)_兩␺

B

(␯m)

_典

_{⫽0. 共12b兲}

That is, the bound part is orthogonal to all lower-subband states and also to the extending part. Similarly, the extending part is orthogonal to all higher-subband states. Note that in the expansion 共10a兲, the continuous unconfined subbands above the barriers as well as the confined subbands with n ⭓␯ must be included to give a complete set of basis func-tions for the localized function␺_B(␯m).

It is convenient in derivation of formulas to decompose the Hilbert space into two subspaces S_B(␯) andS_X(␯), where

SB (␯)

is the space with all higher-subband states␺nk(n⭓␯) as basis vectors and S_X(␯) is the one with all lower-subband states␺nk(n⬍␯) as basis vectors. A projection operator P␯ is defined to project a state onto the subspaceS_B(␯). Accord-ingly, the total Hamiltonian in Eq. 共1兲 can be written as

H⫽H_U(␯)⫹H_C(␯)⫹共1⫺P_␯兲Vc共1⫺P␯兲, 共13兲 where

H_U(␯)⫽P_␯共H0⫹Vc兲P␯⫹H0共1⫺P␯兲 共14a兲 and

H_C(␯)⫽共1⫺P_␯兲V_cP_␯⫹P_␯V_c共1⫺P_␯兲. 共14b兲 H_U(␯) is the Hamiltonian of the uncoupled system in which there is no intersubspace coupling—i.e.,

具

␺B兩HU (_␯) 兩␺X

典

⫽0, 共15兲 if ␺B苸SB (␯) _{and ␺} X苸SX

(␯)_{. In writing Eq. 共14a兲, we have} made use of the fact that P_␯ commutes with H0 such that (1⫺P_␯)H₀(1⫺P_␯)⫽H₀(1⫺P_␯). Here H_C(␯)is the term for intersubspace coupling. The third term in Eq.共13兲 is for scat-tering within subspaceS_X(␯)and gives a second-order smaller effect if the impurity states and the resonant coupling are of main interest. This term will thus be neglected. From the physical viewpoint, the bound part of the resonant state ␺B

(␯m)

should be stationary if the coupling H_C(␯) were switched off. That is, ␺_B(␯m) is an eigenstate of H_U(␯). Since the lower-subband states ␺nk苸SX

(_␯)

are also eigenstates of H_U(␯), as can be easily seen from Eq. 共14a兲, the uncoupled Hamiltonian, which will be regarded to be zeroth order, is

diagonal in the representation for which ␺_B(␯m) and all the lower-subband states ␺nk苸SX

(␯)

serve as basis functions. Note that the localized states␺_B(␯m)attached to different sub-bands␯ are eigenstates of different Hamiltonians H_U(␯). The orthogonality between those localized states is not guaran-teed. In some cases, there is more than one localized state which is attached to a common subband and has the same symmetry of axial rotation. An additional index—say, j—is required as well as the indices (␯m) to characterize these states. These states ␺_B(␯m j) must be mutually orthogonal be-cause they are distinct eigenstates of the same Hamiltonian HU

(␯) .

To find the bound part of a resonant state, we imagine that the resonant coupling is at present switched off. Define a space S_B(␯m) in which each element can be expressed in the form of the right of Eq.共10a兲 with the specific indices (␯m). All the localized states␺_B(␯m j), j⫽1,2, . . . , characterized by the same (␯m) are therefore inSB

(␯m)

. We first restrict our attention to ␺_B(␯m1) which is the lowest of all the states ␺B

(␯m j)

. The state ␺_B(␯m1) is the ground state in the space

SB (␯m)

of the uncoupled Hamiltonian H_U(␯) even if it may not be the ground state in the spaceS_B(␯)and in the whole Hilbert space. What the problem becomes now is to find a set of Y_n(␯m)(␳) (n⭓␯) in Eq. 共10a兲 such that the expectation en-ergy E_I(vm)⫽

具

␺_B(␯m)兩H_U(␯)兩␺_B(␯m)

典

, which is also equal to

具

␺B (␯m)_兩H兩␺

B (␯m)

_典

, has the lowest energy. To this end, we use the variation method with trial functions for Y_n(vm)(␳) in the following form: Y_n(␯m)共␳兲⫽␳兩m兩

