Resonant hydrogenic impurity states and 1s-2p(0) transitions in coupled double quantum wells

Resonant hydrogenic impurity states and 1s

À2p

₀

transitions in coupled double quantum wells

S. T. Yen*

Department of Electronics Engineering, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China 共Received 8 February 2003; revised manuscript received 2 June 2003; published 21 October 2003兲

Calculations of the 1s and resonant 2 p0states of a shallow donor in double-quantum-well structures are

performed. The variational method is used to calculate the localized part of the impurity states, taking into account the effect of intersubband mixing. The resonance coupling of the 2 p0state with the first subband is

then dealt with using the Green function technique. The results show that for an asymmetric double-quantum-well structure the 1s state has a maximum binding energy as the donor is around the center of the wider double-quantum-well while the 2 p0 state has in general a maximum binding energy as the donor is in the narrower well. The

resonant coupling of the 2 p0state is stronger for the structures with a stronger intersubband mixing, where the

2 p0energy level is closer to the first subband bottom. The resonance-induced broadening of the 2 p0state can

be as large as 6 meV, corresponding to a lifetime of⬃0.1 ps. The resonance in general causes a negligibly small blueshift but can give a redshift of the order of 1 meV when the resonance is strong. A phase transition of the 2 p0 state can occur from the resonant nature to the bound nature by modulation of the interwell

coupling. The 1s⫺2p0transition energy is also calculated. The possibility of population inversion between the

1s and 2 p0states is discussed.

DOI: 10.1103/PhysRevB.68.165331 PACS number共s兲: 73.20.Hb

I. INTRODUCTION

In the last two decades, there has been considerable atten-tion given to shallow impurities confined to

quasi-two-dimensional 共quasi-2D兲 systems formed by

GaAs/AlxGa1⫺xAs epitaxial layers 1–22

because of their tech-nological importance in electronic and optoelectronic de-vices as well as their intrinsic physical interest. Numerous theoretical investigations have been performed with6,16 –22or without1–5,7–15 an external magnetic field to understand the physics of the hydrogenic impurity states in quantum wells 共QW’s兲. Due to the quite large difference in symmetry be-tween the impurity Coulomb potential and rectangular QW potential, the problem of impurity states in QW’s may not be exactly solvable and its solutions are in general obtained by variational calculations with localized trial functions. The calculations have been mostly limited to the ground states 1s and some low-lying excited states such as 2 p0, 2 p⫾1, 3d0, 3d⫾1, 3d⫾2, etc., in the usual

spectro-scopic notation.20 The states 1s, 2 p_⫾1, 3d_⫾2, etc., have zero nodes along the growth axis共the z axis兲, in similarity to the states in the lowest subband. In the absence of magnetic fields, such states always lie below the lowest subband 共re-gardless of the QW width兲 and therefore have been regarded as associated with the lowest subband. Similarly, the 2 p0and

3d_⫾1states are associated with the second subband because of their nodal property along the z axis, similar to the states in the second subband. With the QW thickness changing, the energy levels of the states move with the associated subband and may overlap with the lowest subband if the QW is suf-ficiently narrow.3–5In the same rule, the 3d₀state, associated with the third subband, may overlap in energy with the low-est two subbands. In a real case these states are unbound and are describable by the Fano formulations23in which the reso-nant interaction is treated between a discrete energy level and one or more continua of states that overlap with the energy level. The Fano resonance is not a new problem; it

has been frequently studied in various fields of physics, par-ticularly in atomic, molecular, and solid-state physics. Such a resonance can lead to an energy shift and broadening of the resonant state. The resonant impurity states in bulk semicon-ductors have been studied, for instance, by Bassani et al.24

The resonance of impurity states confined to QW’s was first pointed out by Priester et al.10 and later was investi-gated, both theoretically12–14and experimentally,11in several works. In the studies of Priester et al.10 and Jayakumar

et al.,12 the calculations were limited to the on-center

impurity—the impurity at the center of the QW. The local-ized part of each impurity state was determined by a single subband to which the impurity state is attached, with the intersubband mixing neglected.10,12The resonant effect was then considered using the Green function technique, equiva-lent to the treatment of the Fano model.23 Because of ne-glecting intersubband mixing, their calculations will be valid quantitatively only for the impurity in a sufficiently narrow QW where its binding energy is much smaller than the en-ergy separation between its corresponding subband and the nearest higher one. For the on-center impurity, the parity difference causes no resonant coupling between the 2 p0state

