Population inversion in a p-doped quantum well with reduced photon energy

Population inversion in a p-doped quantum well with reduced photon energy

Chi-Ken Lu,1 Hsin-Fei Meng,1,*and Pin Han2

1_{Institute of Physics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China} 2_{Institute of Precision Engineering, National Chung Hsing University, Taichung, Taiwan, Republic of China}

共Received 15 February 2005; revised manuscript received 4 May 2006; published 21 July 2006兲

We study a multilayer silicon-germanium quantum well structure doped with acceptor impurities for resonant-state lasers capable of emitting photons of energy below 4 meV共1 THz兲. Unlike previous proposals on terahertz lasers in doped silicon-germanium quantum wells, the emitted photon energy does not need to exceed the acceptor binding energy, which is tens of meV. The energy constraint is relaxed by placing the acceptor impurity levels and the quantum well subband continuum in separate layers of different germanium compositions. We calculate the nonequilibrium behaviors of the holes in detail and demonstrate that population inversion between strain-split impurity levels can be built for sufficiently high acceptor densities under the application of a moderate dc electric field at about ten kelvins.

DOI:10.1103/PhysRevB.74.035328 PACS number共s兲: 72.10.⫺d, 42.55.⫺f, 78.45.⫹h

I. INTRODUCTION

Electromagnetic radiation in the 1 – 10-THz range, corre-sponding to wavelengths of 30– 300␮m, has important ap-plications in optical communication, medical diagnosis, and radio astronomy. There has been significant progress in the realization of THz radiation sources.1 _{For example,} broad-band THz radiation with output power up to 20 W is gener-ated from subpicosecond electron bunches in an accelerator. It is applied to image the distribution of a specific protein or water in tissue and buried layers in semiconductors.2Despite the high power output this approach is not very convenient for compact applications. Owing to rapid advances in semi-conductor technology, many earlier theoretical ideas based on solid-state THz sources can now be implemented. For instance, the Bloch oscillation共BO兲 is not possible for bulk crystals because of scatterings. The structure of superlattices makes it possible for BO to take place due to a shrinking zone boundary. By precise control of the period of the super-lattice, emission at 1.7 THz was observed in a GaAs/ AlGaAs superlattice due to BO.3_{However, BO has not been} proved to be viable for a THz laser. In addition, there is growing interest in quantum cascade lasers 共QCL’s兲 where THz radiation results from intersubband electrolumin-escence.4 QCL’s have been demonstrated to emit cw THz radiation at liquid-nitrogen temperature in GaAs/ AlGaAs.5 However, to date there is no QCL able to emit radiation below 2.9 THz.6

One promising way to realize semiconductor THz lasers is resonant-state lasers7,8 共RSL’s兲 based on doped quantum well 共QW兲 structures,9,10 _{whose operation involves} strain-induced resonant states and pumping by an electric field. A THz transition between higher and lower acceptor states has been observed.9,10 _{In RSL’s with one single QW, the two} degenerate valence bands are split by symmetry-lowering ex-ternal strain caused by exex-ternal pressure or lattice mismatch. The splitting also removes the degeneracy of the hydrogen-like acceptor localized states, and therefore two localized states are formed with energy levels attached to each split band. As the strain is so strong that the energy splitting ex-ceeds the binding energy of the acceptor, one of the two

localized states becomes resonant with the band to which the other localized state is attached. The coincidence in energy leads to resonant scattering between the continuous and lo-calized states. A population inversion between the two local-ized states can be achieved by resonant capture of the holes under an electric field.

In the previous approach to RSL’s7–10 _{there is a serious} constraint on the emitted photon energy. In single QW’s the acceptor level splitting needs to be greater than the impurity binding energy in order to have a resonant state. The photon energy therefore must be larger than the binding energy, which is several tens of meV关15 meV for Ge and 50 meV for Si共Ref.11兲兴 corresponding to more than 10 THz. In this paper we present a concept of silicon-germanium QW RSL’s which is free of such a constraint. Instead of one single QW, in our structure the continuous and localized states are in different layers and the resonance can be controlled by inde-pendent strains in different layers. Therefore resonant scat-tering can occur even if the energy splitting is smaller than the acceptor binding energy. Silicon-germanium alloy is cho-sen as the material system for this concept because of its low absorption in the THz range and easy integration with Si electronics. We calculate the energies of the localized accep-tor levels and continuous subband levels共indicated as “con-tinuum” below兲 and give the relation between the emitted photon energy and the structure parameters. In order to show that population inversion can be realized under practical ex-perimental conditions, we construct a comprehensive theo-retical model for the nonequilibrium behaviors of holes in the QW structure and study in detail the the dynamical be-haviors of the holes with an external pumping field. Our results indicate that emission as low as 1 THz can be ob-tained in the QW structure with reasonable germanium com-positions under a dc electric field of about 100 V / cm at 10 K.

The paper is organized as follows. Section II introduces the QW structure featuring the flexible control of lasing fre-quency by germanium compositions. In particular we show how the QW structure is able to emit radiation of frequency at 1 THz. In Sec. III the wave functions and resonant transi-tion between localized states and subband continuum are cal-culated. In Sec. IV the Boltzmann equation and rate equation

are employed to study the population inversion. In Sec. V we present the final results on the conditions for the formation of a population inversion in the QW structure. Section VI draws the conclusion.

II. QUANTUM WELL STRUCTURE AND PUMPING MECHANISM

The profile of valence band edge diagram along crystal growth direction共z direction兲 of the proposed QW structure is shown in Fig. 1. For clarity the sign of the energy is reversed. The splitting of heavy-hole and light-hole band edges is due to strain caused by lattice mismatch between SiGe alloy and Si. The strain can be linearly related to the germanium compositions in the alloy. The two Si1−xGex lay-ers sandwiching the central Si1−yGey layer have identical profiles and are␦ doped with identical acceptor density na. The profile has been designed to be symmetric for simpler theoretical treatments.