兺

l C_nl(␯m)e⫺␣l (␯m) ␳2 共16兲 for n⭓␯. Such an expression has the asymptotic behavior that Y_n(␯m)→␳兩m兩 as␳→0 and Y_n(␯m)→0 as␳→⬁, in consis-tency with the asymptotic behavior of the solution of the uncoupled Schro¨dinger equation. The coefficients C_nl(␯m) are linear variational parameters, and␣_l(␯m) can serve as nonlin-ear variational parameters. For a given set of␣_l(␯m), a matrix eigenvalue problem for the coefficients C_nl(␯m) can be ob-tained by means of variations applied to the expectation en-ergy E_I(␯m) with respect to C_nl(␯m). The coefficients are then obtained simply by matrix diagonalization. By varying the nonlinear parameters ␣_l(␯m), we have more degrees of free-dom to obtain a lower E_I(␯m) which corresponds to a more accurate solution. The set of␣_l(␯m) leading to the minimum energy gives the best solution in the framework of the trial function in the form of Eqs.共10a兲 and 共16兲. The other states ␺B

(␯m j)_{, j⫽1, can be obtained simultaneously with the} low-est state ␺_B(␯m1) by the matrix diagonalization. The binding energy of the localized state ␺B

(␯m j) is defined to be EB (␯m j) ⫽E_␯0⫺EI (␯m j)_{, where E} I

(␯m j) _{is the energy of the states} ␺B

(␯m j)

. Since the total Hamiltonian H depends on the QW structure and the impurity position zi, the resulting energy E_I(␯m j) and parameters C_nl(␯m j) and ␣_l(␯m j) 共and thus ␺_B(␯m j)) are also functions of the QW structure and zi. The Gaussian orbitals chosen for composition of the trial function make

tractable the integrals encountered in calculations of the ma-trix elements of the eigenvalue problem. However, the mul-tisubband trial function in the form of Eq. 共10a兲 has the shortcoming that an accurate solution requires a large amount of higher subbands to be included in the variational method for an impurity state in a wide well, regardless of the impurity position. Since we are mainly interested in the reso-nant impurity states which appear only in a sufficiently nar-row well, we will not encounter the difficulty in the present paper.

In the expression 共10a兲, we have included all the higher subbands (n⭓␯) for the trial function of the localized states ␺B

(␯m)_{. The more subbands the trial function contains, the} more accurate the solution is. However, for a QW sur-rounded by finite barriers, the subbands with bottoms above the barriers merge into a continuum, causing difficulty in the treatment of the variational technique. In fact, the inclusion of all higher fn(z) with n⭓␯ in Eq. 共10a兲 serves to form a complete basis set to accurately construct the z-dependent component of the localized states ␺_B(␯m)(␳,␾,z) for an arbi-trary ␳. To solve this problem, we discretize the continuum of subbands by putting two infinite potential walls enclosing the QW structure in the z direction as long as the walls are far away from each other such that the impurity states of interest will not be affected by the enclosure. The resulting discrete subbands can thus mathematically serve as a basis set to accurately construct the z-dependent component of the localized states but do not influence the physical results of interest. The distance between the infinite walls should be chosen with caution to avoid compression of the impurity wave functions by the walls. On the other hand, it will re-quire inclusion of more subbands in the calculation to obtain an accurate binding energy of an impurity state in a larger space enclosed by the walls.

Now we switch on the resonant coupling H_C(␯) between the localized state ␺_B(␯m j) and the subspace S_X(␯). The reso-nant states can be treated with the technique of resolvent operators. Let G(0) and G be the resolvent operators for the uncoupled system H_C(␯)and the total system H, respectively. We have

G(0)⫽共E⫺H_U(␯)⫹i0⫹兲⫺1 共17a兲 and

G⫽共E⫺H⫹i0⫹兲⫺1. 共17b兲

The DOS spectrum of the single impurity state␺(␯m j)can be expressed as

n_B(␯m j)共E兲⫽⫺1

␲Im关GBB兴, 共18兲

where G_BB⫽

具

␺_B(␯m j)兩G兩␺_B(␯m j)

典

. To find the matrix element G_BB, the two resolvent operators are related to each other through the Dyson equation

G⫽G(0)⫹G(0)H_C(␯)G. 共19兲 Since H_U(␯)is diagonal in the representation for which␺_B(␯m j) and all␺_nk苸S_X(␯) are basis functions, G(0) is also diagonal

in the respresentation. Making use of this property and van-ishing intrasubspace coupling of HC

(␯)

, we obtain the follow-ing equalities from Eq.共19兲:

GBB⫽GBB (0)_⫹G BB (0)