and the lowest subband; consequently, the 3d₀ state is the lowest resonant state in resonance with the lowest subband. Calculated results showed that the resonance effect causes a negligibly weak shift and broadening of the 3d0 state 共both

are of the order of 0.1 meV兲. The resonance of the 2p0state

was first theoretically investigated by Yen, with varying the impurity position in the QW and taking into acount the in-tersubband mixing.14 The resulting energy broadening has been found capable of ⬃3meV 共corresponding to a lifetime of ⬃0.1ps) for the impurity midway between the center and the edge of a 400-Å QW. It is expected that such a predomi-nant resonance would allow a more resolvable experimental observation, compared to the resonance of 3d0 state, under

the environment of various intrinsic scatterings.

coupled double quantum wells共CDQW’s兲 consisting of two QW’s separated by a thin barrier.15–19 These investigations were totally focused on the localized states (1s and 2 p_⫾1) associated with the lowest subband, in no external field,15in a magnetic field,16 –18or simultaneously in an electric and a magnetic fields.19In a CDQW structure, concurrent effects of quantum confinment and interwell tunneling on the impurity states lead to a rich variation of the binding energy, depend-ing on the overlap between the impurity Coulomb potential and the distribution of the electron density which is a func-tion of the impurity posifunc-tion, the well thickness, the barrier thickness, the ratio of the thicknesses of the two QW’s, and also the external fields.

In this paper, the 2 p0 resonant donor state in

GaAs/AlxGa1⫺xAs CDQW’s is investigated. We calculate,

based on the theory in Ref. 14, the variations of the lifetime broadening, the energy shift 共due to the resonance effect兲, and the binding energy of the 2 p0state in CDQW structures,

as functions of the impurity position and well thickness, for different interwell coupling. Because of the strong resonant coupling of the 2 p0 state with the continuum of the lowest

subband, it is expected to be a potential candidate for hot electron traps in the mechanism leading to a negative differ-ential conductance24and serve as the upper state for intraim-purity population inversion between the 1s and 2 p0 states

for terahertz-stimulated emission.25Therefore, we also calcu-late the 1s⫺2p0 transition energy for the donor in CDQW

structures as a function of the impurity position and the well width. In Sec. II we present briefly the calculation techiques. Calculated results and discussion are presented in Sec. III. Finally, a summary of the results and a conclusion are drawn in Sec. IV.

II. CALCULATION METHOD

The CDQW structure considered in this study, as sche-matically shown in Fig. 1, consists of two GaAs wells, the left one being of thickness wL and the right one being of

thickness wR, saparated by an AxGa1_⫺xAs barrier of

thick-ness b. The whole structure is assumed sandwiched by two semi-infinite AlxGa1⫺xAs barriers. The origin of the

coordi-nates is taken to be at the midpoint of the interwell barrier and z_idenotes the z coordinate of the hydrogenic donor at an arbitrary position. The in-plane radial coordinate of the do-nor location is assumed to be ␳⫽0.

The problem of impurity states in CDQW’s is dealt with mainly based on the approaches in Ref. 14 with a slight modification. According to the multisubband theory in Ref. 14, the 1s donor state␺1s in a 2D system is totally localized

while the 2 p0 state ␺2 p₀ may be unbound due to the

reso-nance interaction. Therefore, ␺2 p₀ is expressed in a

combi-nation of a localized part ␺_{2 p}

0

(B) _{and an extending part}

␺2 p₀ (X) : ␺2 p₀⫽␺2 p₀ (B)_⫹␺ 2 p₀ (X)

. The localized functions ␺1s and

␺2 p₀

(B) _{are obtained by the variational method. To find proper}

trial functions for ␺1s and␺2 p₀ (B)

, we note first from the mul-tisubband theory14that␺_1s is an element of a function space

S⭓1which is a Hilbert space with all subband state functions ␺nkof the corresponding impurity-free 2D system共where n

is the subband index and k is an in-plane wave vector兲 as the basis vectors. Also,␺_{2 p}

0 (B) is an element ofS_⭓2and␺_{2 p} 0 (X) is an element ofS1, whereS⭓␯represents the function space with

the subband state functions ␺nk(n⭓␯) above the (␯⫺1)th

subband as the basis vectors andS_␯ represents the function space with the functions␺_␯kin the␯th subband as the basis vectors. Therefore,S_⭓1⫽S_⭓2⫹S1. As a matter of fact,␺1s

must be the extremal giving the global minimum of the func-tional E关␺兴 of the expectation value of the energy E关␺兴 ⫽