As can be seen in the profile of the heavy-hole band edge in Fig. 1, holes are confined in the central layer in the z direction due to potential barriers constituted by the two identical Si_1−xGexlayers at two sides. Series of subbands are formed due to the confinement. We label the energy mini-mum of the first heavy-hole subband共HH1兲, which is a func-tion of the well width W, by the dash-dotted line in the cen-tral layer. In addition there is a series of localized acceptor levels attached to the heavy-hole band edge in each␦-doped layer. We focus on the low-lying heavy-hole 2p±1 level

共HH2P±兲, labeled by a short dashed line, and the light-hole acceptor 1s level共LH1S兲, labeled by a short solid line. LH1S and HH2P± have opposite parity and hence are expected to give the strongest intensity of radiation among all possible transitions. Besides the acceptor 1s level attached to the heavy-hole band edge is the very lowest state for holes in the system and is labeled by HH1S, which is shown below HH2P± in Fig.1. With precise control of germanium

com-positions x and y in the QW structure, the heavy-hole and light-hole band edges as well as the localized acceptor levels can be adjusted to have the relative energies required for THz lasers.

Now we consider the pumping mechanism of the holes under an external electric field F 共strength F兲 perpendicular to the z direction—say, the x direction. The physical picture is shown in Fig.2. HH2P± is below LH1S and the minimum of HH1 by␦and E0, respectively. Note that in our problem␦

must be larger than E0to have resonance between HH1 and

LH1S. At zero temperature all the holes stay in the low-lying HH1S state without field. When the external electric field is applied, some holes on HH1S are initially field ionized. Then more holes are excited to HH1 through impact ionization and acquire more kinetic energy until the occurrence of phonon scattering. The processes of field ionization and pumping of holes are denoted by process 1 and 2, respectively, in Fig.2. Another channel for slowing down the holes in HH1 is the resonance capture by LH1S. The transition between heavy-hole and light-heavy-hole states, denoted by process 3, is facilitated by the off-diagonal matrix element8 _{of the Luttinger-Kohn} Hamiltonian to be discussed below. As the occupation of higher LH1S grows with increasing external field and the lower HH1S and HH2P± are gradually depleted by impact ionization, a population inversion is expected. Emission of THz photons, indicated by process 4, will take place due to the radiative decay of holes from LH1S to HH2P±.

The resonance of the localized state and the continuum is achieved by raising the strain of the lattice so that the local-ized state is lifted to immerse within the continuum. In the previous works on QW RSL’s9,10 this acceptor impurity is doped in the same layer as the continuous states, so E0 is

simply the binding energy. Apparently in such a case the strain splitting ␦ must exceed the binding energy, corre-sponding to a lower bound of photon energy. In this work we

FIG. 1. The band edge profiles for light hole共LH, solid line兲 and heavy hole 共HH, dashed line兲 of the proposed QW structure are shown. Both x and y directions are perpendicular to the crystal growth direction z. x, and y are the germanium compositions for the well and barrier layers, respectively. W is the well width. The en-ergies of the strain-split acceptor levels LH1S, HH1S, and HH2P± relevant for the THz laser are also shown. d is the distance between the dopant and the boundary of the well. The HH1 minimum is indicated by the dash-dotted line. The inset defines the directions in the system.

FIG. 2. Schematic four-level operation of the THz laser. The operation involves three acceptor states LH1S, HH1S, and HH2P± as well as the HH1 continuum 共shaded region兲. The four major processes are also indicated. Process 1 indicates the field ionization of HH1S through the barrier to reach the HH1 minimum. The low-energy holes in HH1 are pumped toward resonant states by an elec-tric field in process 2. Process 3 represents resonant capture of continuum holes of energy Er to meta-stable LH1S. The radiative

decay by stimulated emission into the lower localized state HH2P± is denoted by process 4.

spatially separate the acceptor impurity and quantum well, so the relative energy between the localized state and con-tinuum has a much higher flexibility by adjusting the com-positions x and y and the well width W. As a result no matter how small␦is we can always adjust the QW structure such that E0⬍␦. However, the only lower bound for the emitted

photon energy is the energy shift of the resonant state caused by the perturbation of the continuum as discussed in Sec. III A. Hence our proposed structure is able to emit photons of energy less than the binding energy which is usually sev-eral tens of meV共12 THz in the case of Si兲 and is expected to fulfill the needs of solid-state optical sources of several THz or even sub-THz range. Because the relative energies of the localized and continuous states are crucial to the laser operation, below we calculate the quantitative relations be-tween the relevant levels in the QW structure and QW pa-rameters like the width W and germanium compositions. Even though the acceptor levels are outside the central well, there is no difficulty for the holes in the central well to be resonantly captured by the acceptor as long as there is an overlap between the wave functions of the acceptor levels and HH1. In order for the above picture to be valid, it is important to choose an intermediate value for the distance between the dopants and quantum well. The distance should be neither so large relative to the acceptor Bohr radius that there is no overlap between the acceptor level and quantum well level nor so small that the acceptor level itself becomes heavily influenced by the well.

III. LOCALIZED AND CONTINUOUS STATES A. Subband and impurity wave functions

In this subsection we calculate the wave functions and energies of the relevant states. We first consider a perfect crystal. The wave functions for the heavy-hole and light-hole bands can be represented by the eigenfunctions of the Luttinger-Kohn Hamiltonian12_H

LKin the Bloch function ba-sis 兵u_3/2, u_1/2, u_−1/2, u_−3/2其, which is the periodic sum of the atomic orbitals with total angular momentum quantum num-ber j =3₂. The subscripts stand for their z component jz of total angular momentum j. The column vector⌿ formed by the envelope functions 兵␸3/2共r兲,␸1/2共r兲,␸−1/2共r兲,␸−3/2共r兲其 is

the eigenfunction of HLK. The true wave function␺共r兲 of the state is given by ␺共r兲=⌺_␶␸_␶共r兲u_␶. The Luttinger-Kohn Hamiltonian can be written as

HLK= ប2 2m0

冢

aˆ₊ bˆ cˆ 0 bˆ† aˆ− 0 cˆ cˆ† 0 aˆ− − bˆ 0 cˆ† − bˆ† aˆ+

冣

jz= 3 2 jz= 1 2 jz= − 1 2 jz= − 3 2 , 共1兲

and the matrix elements are

aˆ+= − kˆz共␥1− 2␥兲kˆz−共␥1+␥兲共kˆx 2 + kˆy 2_{兲, 共2兲} aˆ₋= − kˆz共␥1+ 2␥兲kˆz−共␥1−␥兲共kˆx2+ kˆy2兲, 共3兲 bˆ =