兺

n⬍␯,k VB,nkGnk,B 共20a兲 and Gnk,B⫽Gnk,nk (0) _V nk,BGBB, 共20b兲 where G_BB(0)⫽

具

␺_B(␯m j)兩G(0)兩␺_B(␯m j)

典

⫽共E⫺E_I(␯m j)⫹i0⫹兲⫺1, 共21a兲 Gnk,nk (0) _⫽

_具

_␺ nk兩G(0)兩␺nk

典

⫽共E⫺Enk⫹i0⫹兲⫺1, 共21b兲 Gnk,B⫽

具

␺nk兩G兩␺B (␯m j)

_典

, 共21c兲 V_B,nk⫽V_nk,B* ⫽

具

␺_B(␯m j)_兩V c兩␺nk

典

. 共21d兲 Substituting Gnk,Bin Eq.共20b兲 into Eq. 共20a兲, we can have GBB⫽关F共E兲⫹i⌫共E兲兴⫺1, 共22兲 with F共E兲⫽E⫺E_I(␯m j)⫺

兺

n⬍␯,k P兩VB,nk兩 2 E⫺Enk ⫽E⫺EI (␯m j)_⫺1 ␲ n

兺

⬍␯ P

冕

E_n0 ⬁ _⌫n共E

_⬘

_兲 E⫺E

⬘

dE

⬘

, 共23兲 ⌫共E兲⫽␲

兺

n⬍␯,k兩VB,nk兩 2_␦共E⫺E nk兲⫽

兺

n⬍␯ ⌫n共E兲, 共24兲 and ⌫n共E兲⫽ A 4兩VB,nk兩 2 _{with k⫽}

冑

_E⫺E n0. 共25兲 The area A in Eq. 共25兲 will be canceled out with that in 兩VB,nk兩2. Here Px⫺1 means to take the Cauchy principal value of x⫺1. The resonance energy E_R(␯m j) of the impurity state is obtained by finding the peak position of the DOS spectrum n_B(␯m j)(E), and the spectrum width is approximated by⌫(E_R(␯m j)), which reflects the capture time of an electron into the localized state ␺_B(␯m j) from the subband states ␺nk苸SX

(␯) _{and also the escape time of an electron from the} localized to the subband states. The time is estimated using the relation ␶⫽ប/2⌫(E_R(␯m j)). The resonant coupling causes an energy shift of⌬E(␯m j)⫽E_R(␯m j)⫺E_I(␯m j). It is noted that the DOS spectrum nB

(␯m j)

(E) is due to a single resonant state and has dimension of inverse energy. For a low doping con-centration, impurities are far from each other so that the in-teraction between the impurities is negligible. Consequently, the total DOS spectrum due to all the impurities is the sum of all the impurity spectra, which is simply the single-impurity spectrum multiplying the number of impurities.

III. NUMERICAL RESULTS AND DISCUSSION

We present numerical results to give an illustration of the resonant states. Calculations are performed for the 2 p0 or ␺(201)_{resonant impurity state (␯⫽2, m⫽0, j⫽1) in a QW} structure consisting of a GaAs well sandwiched by Al0.2Ga0.8As barriers. The 2 p0 state lies above the first-subband edge and attached to the second first-subband for an on-center impurity in the QW well with a width less than some critical value (⬃60 nm).3 In the calculation of the bound part of the 2 p₀ state, we use the trial functions共10a兲 and共16兲 consisting of nine lowest subbands in subspace S_B(2)

(n⫽2 –10) and five Gaussian orbitals (l⫽0 –4). Two infi-nite potential walls 60 nm apart are artificially imposed to surround the QW structures to discretize the continuum of the subbands above the Al0.2Ga0.8As barriers. It has been made sure that the walls do not significantly compress the 2 p0 state and the first subband states. The problem becomes a 45⫻45 matrix eigenvalue equation for each set of the pa-rameters ␣_l(20). The accuracy of the numerical results has been confirmed by observation of the convergence of the data with increasing the number of terms used in the expres-sion of the trial function. The five nonlinear variational

pa-FIG. 1. Waterfall plots of共a兲 the binding energy, 共b兲 the DOS spectrum width, and 共c兲 the shift of resonance energy of the 2p0state in

GaAs-Al0.2Ga0.8As QW structures as a function of the impurity position for various well widths. Each of the numbers shown in the figures

rameters ␣_l(20) are under the restriction of ␣_l(20)⫽␣⫹␤_l where␣is treated as a nonlinear variational parameter but␤_l are fixed to be ␤l⫽0.25l

3

for l⫽0 –4. In the calculations, the effective mass and the low-temperature band gap for the Al_xGa_1⫺xAs material system are assumed to be (0.067 ⫹0.083x)me and 1.519⫹1.447x⫺0.15x2 eV, respectively, according to Casey and Panish,19 where m_e is the free elec-tron mass. The barrier height is assumed to be 0.65 of the band gap difference. The static dielectric constant of AlxGa1⫺xAs is adopted by a linear interpolation between those of GaAs共12.58兲 and AlAs 共10.06兲.