具

␺兩H兩␺

典

/

具

␺兩␺

典

for␺苸S_⭓1, since␺1s is the ground state

of the 2D system with the impurity, which is described by the Hamiltonian H where H⫽H0⫹Vc共r兲, 共1a兲 Vc共r兲⫽⫺ 2 ⑀

冑

␳2_⫹共z⫺z i兲2 . 共1b兲

Here H0 is the impurity-free Hamiltonian of the CDQW, Vc

is the Coulomb potential energy due to the positive charge of the donor, and ⑀is the static dielectric constant of the mate-rial. We neglect the difference in dielectric constant between the well and barrier layers. The electron is at the position r ⫽(␳,z) where ␳ is the in-plane radial vector and z is the

z-component coordinate of the electron. The effective atomic

units have been adopted in Eq.共1兲. The energy and length are in units of the effective Rydberg and effective Bohr radius, respectively, of the material 共GaAs兲 making up the well.14 Similarly, the localized function ␺_{2 p}

0

(B)

is the extremal giving the minimum of the expectation value of the energy in the space S_⭓2: E关␺兴⫽

具

␺兩H兩␺

典

/

具

␺.兩␺

典

for ␺苸S_⭓2. For ad-missible functions restricted withinS_⭓2, ␺_{2 p}

0

(B) _{can therefore}

be regarded as the ground state of the Hamiltonian.14On the contrary, for functions in the total space S_⭓1, ␺_{2 p}

0

(B)

is no longer stationary and must be corrected with the part␺_{2 p}

0

(X)_.

Based on the argument above, we calculate variationally

␺1sand␺2 p₀ (B)

using the trial functions which are elements of

S⭓1andS⭓2, respectively. We take the trial functions in the

form14 ␺␯⫽

兺

n⭓␯

兺

l Cnle⫺␣l␳ 2 fn共z兲, 共2兲

FIG. 1. Schematic illustration of the GaAs/AlxGa1⫺xAs CDQW

structure consisting of a GaAs left well of thickness wLand a GaAs

right well of thickness wR, separated and sandwiched by

AlxGa1⫺xAs barriers. The interwell barrier is of thickness b. The

origin is taken at the midpoint of the interwell barrier. The donor is at z⫽zi.

where ␯⫽1 for ␺_1s and ␯⫽2 for ␺_{2 p}

0

(B)_{. Here f}

n(z) is the

z-dependent part of the subband function ␺nk

⫽eik•␳_f

n(z)/

冑

A (A being the area of the CDQW兲 which

satisfies the impurity-free Schro¨dinger equation H0␺nk

⫽Enk␺nk, Enk being the corresponding eigenvalue.

Obvi-ously, ␺1苸S⭓1 and␺2苸S⭓2. It should be pointed out that

the trial functions␺1 and␺2 in the form共2兲 are not

neces-sarily orthogonal since␺₂is the trial function for␺_{2 p}

0

(B) _which

is only the localized part of the 2 p0 state ␺2 p₀. A more

detailed discussion about the orthogonality between the 1s and 2 p0states can be found in Ref. 14. For a given set of␣l,

minimizing E关␺_␯兴 by varying Cnl gives a set of eigenvalue

equations ⳵E/⳵Cnl⫽0 (n⫽␯,␯⫹1, . . . ; l⫽1,2, . . . ),

which determines the minimal expectation values and simul-taneously the corresponding coefficients Cnl. By further

varying the nonlinear parameters␣l, we have more degrees

of freedom to obtain a lower minimal E which corresponds to a more accurate solution. The set of ␣_l leading to the minimum energies, E_1s or E_{2 p}

0

(B)_⫽min

␺_␯E关␺␯兴, gives the best

solutions for␺1sand␺2 p₀ (B)

in the framework of the trial func-tion 共2兲. The binding energies of the 1s and 2p0 states are

then determined by the energy differences EB,1s⫽E1,k⫽0 ⫺E1s and EB,2p₀⫽E2,k_⫽0⫺E2 p₀

(B)_{, respectively.}

It is worth giving a discussion about the applicability re-strictions of the trial function 共2兲. The localized functions