冑

3共kˆx− ikˆy兲共␥kˆz+ kˆz␥兲, 共4兲 cˆ =

冑

3␥共kˆx− ikˆy兲2. 共5兲

m0is the free electron mass and kˆi= i_⳵x⳵_i, i = x , y , z, are opera-tors for the envelope functions.␥1,␥2, and␥3are

material-dependent Luttinger parameters, and ␥=共2␥₂+ 3␥₃兲/5. For crystals with translational invariance the envelope functions are all proportional to plane waves eik·rand the above opera-tors turn into c numbers. Diagonalization of the matrix gives the spectrum E±共k兲 which possesses fourfold degeneracy at

the band edge. The sign ⫾ indicates that there are two branches: the heavy-hole and light-hole bands. The spectrum

E±共k兲 is given by E±共k兲 = ប2 m0

冋

␥1 k2 2 ±

冑

␥2 2_k4_{+ 3共␥} 3 2_−␥ 2 2_兲共k x 2 ky 2 + ky 2 kz 2 + kz 2 kx 2_兲

册

. 共6兲 When the perfect crystal is subject to a stress due to ex-ternal strain or lattice mismatch the crystal symmetry is low-ered and the fourfold degeneracy at the valence band edge is split into two twofold degeneracies. If the strain is along the 关001兴 axis, which is parallel to the z direction, this effect is to add a strain term Vst to the Hamiltonian.13 It can be repre-sented by the diagonal matrix

Vst=

冢

␨ 0 0 0 0 −␨ 0 0 0 0 −␨ 0 0 0 0 ␨

冣

. 共7兲

The coincidence of heavy-hole and light-hole bands at the band edge is split by the strain factor␨which is proportional to external force and dependent on the direction of strain. In QW’s the strain results from the lattice mismatch between Si and SiGe alloy. In epitaxially grown SiGe QW structure on Si substrate, the lattice constant of the whole structure is fixed by the Si lattice constant. Because the natural lattice constant of SiGe alloy is different from Si, there must be a strain in the alloy to force the lattice constant to match Si. The relation between valence band splitting␨ due to strain and the germanium composition t in Si1−tGetalloy was stud-ied before.14 _{The expression in eV is ␨共t兲=0.01+0.2t} −1₄

冑

0.0016+ 0.0074t + 0.24t2_{. In our proposed QW structure}

the germanium compositions vary in the z direction and the hence the strain factor␨is a function of z.

For an acceptor in the stressed crystal we shall add the Coulomb potential VI due to the charged center:

VI共r兲 = vI共r兲I = 1 4␲⑀

e

rI, 共8兲

where⑀is the dielectric constant. r is the distance from the acceptor. I represents the 4⫻4 identity matrix. In the high-strain limit the off-diagonal coupling bˆ and cˆ can be consid-ered as perturbations and HLKbecomes approximately diag-onal with twofold degeneracy for heavy and light holes. The

resultant localized states can also be divided into two sub-groups like the band states.

After reviewing the bulk crystals we can extend the dis-cussions to the states in QW structures shown in Fig.1. Even without strain the valence band edge depends on the ger-manium compositions,15 _{described by V}

b共z兲=vb共z兲I. For Si_1−tGetalloy grown on Si, the valence band offset in eV can be written asvb= 0.84t. The total band edge profile in Fig.1 comes from the sum of Vb共z兲 and Vst共z兲. The Luttinger pa-rameters have different values in different layers; hence, they are functions of z. The homogeneity of those parameters is assumed within each silicon-germanium layer, and their val-ues are determined by linear interpolation between pure Si and pure Ge. The heavy-hole and light-hole subbands in the structure can be expressed by the total Hamiltonian H:

H = HLK+ Vb共z兲 + Vst共z兲. 共9兲 Note that z = 0 is at the center of well, so there is a parity symmetry with respect to z→−z in this problem. Here we separate HLKinto diagonal and off-diagonal parts, labeled by

HLK

0

and HLK

1

, respectively. The wave functions for HH1 emerge from eigenfunctions of the diagonal parts H0_{= H}

LK

0

+ Vb共z兲+Vst共z兲 of the full Hamiltonian H. The off-diagonal heavy-hole–light-hole mixing HLK

1

will be considered later as a perturbation. The unperturbed Schrödinger equation can be written as

H0⌿ =⑀⌿. 共10兲

We solve this to obtain the HH1 envelope functions⌿ of the wave functions␺kwith eigenvalues␧共k兲. On the other hand,

the localized states␾_1sLHand␾_2p

±

HH

with respective eigenvalues

E_1sLH and E_2p

±

HH

are eigenstates of the Hamiltonian HLK

0

+ VI共r兲 +关Vb共z兲+Vst共z兲兴z=z₀±. Here z0±⬅ ±

共

W

2+ d

兲

denote the positions

of the acceptors. The implicit assumption is that the Cou-lomb potential VIhas little effect on the subband wave func-tions while the nonuniform strain is irrelevant to the local-ized state. The above approximations are justified by the condition that the distance between the dopant and quantum well boundary d as well as the thickness of outer Si1−xGex layers be both larger than the acceptor Bohr radius. The equations turn out to be the typical one-dimensional potential well problem for the subband and hydrogen atom problem for the localized states. The energy spectra for the relevant states are shown in Fig. 3. The explicit wave functions for HH1 can be expressed as ␺k共␳,z兲 = 1

冑

Ag共z兲e ik·␳ជ_u ±3/2, 共11兲

where the⫾ sign in the wave functions reflects the twofold degeneracy guaranteed by time-reversal symmetry in the ab-sence of a magnetic field and the envelope function g共z兲 has even parity to yield the lowest energy of all subbands. A is the QW area.␳ជ=共x,y兲 is the in-plane coordinate. The accep-tor wave functions localized at z = z₀±and␳ជ= 0 are of the form

␾1s LH_共␳ជ ,z兲 =␸_1s关␳ជ,z − z₀±兴u_±1/2, 共12兲 ␾2p_± HH_共 ␳ជ,z兲 =␸2p_±关␳ជ,z − z0 ±_兴u ±3/2, 共13兲

where⫾ stands for z⬎0 and z⬍0, respectively. We use the hydrogenic trial functions

␸1s共␳ជ,z兲 = 1

冑

␲a2bexp

冉

−

冑

␳2 a2+ z2 b2

冊

, 共14兲 ␸2p_±共␳ជ,z兲 = 1 2␲a4b␳e i␾ exp

冉

−

冑

␳ 2 a2+ z2 b2

冊

. 共15兲 a共in-plane Bohr radius兲 and b 共out-of-plane Bohr radius兲 are

variation parameters for minimizing their energy, and ␾ is the polar angle in the x-y plane.␳ is the modulus兩␳ជ兩. Varia-tional calculations are performed to obtain the acceptor level splitting␦ and the difference E₀ between HH2P± and HH1 minima. Variational calculations are performed to obtain the acceptor level binding energy. The resultant binding energies and the variational Bohr radius of the levels of interest are shown in TableI.