Figures 1共a兲, 1共b兲, and 1共c兲 are the waterfall plots of the binding energy E_B(201), the spectrum width⌫(E_R(201)), and the shift of resonance energy ⌬E(201), respectively, of the 2 p0 state as a function of the impurity position zifor various well widths. The origin is chosen at the center of the QW. The binding energy of the 2 p0 depends on the electronic distri-bution probability around the impurity position. As a conse-quence, the binding energy exhibits two peaks associated with the two maxima of the electronic distribution probabil-ity of the 2 p₀ state. Furthermore, the maximum electronic distribution probability can be enhanced by a reduction of the well width. This causes an increase of the binding energy with the well width decreasing, as can be seen from Fig. 1共a兲. Further reduction of the well width can nevertheless push up the energy level to be close to the barrier height, leading to a widespread electronic distribution and thus a reduction of the binding energy. This explains the lower binding energy for the 7-nm well than for the 10-nm well.

The spectrum width ⌫(E_R(201)) reflects the resonant cou-pling strength, which depends on the impurity position and the resonance energy relative to the lowest subband bottom, i.e., E_R(201)⫺E10, through the coupling matrix elements in Eq. 共24兲. When the impurity is at the well center, the parity difference of the bound ␺_B(201) and the first subband states ␺1k causes no coupling (⌫⫽0), shown in Fig. 1共b兲. The spectrum width goes through a maximum and then reduces toward zero as the impurity moves away from the well cen-ter. The maximum spectrum width of a wider well is larger than that of a narrower well. This is because the energy dif-ference E_R(201)⫺E10 is smaller for a wider well. For a large value of kR⫽

冑

ER

(201)_⫺E10 ,␺1k

R exhibits a rapid oscillation along ␳, resulting in a small overlap integral for the spec-trum width. In Fig. 1共b兲 we show the value of the maximum spectrum width for each of the wells. It can be as large as

⬃3 meV. We also show the value of the capture or escape time associated with the maximum resonant coupling for each well width. It ranges approximately from 0.1 to 1 ps.

The resonance energy shift ⌬E(201) shown in Fig. 1共c兲 exhibits more complicated variations with the impurity posi-tion. As can be seen from Eq. 共23兲, it depends on the reso-nance energy level E_R(201) and the coupling strength ⌫(E) over a range of energy. Therefore, a wider well can virtually have a more significant energy shift than a narrower one. Since ⌫(E) in principle decays with E, the coupling is ex-pected to cause a blueshift of the impurity level, i.e., ⌬E(201)_{⬎0. As can be seen, the maximum shift is of the} order of 0.1 meV. For wide wells, such as those of width ⬎30 nm, the resonance energy ER

(201)

is close to the first-subband bottom E10. This may cause reduction of the blue-shift or even a redblue-shift of the impurity level (⌬E(201)⬍0), which does not show in the waterfall plot of Fig. 1共c兲. Ac-cording to the calculated results, the redshift can be as large as tenths of meV for a wide well. The energy shift is in general much smaller than the binding energy.

IV. CONCLUSION

We have presented the formulas for calculations of the characteristics of resonant impurity states in QW structures by dividing the total system into an uncoupled system and the resonant coupling. The division depends on which sub-band the impurity state of interest is attached to. The bound part of the resonant state is a stationary state in the un-coupled system and has been obtained by the variational method with a multisubband trial function. After having found out the representation in which the uncoupled Hamil-tonian is diagonal, we treat the resonant coupling with the technique of the resolvent operators and obtain the spectrum of the resonant impurity state. The numerical results of the binding energy, the DOS spectrum width, and the energy shift of the 2 p₀ state have been presented. The spectrum width can be as large as several meV, causing a capture or escape time of ⬃0.1 ps. The energy shift is in general less than 0.4 meV, much smaller than the binding energy.

ACKNOWLEDGMENT

This work is supported by the National Science Council of ROC under Contract No. NSC 90-2112-M-009-054.

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