␺1s and␺2 p₀

(B) _{can be described by the trail function共2兲 with}

an accuracy depending on the numbers of Gaussian orbitals

e⫺␣l␳2 _{and subband functions f}

n(z) that are included in the

trial function 共2兲. For an impurity in a very wide (⬎400 Å ) or very narrow (⬍30 Å ) well such that a lot of subbands are close in energy to the impurity state of interest, one must include many subband functions fn(z) in the trial

function共2兲 to obtain an accurate result. Fortunately, increas-ing the number of subband functions means an increase in the number of linear variational parameters Cnl. In this case,

one has simply to diagonize larger-rank matrices. Once the localized part ␺_{2 p}

0

(B)

is obtained, the resonance coupling between␺_{2 p}

0

(B) _{and the continuum of the lowest}

sub-band states␺1,kis treated with the Green function technique.

The resulting partial density of states共DOS兲 of the 2p0

reso-nant state ␺2 p₀ can be written in the form of14

n2 p₀共E兲⫽⫺ 1 ␲Im 1 F共E兲⫹i⌫共E兲, 共3a兲 F共E兲⫽E⫺E_{2 p} 0 (B)_⫺1 ␲P

冕

E_1,k⫽0 ⬁ _⌫共E

⬘

_兲 E⫺E

⬘

dE

⬘

, 共3b兲 ⌫共E兲⫽␲

兺

k 兩

具

␺2 p0 (B)_兩V c兩␺1,k

典

兩2␦共E⫺E1,k兲 ⫽

再

A 4兩

具

␺2 p₀ (B)_兩V

c兩␺1,k⫽冑E⫺E_1,k⫽0

典

兩2 for E⭓E1,k⫽0

0 for E⬍E_1,k⫽0.

共3c兲

Here Px⫺1means to take the Cauchy principal value of x⫺1. The resonance energy E2 p₀ of the 2 p0 state is obtained by

finding the peak position of the partial DOS spectrum

n_{2 p}

0(E). The broadening width of the 2 p0 state is here

de-fined as the full width of half maximum 共FWHM兲 of the partial DOS spectrum, different from that in Ref. 14. The DOS spectrum is of a Lorenzian shape if its width is much smaller than the energy difference E2 p₀⫺E1,k⫽0; in this case

the FWHM can be well approximated by 2⌫(E2 p₀). It

re-flects the capture time of an electron from the lowest sub-band states␺1,kinto the 2 p0localized state␺2 p₀

(B)

and also the escape time of an electron from the localized to the subband states. The resonance lifetime is estimated using the relation

␶⫽ប/FWHM. The resonant coupling also causes an energy shift of ⌬E⫽E2 p₀⫺E2 p₀

(B)_.

III. RESULTS AND DISCUSSION

In this section we present the calculated results for the donor states in GaAs/AlxGa1⫺xAs CDQW structures with an

interwell barrier of thickness b⫽2 nm. In the calculations, the effective mass of AlxGa1⫺xAs is assumed to be

0.0665(1⫹1.25x)m_ewhere m_eis the free electron mass. The barrier height at the GaAs/AlxGa1⫺xAs interfaces is

deter-mined from the relation⌬E_g⫽1.36x⫹0.22x2 _{eV with 60%}

conduction-band offset. The static dielectric constant ⑀ is taken as 12.5. These parameters are intentionally taken the same as those in Ref. 19 to compare some of our results with those in Ref. 19.

In the variational calculation, we use nine subbands fn(z)

(n⫽1 –9 for ␺1s and n⫽2 –10 for␺2 p₀

(B)_{) and five Gaussian}

orbitals e⫺␣l␳2 _(l⫽1 –5) for the trial functions expressed in

Eq. 共2兲. The nine subbands are all discrete even if some of them are above the barrier in energy since we use the CDQW structures bounded by two infinitely high potential walls共60 nm apart兲 to simulate the real structures.14It has been previ-ously checked that the infinitely high potential walls are far apart from each other such that the impurity states of interest will not be affected by the artificial enclosure. The five pa-rameters ␣l are taken in the form of ␣l⫽␣⫹0.25(l⫺1)3

where␣ is determined variationally.14This set of␣lis found

capable of giving accurate binding energy by comparing the binding energies obtained using a variety of sets of␣lin the

variational calculation.