Next we consider the corrections to the impurity states resulting from the QW confinement potential as well as the off-diagonal couplings with the HH1 continuum. Such cor-rections are necessary for having a more precise prediction on the emitted photon energy. Here we focus on the correc-tions to the binding energy of LH1S, which is resonant with the continuum. Note that the binding energy is relative to the barrier, not the quantum well continuum. The expressions for the corrections⌬E1sare given below, and the details of

deri-vation are presented in Appendix A:

FIG. 3. Spectrum of the diagonal part H0_{of the full Hamiltonian}

H. The acceptor states of interest and the continuous HH1 are shown. The binding energy for HH2P± and the emitted photon energy are denoted by E₀ and ␦, respectively. Their values and corresponding variational Bohr radius are shown in TableI.␧共k兲 is the spectrum of the subband HH1.

TABLE I. Binding energies and variational parameters of the localized acceptor states.

Level Binding energy共meV兲 Variational Bohr radius共nm兲

HH1S 20.9 a = 3.6, b = 3.0

HH2P± 4.9 a = 4.4, b = 3.9

⌬E1s=⌬ + P A 共2␲兲2

冕

dk 兩␣k兩2 E1s−␧k . 共16兲

⌬ is the correction due to the confinement potential while the integral due to the coupling with the continuum. P stands for the Cauchy principal-value integration.␣k and⌬ are given

by

␣k=具␸1s兩cˆ兩␺k典, 共17兲

⌬ = 具␸1s兩关vC共z兲 − vC共z0兲兴兩␸1s典. 共18兲

Note that only the off-diagonal elements involving kxand ky are considered because the resonance requires a large in-plane momentum. The confinement potentialvC is the diag-onal element of Vb+ Vst belonging to light-hole states. The correction due to the confinement potential⌬ is negligible in the present case because a very small portion of the impurity wave function for the impurity falls on the QW region and the confinement potential is small compared to the impurity binding energy. In fact our calculation shows that this cor-rection on E1sis less than 0.1%. However, this effect for the

case of smaller binding energy is important such as the shal-low donors located in the barrier near the quantum well.16 The second term resembles the formula for a second-order perturbation. Even though still only about 10% of E1s, it

provides significant corrections in case of a small emitted photon energy. The smallness of the corrections is reasonable since the light-hole localized states and the heavy-hole con-tinuum have a small overlap and they can couple to each other only though the off-diagonal elements of HLKwhich is treated as a perturbation in the high-strain limit.8 _{The QW} continuum and the HH2P± are assumed to be unaffected by the perturbation.

To be specific we consider the case which gives a radia-tion frequency of 1 THz共4 meV兲. We set␦= 6 meV in Fig. 2.␦determines the germanium composition x in the Si1−xGex layer in Fig.1. The corresponding x is 0.088. Such an ar-rangement gives an unperturbed binding energy of LH1S 21 meV by a variational calculation. This value is shifted to 23 meV after the correction in Eq. 共16兲 is put in. Conse-quently the real emitting photon energy is 4 meV as ex-pected. Next we determine the width W and germanium composition y of the central layer such that the HH1 mini-mum lies equally between LH1S and HH2P⫾. In other words E0is set to 2 meV in Fig.2. To meet the requirement

we choose the composition y by coinciding the HH band profile in the Si1−yGey layer 共dashed line in Fig. 1兲 with HH2P± in the Si1−xGexlayers. This gives y = 0.094. W is so determined such that the HH1 minimum共dash-dotted line in Fig.1兲 is 2 meV above due to the spatial quantization. By solving the potential well problem with the barrier height given by the valence band offset, the well width W is 11.7 nm. The energy levels relevant to the laser operation are shown as functions of germanium composition x in Si1−xGex layers for fixed well width W in Fig. 4. The valence band edges in Si layers are taken as zero. The parameters used in the calculation are summarized in TableII. The x dependence of the LH and HH band edges is due to the collective con-tributions from the x-dependent intrinsic band edge offset of

SiGe alloy and x-dependent strain. The LH1S and HH2P± acceptor levels are downward shifted from the band edges by the binding energies calculated by the variational trial wave functions in Eqs. 共14兲 and 共15兲. The HH1 minimum in the central well depends on x through the QW barrier height. The lasing operation is possible only when the HH1 mini-mum lies between HH2P± and LH1S, as indicated by the arrow in Fig.4. After having all the levels in the right order of energy, LH1S is immersed within the HH1 continuum as the resonant state.

FIG. 4. Unperturbed LH1S 共upper dotted line兲 and HH2P± 共lower dotted line兲 energy levels in Si1−xGexbarrier layers are

plot-ted versus germanium composition x. The solid line stands for the LH1S energy with the corrections in Eq.共16兲. The valence band

edge in outer most Si layer is taken as zero energy. The HH1 mini-mum is also shown as a dashed line and the difference from the perturbed LH1S is denoted by resonance energy E_r. The heavy-hole 共diamond兲 and light-hole 共circle兲 band edges in barrier layers are also plotted for references. The QW width W is 11.7 nm. y = 0.094. LH1S becomes a resonant state if the HH1 minimum lies between the two localized states HH2P± and LH1S. The arrow indicates that the resonance condition for the THz laser is satisfied in the region right to the vertical line.

TABLE II. Useful values in the calculation are listed with ref-erences. Luttinger parameters and optical phonon energy used here are obtained by interpolation with values between Si and Ge.

Parameter Value Description 共␥1,␥2,␥3兲 共4.22,0.39,1.44兲a/

共13.38,5.69,4.24兲b

Luttinger parameters in Si/ Ge ប␻0 53/ 37c Optical phonon energy

for Si/ Ge共meV兲 ␯A 5b Average optical phonon

emission rate共1012_s−1_兲

2.328c Mass density共g/cm3兲 c 9040d _{Sound velocity共m/s兲}

⌶ 9d Deformational potential共eV兲

a_Reference17. b_Reference7. c_Reference11. d_Reference18.

B. Resonant transition

The hybridization of the localized LH1S and HH1 con-tinuum via the off-diagonal perturbation HLK

1

leads to a new set of resonant states兵⌿E其 labeled by its complex energy E + i⌫E

2. The imaginary part is given by

⌫E 2 =␲

A

共2␲兲2

冕

dk␦共E − ␧k兲兩␣k兩2. 共19兲

The nonzero imaginary energy⌫Ehere represents that⌿Eis a quasistationary state. More precisely speaking the HH1 holes of momentum k can be captured by LH1S with the transition rate Wk

res

for a time interval ប/⌫. The transition rate and the time interval are determined in a self-consistent manner—that is, Wk res = 2 ប兩␣k兩2 ⌫/2 关␧共k兲 − E1s兴2+⌫2/4 , 共20兲 ⌫ ប=

兺

k Wk res . 共21兲

The center of the Lorentzian corresponds to E_1sbecause the resonant state⌿E_1scontains the maximum component of the localized LH1S. For simplicity we regard the unknown⌫ in Eq.共20兲 as close to zero and the Lorentzian is reduced to a␦ function. As long as the resultant ⌫ from Eq. 共21兲 is small compared to the resonance energy Er⬅E1s−␧共k=0兲 共see

Figs. 2 and 3兲 of LH1S, this method is self-consistent to obtain⌫.

Next we work out⌫ in the small-⌫ limit. In other words the resonant transition rate Wk

res

can be given simply by the Fermi golden rule

Wk

res =2␲

ប 兩␣k兩2␦关␧共k兲 − E1s兴. 共22兲

In order to obtain an explicit expression for the transition rate we need to calculate the overlap integral具␸1s兩␺k典.

As-suming that the main contribution to this integral comes from the region in the barrier—i.e.,兩z兩⬎W₂—we arrive at

具␸1s兩␺k典 = 1

冑

A 1

冑

␲a2_b

冕

dzg共z兲

冕

dxdye ikជ·␳ជ ⫻exp

冉

−

冑

␳ 2 a2+ z2 b2

冊

=Ꭽ

冑

16␲ab 2 A 1 ␩2

再

e −␩d/b 1 ␬b −␩

冋

␬b − 2␩ ␩共␬b −␩兲+ d b

册

+ e−␬d

冋

共2␩−␬b兲 ␩共␬b −␩兲2+ 共2␩+␬b兲 ␩共␬b +␩兲2

册

冎

. 共23兲

The dimensionless quantity␩⬅

冑

1 + a2k2is introduced.Ꭽ is to normalize the envelope function g共z兲 as 兰Ꭽ2_{兩g共z兲兩}2_{dz = 1.␬}

is the decay constant of g共z兲 in the barriers.

⌫ as a function of resonance energy Eris plotted in Fig.5 for various acceptor in-plane Bohr radii a and separations d. The effect of coupling with the continuum can be

investi-gated through⌫. For a distance d much larger than the Bohr radius, the coupling is diminished due to the decreasing over-lap between the impurity state and the continuum. In such a case the formation of a resonant state is impossible. How-ever, the dependence of the coupling on the Bohr radius is determined by two competing factors. Namely, in the z di-rection the envelope function g共z兲 of the continuum has a larger overlap with the localized impurity state of larger Bohr radius, while in the x-y plane the continuum of higher kinetic energy can only be coupled to the impurity state of smaller Bohr radius because such a localized state has larger Fourier momentum components. For larger Bohr radius, it is shown in Fig. 5 that ⌫ is larger at lower Er while it is smaller at higher Er.

IV. HOLE DISTRIBUTION AND POPULATION INVERSION

So far there is no comprehensive theoretical model for the nonequilibrium behavior of acceptor levels interacting with a subband in QW’s. In order to make quantitative predictions of the conditions for hole population inversion, below we construct a model which takes into account all of the relevant physical processes for such a system.

A. Hole statistics at equilibrium

The occupation probabilities of LH1S, HH2P⫾, HH1S, and HH1 states are indicated by f₁, f₂, fg, and fk,

respec-tively. In thermal equilibrium the occupation ratio of LH1S to HH2P± is given by the Boltzmann factor—i.e., f2/ f1

= exp共−␤␦兲.␤is the inverse of the product of the Boltzmann constant kB and temperature T. Moreover, at equilibrium the hole densities are determined by assigning each level its Boltzmann weighting. Note that all the holes are provided by the lowest localized level, and hence we have the following normalization of total holes:

FIG. 5. The energy width⌫ of the resonant state is shown as a function of the resonant-state energy E_rmeasured from the HH1 minimum. Symbol curves correspond to various in-plane Bohr ra-dius a of LH1S orbital with fixed d = 6 nm. Dashed and dotted lines correspond to various distances d between the acceptor and bound-ary of the central well with a = 2.7 nm.

nafg+ naf1+

1

A

兺

k

fk+ naf2= na. 共24兲 When the electric field is turned on holes acquire kinetic energy from the external field and the distribution of holes deviates from the Boltzmann distribution. In order to give a quantitative account of how the nonequilibrium populations depend on the parameters共e.g., field strength F, temperature

T, and acceptor density na兲, we need to study the micro-scopic kinetics governing the transitions among the states.

B. Boltzmann kinematic equation

The strategy for obtaining the nonequilibrium populations is as follows. First we neglect the low-lying HH2P± and HH1S temporarily and solve the kinetics of the subsystem containing HH1 and LH1S in order to obtain the relation between f2and fk, with considerations of phonon scattering

within HH1 and the resonant transition between the continu-ous HH1 and the localized LH1S. This is justified because the resonant scattering is much faster than the decay through spontaneous emission from LH1S to HH2P⫾.7 _Afterwards the occupation probability f1of HH2P± is determined by the

its balance with the nonequilibrium subband distribution fk

through impact ionization, thermal recombination, and their inverse processes Auger recombination and thermal excita-tion. Detailed calculations are given below.