Figure 2 shows the binding energies of the 1s state (E_B,1s) and 2 p₀ state (E_B,2p

0) as functions of the impurity

position z_i in various CDQW structures. In Fig. 2共a兲, the curves are for the structures consisting of a GaAs left well of thickness wL⫽10 nm and a GaAs right well of thickness

varying from 3 to 10 nm (wR⫽3, 5, 7, 9, 10 nm), separated

by a 2-nm Al0.2Ga0.8As interwell barrier, and then

sand-wiched by Al_0.2Ga_0.8As barriers. 共See Fig. 1.兲 For Fig. 2共b兲, the structures are similar to those for Fig. 2共a兲 except that Al0.3Ga0.7As is used for the barriers. Also, we replace the

curves for (wL,wR)⫽(10,7) nm with those for (5,2.5) nm

to make a comparison of the 1s binding energy of the (5,2.5)-nm CDQW with that in Fig. 4共a兲 of Cen et al.19The

results are in excellent agreement. It is found that the 1s state has a maximum binding energy as the impurity is around the center of the left wider well for asymmetric CDQW struc-tures. This is because there is a larger likelihood of finding the electron of the lowest subband states in the wider well. Since the 1s state is mostly composed of the lowest subband states ␺1,k(⬀ f1), the impurity in the wider well enhances

the Coulomb potential compared to the impurity in the nar-rower well. A larger overlap between the electron distribu-tion and the Coulomb potential leads to a larger binding en-ergy. This can also explain that the peak binding energy of the 1s state increases as the right well thickness decreases. With the right narrower well thickness decreasing, the prob-ability of finding the electron in the right well reduces and part of it transfers to the left well through interwell tunnel-ing; this enhances the 1s binding energy as the impurity stays in the left wider well.共The change in the distribution of the electron probability density can also be understood from the viewpoint of intersubband mixing between the first and second subbands.兲 For the symmetric structure 关(wL,wR)

⫽(10,10)nm兴, the electron density has a symmetric

distribu-tion, resulting in two lowest peaks of the binding energy. Similarly, the 2 p0 state has a maximum binding energy as

the impurity is around the center of the right narrower well, except for the structures (10,3) nm, since the 2 p0 state is

mostly composed of the second subband states␺2,k(⬀ f2) in

which the electron is more likely to stay in the narrower well. For the symmetric structure, there is equal likelihood of finding the electron of the second subband states in the two wells. Thus we have a symmetric variation of the 2 p0

bind-ing energy with the impurity position. With the right nar-rower well thickness decreasing, the electron density兩 f2(z)兩2

initially has a tendency to being localized in the right well because of the reduction of the intersubband mixing and, consequently, the 2 p₀peak binding energy increases. Further decreasing the right well thickness then causes the electron density to spread out of the right well into the barriers and the left wider well since the second quantized level is pushed up close to the third and other higher subbands共a peculiarity in CDQW兲, increasing the mixing of the second subband with the higher subbands. This explains the reduction of the 2 p0 binding energy as the right well thickness decreases

from (wL,wR)⫽(10,5) to (10,3)nm. Furthermore, the 2p0

peak binding energy for the (10,3)-nm structures occurs at a position deviating significantly from the center of the right narrower well. The results indicate that the effect of intersub-band mixing is important in accurate calculations of the 2 p0

state in a CDQW structure. We also find from Fig. 2共b兲 that the effect of intersubband mixing is smaller on the 2 p0state

in the (5,2.5)-nm structure than in the (10,3)-nm structure because of the difference in the left wider well thickness.

Figure 3 shows the FWHM of the patial DOS spectrum of the 2 p0 state induced by the resonance coupling as a

func-tion of the impurity posifunc-tion z_i for the CDQW structures consisting of a 10-nm left well and a right narrower well of various thicknesses (wR⫽3,5,7,9,10 nm), separated and

sandwiched by共a兲 Al_0.2Ga_0.8As barriers and共b兲 Al_0.3Ga_0.7As barriers. The interwell barrier layers are still of thickness b ⫽2 nm. It is found from Fig. 3共a兲 that for the structures with Al0.2Ga0.8As barriers each curve exhibits two peaks as the

impurity is around the centers of both the left and the right wells, except for the (10,3)-nm structure. The variation of the 2 p0 spectrum width can be well explained by means of

the matrix elements in Eq.共3c兲. Since the matrix element is of the Coulomb potential between␺_1,kand␺_{2 p}