For a given number of holes in the subsystem containing HH1 and LH1S, the nonequilibrium distribution fkin HH1

and occupation of LH1S f₂are studied by solving the Bolt-zmann kinetic equation numerically for various electric fields and acceptor densities. In the subsystem the holes in HH1 acquire kinetic energy from the constant electric field F ap-plied along the x axis. For moderate electric field and low temperature, it is adequate to adopt the concept of streaming

motion19 _{in which the only significant scattering is due to} optical phonons共energy ប␻0兲. This is implemented by

intro-ducing a particle drain in momentum space such that once a specific hole drifts with velocity eF /ប through the energy surface␧=ប␻0共denoted by ⌸兲 in the momentum space, the

hole will experience an optical phonon scattering and simul-taneously reemerge as a hole of energy less than⑀0.7,8Hence fk= 0 for ␧共k兲艌ប␻0. The energy ⑀0 is determined by the

requirement that in the presence of constant electric field F the probability for a hole being able to drift beyond the con-stant energy surface␧=ប␻0+⑀0without emitting one optical

phonon be negligibly small. The quantity⑀0 is equal to the

product of the external force, eF, carrier velocity

冑

2m*ប␻0/ m*, and inverse of the average optical phonon

emitting rate,␯A. m*stands for the effective mass. Note that the energy-independent optical-phonon-emitting rate is due to the constant density of states in two dimensions. Therefore the excess energy can be expressed as

⑀0= eF

␯A

冑

2ប␻0

m* . 共25兲

The reemerging holes can be modeled as a particle source7,8

S共k,t兲 = e

ប

冋

冕

_⌸fk共t兲F · dS

册

冋

冕

⌰„⑀0−␧共k

⬘

兲…d2k

⬘

册

⌰„⑀0−␧共k兲…, 共26兲

where ⌰ is the step function. The meaning of the above expression is that the hole reemerging rate is uniform for energy within⑀0and the total reemergence rate must match

the collection of the outward carrier flux e_បFfk passing

through the surface⌸ in momentum space.

In order to properly account for the temperature effects, we include the acoustic phonon scattering. The acoustic pho-non scattering rate W_k,kacu_⬘is of the form20

W_k,kacu_⬘=2␲⌶ 2_q2
_␻qWA

冉

nq+ 1 2⫿ 1 2

冊

␦„␧共k

⬘

兲 − ␧共k兲 ⫿ ប␻q…, 共27兲 where
is the mass density of solid lattice and ⌶ is the lattice deformation potential. The acoustic phonon involved in the transition has wave number q = k

⬘

− k and its disper-sion is given by␻q= cq where c is the sound velocity in the

solid. Emission and absorption of phonons in the processes correspond to⫹ and ⫺, respectively. The product WA rep-resents the QW volume.

We assume homogeneity in the x and y directions so that the distribution is a function of the variables kxand kyonly. The set of kinetic equations can be written as

⳵fk ⳵t + eF ប · ⳵fk ⳵k = Sk− Dk+ C1关fk, f2兴, 共28兲 ⳵f₂ ⳵t = C2关fk, f2兴. 共29兲 Ci关fk, f2兴, i=1,2, represent the collision terms for the

acous-tic phonon and resonant scattering. They are functionals of the distribution functions. The explicit expressions for the collision terms are

C1关fk, f2兴 = naA兵Wk res_共f 2− fk兲其 +

兺

k⬘ 兵Wk⬘k acu fk⬘− Wkk⬘ acu fk其, 共30兲 C2关fk, f2兴 =

兺

k Wk res_共f k− f2兲. 共31兲

The kinetic equations 共28兲 and 共29兲 are solved numerically by starting with the equilibrium distribution and then inte-grating forward in time until a steady state is reached. Note that the sum of densities, naf2+

1

A兺kfk, is a conserved

quan-tity in the time evolution, guaranteed by cancellation of col-lision terms and the boundary conditions at the surface⌸. In this way not only the steady state but also the transient of the system can be modeled. The occupations of LH1S f₂and the HH1 fkare obtained up to an arbitrary total number of holes

in the subsystem. In particular the relation between f2and fk

at steady state can be readily seen by setting the left-hand side of Eq.共29兲 equal to zero:

f2=

兺

_k Wk res fk

兺

_k Wk res =

冕

d␧␦关␧共k兲 − Er兴fk. 共32兲 Now we consider the special case with no electric field. The subsystem is in thermal equilibrium. The occupations of HH1 and LH1S obey the Boltzmann statistics guaranteed by the presence of a ␦ function in the expression for resonant scattering as well as the fact that the scattering between HH1 states k and k

⬘

due to acoustic phonon emission and absorp-tion satisfies the relaabsorp-tions

W_kacu_⬘k Wkk⬘ acu = 1 + nq nq = exp兵−␤关␧共k

⬘

兲 − ␧共k兲兴其. 共33兲 ␧共k

⬘

兲⬎␧共k兲 is assumed without loss of generality, and q is the wave vector of the phonon involved in the process. Therefore in equilibrium f₂ is given by

f2= N/A 1 A

兺

k e−␤␧共k兲+ nae−␤Er e−␤Er_, 共34兲

where N represents the total number of holes in the sub-system.

In order to describe the effect of the electric field on the distribution, we define a dimensionless parameter␭共F,T兲 by

␭共F,T兲 ⬅ 1 A

兺

k fk naf2+ 1 A

兺

k fk = ns n2+ ns . 共35兲

␭共F,T兲 is the fraction of holes in HH1 for the subsystem. For low temperature at equilibrium virtually all holes stay near the HH1 minimum, so␭ is close to unity. In the presence of the electric field the population of LH1S increases as a con-sequence of Eq. 共32兲, since holes in HH1 acquire kinetic energy from the field, so the nonequilibrium distribution fk

has a larger value at ␧共k兲=Er. Therefore, for a given na, ␭共F,T兲 is expected to decrease as the electric field increases. An increase of the acceptor density naalso raises f2because

the distribution in HH1 becomes more concentrated on ␧共k兲艋Er. This is because the stronger resonance scattering inhibits the holes from acquiring energy higher than the reso-nance energy Er.

C. Impact ionization and thermal recombination rates

Next we turn to the interactions between HH1 and low-lying localized states including HH1S and HH2P±. The in-teractions are dominated by impact ionization and the ther-mal recombination as well as their inverse processes. In the impact ionization process one energetic hole in HH1 with momentum k scatters with one hole in the low-lying local-ized states␾bin the barrier through the Coulomb interaction such that they both come out as free holes in HH1. The transition rate is given by

wip共k兲 =2␲ ប k

兺

₁,k2

冏

具k1,k2兩 e2 r兩k,b典

冏

2 ⫻␦„␧共k兲 − Eb−␧共k1兲 − ␧共k2兲…, 共36兲

where r is the separation between the incident hole and lo-calized hole. The summation is over all final two-particle Bloch states共k1, k2兲. The first task is to evaluate the

scatter-ing matrix element

具

k₁, k₂

兩

e_r2

兩

k , b

典

. Substituting the explicit expressions for those localized wave functions and Coulomb potential into the scattering matrix element, it becomes

冕

d3_r 1d3r2 1

冑

Ae −ik1·␳ជ1_f*共z 1兲 1

冑

Ae −ik2·␳ជ2_f*共z 2兲 ⫻V共兩r1− r2兩兲 1

冑

Ae ik₁·␳ជ₁ f共z1兲␾b共r2兲, 共37兲

where the dummy coordinates ri=共␳ជi, zi兲, i=1,2, are to be integrated out to obtain an impact ionization rate as a func-tion of the momentum k of the incident hole. The integral is complicated by the entanglement of the dummy variables r₁ and r2 but it can be eased by replacing the Coulomb

inter-action with its representation in Fourier expansions 1 4␲⑀ e2 兩r1− r2兩 =e 2 ⑀

冕

d3_q 共2␲兲3 1 q2e iq·r1_e−iq·r2_. 共38兲

Similar to Eq.共24兲, the overlap between the HH1 and LH1S, the major contributions to the matrix element come from 兩z兩⬎W

2. After some algebra the scattering amplitude M

ar-rives at the expression

M共k;k1,k2兲 = 具k1,k2兩 e2 r兩k,b典 = 1 A3/2 e2 ⑀

冕

dq_⬜ 2␲ 1 q储2+ q⬜2

冕

dz₁兩f共z₁兲兩2_eiq_⬜z₁ ⫻

冕

dz2f*共z2兲I共z2,q

⬘

兲e−iq⬜z2. 共39兲 q储=兩k−k1兩, and the expression for I共z兲 is given by

I共z,q

⬘

兲 =

冑

4␲a 2 b

冕

0 ⬁ d␳h共␳兲exp

冋

−

冉

␳2+ 共z − z0 ±_兲2 b2

冊

1/2

册

, 共40兲 where the function h共␳兲 is␳J0共aq

⬘

␳兲 for the case of HH1S as

initial state and is

冑

1₂␳2_J

1共aq

⬘

␳兲 for the case of HH2P± as

initial state. J stands for the Bessel functions. q

⬘

=兩k−k1

− k2兩 and␹=

冑

1 + a2q

⬘

2. The upper script⫾ is for z⬎0 and z⬍0, respectively. Note that I共z,q

⬘

兲 decreases with the

mo-mentum transfer q

⬘

as a consequence of localization of the initial acceptor state. The scattering amplitude is expected to decrease rapidly when the momentum transfer q

⬘

is larger than the inverse of the Bohr radius a of the localized orbital. Hence we simplify the expression, Eq.共36兲, as

wip_{共k兲 =}2␲ ប 兩M¯ 兩2k

兺

₁,k2 ⌰

冉

1 a −兩k − k1− k2兩

冊

⫻␦„␧共k兲 − E0−␧共k1兲 − ␧共k2兲… =2␲ ប 兩M¯ 兩2␴共k兲. 共41兲

M¯ stands for the maximum scattering amplitude which

oc-curs when k1= k2, and the angle between k and k1is equal to

that between k and k₂. The summation in the above expres-sion gives the effective phase-space volume␴共k兲 available for this scattering process given that the incident momentum is k. Carrying out k1and k2 integral one obtains

␴共k兲 = A2

冕

d 2_k 1 共2␲兲2 d2k2 共2␲兲2⌰

冉

1 a−兩k − k1− k2兩

冊

⫻␦„␧共k兲 − E0−␧共k1兲 − ␧共k2兲… =

冉

A 共2␲兲2

冊

2

冕

d2u 2 d 2_v⌰

冉

1 a− u

冊

⫻␦

再

ប2 2m 1 2共兩u + k兩 2_+v2_{兲 − 关␧共k兲 − E} 0兴

冎

= A 2 2␲ m ប2

冕

d2u⌰

冉

1 a− u

冊

⫻⌰

冋

− ប 2 4m兩u + k兩 2_{+共␧共k兲 − E} 0兲

册

, 共42兲

where the phase-space dummy variables共k₁, k₂兲 were trans-formed into the new coordinates 共u,v兲=共k₁+ k₂, k₁− k₂兲 with the corresponding Jacobian equal to one-half. After in-tegrating out the variable v the evaluation of ␴共k兲 can be obtained through counting the overlapping area of one circle centered at the origin with radius 1 / a and another circle centered at −k on the x axis with radius 冑4m关␧共k兲−E_ប 0兴. The resultant rate wip_{is plotted in Fig.6as a function of kinetic}

energy ប_2m2k2*. The reverse process of impact ionization is Au-ger recombination, in which two HH1 holes collide and re-sult in one localized hole and one HH1 hole with higher kinetic energy. The Auger process must be taken into account as well.

The holes impact-ionized to the HH1 can go back to the low-lying localized states by acoustic phonon emission—i.e., thermal recombination. The thermal recombination rate is given by21 wtr共k兲 = 210␲c l₀ E₀4m*_c2 关␧共k兲 + E0兴5 a3兩g共z₀±兲兩2共Nq+ 1兲, 共43兲

where c is the sound velocity. Nqis the number of phonons

involved in the scattering, and q is the wave vector of the phonon satisfying conservation of energy given by q =关␧共k兲+E0兴/បc. l0 is the characteristic length for acoustic

phonon scattering,

l₀= ␲ប

4

2m*3_⌶2, 共44兲

where
and⌶ are mass density of the lattice and deforma-tion potential as mendeforma-tioned previously. The reverse process of thermal recombination is the thermal excitation of holes in the low-lying localized states by acoustic phonon absorption. Between the two localized levels HH1S and HH2P±, the thermal capture/generation rates ta/e are given by

ta/e= 210 c l0 mc2 ⌬⑀

冉

Nq+ 1 2⫿ 1 2

冊

. 共45兲

⌬⑀denotes the energy difference between the localized lev-els. The subscripts a and e indicate that these processes are accompanied by phonon absorption and phonon emission, respectively.

D. Hole population in subband and lower localized acceptor states

From Sec. IV B we are able to deal with the nonequilib-rium occupations fkand f2with normalization up to an

arbi-trary total number of holes. Using the impact ionization and phonon emission rates we are now able to deal with the occupations in the subsystem consisting of lower localized levels and HH1. To be precise, we adopt the normalization given by Eq.共24兲 where the total hole density of subsystem consisting of LH1S and HH1 is equal to the vacancy density in HH2P± and HH1S. Since the occupation probability f2is

completely determined from fk, it is convenient to write the

density of HH1 holes nsas

ns=

ns

ns+ naf2

共ns+ naf2兲 = ␭共F,T兲na共1 − f1− fg兲 共46兲 and the hole density of LH1S as

n2=关1 − ␭共F,T兲兴na共1 − f1− fg兲. 共47兲 The dimensionless parameter␭共F,T兲, given by Eq. 共35兲, has values between zero and unity.