0

(B)_{, according}

to their nodal properties along the z axis, we expect that a local minimum of the spectrum width occurs as the impurity is around the midpoint of the whole structure, in agreement with the result in Fig. 3共a兲. This also explains the two peaks of each curve. It is found that the stronger resonance occurs in the (10,9)-nm and (10,10)-nm CDQW structures 共where the intersubband mixing is stronger兲 and the resonance be-comes weak as the right well thickness decreases. The cou-pling strength⌫(E2 p₀) also sensitively depends on the

mag-nitude of the wave vector, k⫽

冑

E2 p₀⫺E1,k⫽0, and thus on

the energy difference E2 p₀⫺E1,k⫽0. For a large value of

k, ␺1,kexhibits a rapid oscillation along␳, leading to a small

overlap integral of 兩

具

␺_{2 p}

0

(B)_兩V

c兩␺1,k

典

兩, as has also been

dem-FIG. 2. The binding energies of the 1s state共dashed curves兲 and the 2 p0state共solid curves兲 vs the impurity position zifor CDQW

structures with共a兲 Al0.2Ga0.8As barriers and共b兲 Al0.3Ga0.7As

barri-ers. The structures considered in 共a兲 have a left well of thickness

wL⫽10 nm and a narrower right well of thickness varying from 3

to 10 nm (wR⫽3, 5, 7, 9, 10 nm). The structures in 共b兲 are similar

to those in共a兲, except that the structure of (wL,wR)⫽(10,7) nm is

replaced by that of (5,2.5) nm. All the structures have an interwell barrier of thickness b⫽2 nm.

onstrated in Fig. 3 of Ref. 10. With the right well thickness decreasing, E2 p₀ moves upwards with the second subband

while the first subband has merely a slight change. As a result, the energy difference E2 p₀⫺E1,k_⫽0increases and the

resonance coupling decreases with the right well thickness decreasing. It is noticed that further decreasing the right well thickness causes strong subband mixing between the second and higher subbands, as has been described above共and there-fore gives a significant probability density of the second sub-band states in the left well兲, enhancing the resonance cou-pling as the impurity is in the left well. This explains that the left peak for the (10,3)-nm structure is higher than the (10,5)-nm structure and the right peak for the (10,3)-nm structure occurs as the impurity is at a position deviating significantly from the center of the right well.

For the CDQW structures with Al0.3Ga0.7As barriers, the

variations of the 2 p0 spectrum width for wR⭐7 nm are

es-sentially similar to those for the structures with Al0.2Ga0.8As

barriers, but for wR⭓9 nm, the behavior is quite different, as

can be seen in Fig. 3共b兲. As the impurity position zi is in the

range from -8 to 8 nm, the 2 p₀ level for w_R⫽10 nm lies below the first subband because the weak interwell coupling results in a small separation of the lowest two subbands. The 2 p₀ state in this case becomes purely localized and the

cor-responding spectrum width becomes zero. Similarly, as the impurity lies at zi⫽2.5⫺7.5 nm, the 2p0 state for wR

⫽9 nm is bound but becomes resonant as the impurity moves out of the range. Accordingly, we expect a transition from the resonant phase to the bound phase of the 2 p₀ state in a CDQW structure by modulation of the interwell cou-pling, which can be easily achieved by an external electric field normal to the interfaces.

Because the resonance coupling depends on the energy separation between the impurity level and the edge of the in-resonance subband, the 2 p0 state can have a much

stron-ger resonance strength than the 3d₀state. The result in Fig. 3 shows that the broadening of the 2 p0state can be as large as

6 meV共corresponding to a lifetime of ⬃0.1 ps), an order of magnitude larger than the 3d0 state.

10,12

Since such a short resonance lifetime is in general less than the lifetime due to other intrinsic scatterings, we expect that the resonance can play a predominant role in a certain of physical processes.

The energy shift of the 2 p0 state arising from the

reso-nance coupling is shown in Figs. 4共a兲 and 4共b兲 as a function of the impurity position zifor the CDQW structures that are

the same as in Figs. 3共a兲 and 3共b兲, respectively. The energy shift exhibits a more complicated variation with zi,

com-pared to the binding energy and the spectrum width. We FIG. 4. The resonance-induced energy shift of the 2 p0state vs

the impurity position zi for CDQW structures with a left well of

thickness wL⫽10 nm and a narrower right well of thickness

vary-ing from 3 to 10 nm (wR⫽3,5,7,9,10 nm). The structures

consid-ered in 共a兲 and 共b兲 have barriers made of Al0.2Ga0.8As and

Al0.3Ga0.7As, respectively. The interwell thickness of the structures

is b⫽2 nm. FIG. 3. The FWHM of the partial DOS spectrum of the 2 p0

state vs the impurity position zi for CDQW structures with a left

well of thickness wL⫽10 nm and a narrower right well of thickness

varying from 3 to 10 nm (wR⫽3,5,7,9,10 nm). The structures

con-sidered in 共a兲 and 共b兲 have barriers made of Al0.2Ga0.8As and

Al0.3Ga0.7As, respectively. The interwell thickness of the structures

know from Eq. 共3兲 that it depends on the 2p0 energy level

E2 p₀ and the coupling strength⌫(E) over a range of energy.

For weak coupling strength⌫(E)⬇0, the peak of the partial DOS spectum is at E2 p₀⬇E2 p₀

(B)

and the energy shift ⌬E ⫽E2 p₀⫺E2 p₀

(B)

is small. This explains the more considerable shift for a wider right well 共with a stronger intersubband mixing兲. Since the function ⌫(E) in general monotonically decreases with E, the integral in Eq.共3b兲 is positive 共i.e., the third term on the right gives a negative contribution兲 if E is not too close to the lower limit E1,k_⫽0 of the integral; as a

result, we have a blueshift for the CDQW structures with a narrow right well (w_R⫽3 –5 nm). On the other hand, for the structures with a wide right well, the energy difference

E2 p₀⫺E1,k⫽0 is small and the energy E we are interested in

is close to the lower limit E_1,k⫽0. This causes the integral in Eq.共3b兲 to be negative, leading to a redshift of the resonance energy. Since the coupling for a wider right well is stronger, the energy shift in such a case is more pronounced. In the case where the 2 p₀ state is a bound state, the coupling with the first subband merely gives a correction in the binding energy, corresponding to a redshift of the 2 p0 level. It is

found that the energy shift is generally of the order 0.1 meV of magnitude for wR⭐7 nm, but for the case of strong

cou-pling (wR⭓9 nm), the shift can be as large as being of the

order 1 meV. Compared with the binding energy in Fig. 2, the energy shift is in general much smaller and can be ne-glected.

The electric dipole transition between the 1s and 2 p₀ states is allowed for electromagnetic fields polarized in the growth direction. Figure 5 shows the 2 p0-1s transition

en-ergy and 2 p₀-E_1,k⫽0 energy difference for the impurity lo-cated at the centers of the wells as a function of the narrower right well thickness wR, with the left wider well thickness

fixed to be wL⫽10 nm. The curves in Figs. 5共a兲 and 5共b兲 are

for the structures with barriers layers made up of Al0.2Ga0.8As and Al0.3Ga0.7As, repectively. It is evident that

the structure with a narrower right well has a larger transition energy and also a larger E2 p₀⫺E1,k⫽0due to the large

sepa-ration of the two lowest subbands. The 2 p0-1s transition

energy lies in the far-infrared or terahertz range. Recently, a mechanism for population inversion between a resonant ex-cited state and the ground state has been proposed for accep-tors in strained bulk semiconducaccep-tors.25In an external electric field parallel to the interfaces, electrons can be accelerated from the band edge upwards to the vicinity of the resonance enegy level. If the resonant coupling is strong, the electrons will likely be captured in the localized part of the resonant state from the band continuum. The excited resonant state is thus occupied and population inversion is achieved if the ground state is emptied due to impact ionization by the elec-tric field. Based on a similar mechanism, we expect the pos-sibility of population inversion between the 1s and 2 p0

states in our CDQW structures by applying an electric field parallel to the interfaces. At sufficiently low temperature such that LO phonon absorption is negligibly weak, electrons can easily reach the 2 p0 energy level through the lowest

subband if the energy difference E2 p₀⫺E1,k⫽0 is less than

the LO phonon energy (ប␻LO⬃36 meV in GaAs兲. On the

other hand, if E2 p₀⫺E1,k⫽0Ⰷប␻LO, population inversion is

difficult to achieve since LO phonon emission occurs before electrons reach the 2 p0 energy level. It is found from the

figures that the CDQW structures with a narrow right well (wR⭐4 nm) are not suitable for observation of the resonant

scattering in transport processes.

IV. CONCLUSIONS

We have presented a calculation of the 1s and 2 p0states

of a shallow donor in a coupled double-quantum-well struc-ture. The variational approach with intersubband mixing is used to calculate the 1s state and localized part of the 2 p0

state. The resonance of the 2 p0 state is considered using the

Green function technique. The results show that the 1s state has a maximum binding energy as the donor is around the center of the wider well of the CDQW structure, while the 2 p₀ state generally has a maximum binding energy as the donor is around the center of the narrower well. The resonant coupling of the 2 p0 state with the lowest subband is strong

when the 2 p0 energy level is close to the lowest subband

bottom. The resonance can give a lifetime of ⬃0.1 ps. By modulation of the interwell coupling, the resonant nature of FIG. 5. The 2 p0-1s transition energy and the 2 p0⫺E1,k⫽0

en-ergy difference vs the right well thickness wR for the impurity

lo-cating at the centers of wells. The CDQW structures have a left well of thickness wL⫽10 nm and barriers made of Al0.2Ga0.8As in 共a兲

and Al0.3Ga0.7As in共b兲. The interwell thickness of the structures is

the 2 p0state can change to the bound nature. The resonance

in general causes a negligibly small blueshift of the 2 p0

energy. However, when the 2 p0 energy level is close to the

lowest subband bottom, it may causes a redshift of the order of 1 meV. The 2 p0-1s transition energy is also presented as

a function of the narrower well thickness. The results are

useful in designing a CDQW structure for experimentally observing the resonance effects of the 2 p0 state.

ACKNOWLEDGMENT

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

*Electronic address: [email protected]

1

G. Bastard, Phys. Rev. B 24, 4714共1981兲.

2_{C. Mailhiot, Y.-C. Chang, and T. C. McGill, Phys. Rev. B 26,}

4449共1982兲.

3_{R. L. Greene and K. K. Bajaj, Solid State Commun. 45, 825} 共1983兲.

4_{S. Chaudhuri and K. K. Bajaj, Phys. Rev. B 29, 1803共1984兲.} 5_{R. L. Greene and K. K. Bajaj, Phys. Rev. B 31, 4006共1985兲.} 6_{R. L. Greene and K. K. Bajaj, Phys. Rev. B 31, 913共1985兲.} 7_{S. Chaudhuri, Phys. Rev. B 28, 4480共1983兲.}

8_{J.-B. Xia, Phys. Rev. B 39, 5386共1989兲.}

9_{S. Fraizzoli, F. Bassani, and R. Buczko, Phys. Rev. B 41, 5096} 共1990兲.

10_{C. Priester, G. Allan, and M. Lannoo, Phys. Rev. B 29, 3408} 共1984兲.

11_{T. A. Perry, R. Merlin, B. V. Shanabrook, and J. Comas, Phys.}

Rev. Lett. 54, 2623共1985兲.

12_{K. Jayakumar, S. Balasubramanian, and M. Tomak, Phys. Rev. B} 34, 8794共1986兲.

13_{A. Blom, M. A. Odnoblyudov, I. N. Yassievich, and K. A. Chao,}

Phys. Rev. B 65, 155302共2002兲.

14

S. T. Yen, Phys. Rev. B 66, 075340共2002兲.

15_{H. Chen and S. Zhou, Phys. Rev. B 36, 9581共1987兲.}

16_{R. Ranganathan, B. D. McCombe, N. Nguyen, Y. Zhang, M. L.}

Rustgi, and W. J. Schaff, Phys. Rev. B 44, 1423共1991兲.

17_{N. Nguyen, R. Ranganathan, B. D. McCombe, and M. L. Rustgi,}

Phys. Rev. B 44, 3344共1991兲.

18_{J. Cen and K. K. Bajaj, Phys. Rev. B 46, 15280共1992兲.} 19_{J. Cen, S. M. Lee, and K. K. Bajaj, J. Appl. Phys. 73, 2848}

共1993兲.

20_{J.-P. Cheng and B. D. McCombe, Phys. Rev. B 42, 7626共1990兲.} 21_{R. Chen, J.-P. Cheng, D. L. Lin, B. D. McCombe, and T. F.}

George, Phys. Rev. B 44, 8315共1991兲.

22_{J. X. Zang and M. L. Rustgi, Phys. Rev. B 48, 2465共1993兲.} 23_{U. Fano, Nuovo Cimento 12, 154共1935兲; Phys. Rev. 124, 1866}

共1961兲.

24_{F. Bassani, G. Ladonisi, and B. Preziosi, Rep. Prog. Phys. 37,}

1099共1974兲.

25_{M. A. Odnoblyudov, I. N. Yassievich, M. S. Kagan, Yu. M.}