Once f1and fgare known, f2can be determined from Eq.

共47兲. f₁ and fg can be calculated from the kinetics between

FIG. 6. Impact ionization rates wip_{as functions of kinetic energy}

␧ of the incident subband hole for HH1S to HH1 and HH2P± to HH1 are respectively shown.

HH1 and the low-lying localized states. Impact ionization and thermal excitation processes cause the upward transi-tions while Auger recombination and thermal recombination processes cause the downward transitions. The respective downward Auger recombination rates from HH1 to HH2P± and HH1S are r_2par= A_2p共T兲n_s2共1− f₁兲 and r_1sar= A_1s共T兲n_s2共1− fg兲, where the coefficients A’s are temperature and acceptor den-sity dependent for the Auger recombination and the factors 共1− f1,g兲 account for the constraint that the process is

forbid-den when the lower acceptor state is filled with a hole. Note that holes in HH1 are not required to have threshold kinetic energy for the recombination process to take place, so we assume the coefficients A’s have a negligible field depen-dence. The holes occupied the continuum can drop to the lower localized states, HH1S and HH2P±, by thermal recom-bination. In our case HH2P± is below the HH1 minimum by 2 meV, which is much smaller than the gap between HH1 and HH1S, 16 meV; here we neglect the latter recombination process since the rate is inversely proportional to the gap. This downward rate from HH1 to HH2P± is proportional to the hole density in HH1 and can be written as r_2ptr = C共F,T兲ns. The coefficient C共F,T兲, dependent on field and temperature, is taken as the average of Eq.共43兲 with respect to fk:

C共F,T兲 =

兺

_k wtr„␧共k兲…fk

兺

_k fk

. 共48兲

We now consider upward transitions. The impact ionization rates for the respective processes, HH1S to HH1 and HH2P± to HH1, are of the expressions r_1sip= B共F,T兲nsfg and r2p

ip = B共F,T兲nsf1. Note that the factors f1 and fg in the expres-sions account for the requirement of an occupied initial lo-calized acceptor state. The coefficients Bi共F,T兲 can be writ-ten as the average

Bi共F,T兲 =

兺

_k wi ip „␧共k兲…fk

兺

_k fk . 共49兲 The subscript of wi ip

in Eq. 共49兲 stands for different rates resulting from different initial localized states in the different collision processes in the present case. There exists a thresh-old of kinetic energy for the hole in HH1 for impact ioniza-tion, and consequently the coefficient B for low field and low temperature is negligibly small. Besides the upward transi-tion caused by the inelastic collision, holes occupying the lower localized states can also be excited to the continuum through phonon emission. Here we also neglect the direct excitation of HH1S holes to HH1 because it requires absorp-tion of phonons of much greater energy. Therefore we are left with the thermal excitation from HH2P± to HH1, and the rate can be expressed as rte_{= D共T兲n}

1. The phonon absorption

coefficient D共T兲 is determined by detailed balance with rtr_at thermal equilibrium.

Since we have to consider two lower localized states in the kinetic problem, we are left with the transition between

HH1S and HH2P±. For simplicity we only consider the ther-mal excitation and recombination. The upward and down-ward transitions among the two levels are given by tangand

ten1. With all the necessary transitions at hand we are ready

to write down the kinetic equations for the populations n1

and ng of the two localized states. Substituting all the for-mula into the relation we have

dn1 dt − tang+ ten1= r2p ar − r_2pip+ rtr− rte = A2pns 2_{共1 − f} 1兲 − B2pnsf1+ Cns− Dn1, dng dt + tang− ten1= r1s ar − r_1sip= A1sns 2_{共1 − f} g兲 − B1snsfg. 共50兲 Now we are left with the determination of the coefficients

A1s, A2p, and D which are assumed to be independent of the

electric field. Since the occupations obtained from the rate equation must be restored to thermal equilibrium when the electric field is set to zero, the requirement of detailed bal-ance at zero field gives A1s, A2p, and D using B1s, B2p, and C:

A_2p共T兲共ns0兲2共1 − f01兲 = B2p共F = 0,T兲ns0f10, A1s共T兲共ns0兲2共1 − f0g兲 = B1s共F = 0,T兲ns0fg0,

D共T兲n₁0= C共F = 0,T兲ns

0

. 共51兲

Note that the zeros as superscripts in fg, f1, n1, and nsstand for the equilibrium values.

V. RESULTS AND DISCUSSIONS

For a given F and T, Eq.共50兲 can be solved to give fgand

f1. Then they can be substituted into Eq. 共47兲 to give f2. f2/ f1⬎1 is the condition for population inversion. Putting

everything together we are now able to obtain the nonequi-librium distribution of holes in all levels under electric field pumping. For the subsystem containing LH1S and HH1, the normalized subband distribution f˜共␧兲⬅ f共␧兲 k2p

共2␲兲2_n

s versus

hole kinetic energy for different acceptor densities is shown in Fig.7 with applied electric field 1 kV/ cm. kp stands for the hole momentum corresponding to the kinetic energy of one optical phonon energyប␻0= 40 meV and the integration

兰兩k兩⬍kpf˜共␧兲d␧ gives unity. For lower acceptor densities na

holes in HH1 are more likely to be pumped to acquire energy exceeding the resonance energy Er. This results in a lower occupation below Er. Higher acceptor densities na lead to higher occupation probability at Er—i.e., larger f˜共Er兲. This phenomenon results from strong resonant scattering for higher acceptor densities. From Eq.共32兲 the occupation f₂of LH1S is consequently enhanced with increasing acceptor density. In other words, for the same hole density in HH1, higher acceptor densities nalead to higher LH1S occupation probabilities f₂. Therefore higher na is advantageous for building population inversion. The effect of electric fields is

shown in Fig.8 by plotting the subband hole fraction␭. At low field, the occupation of LH1S compared to that of HH1 is suppressed by the Boltzmann factor and␭ is near unity. As the field is turned on共between 10−2 and 10−1V / cm兲, holes acquire kinetic energy by field pumping. Hence more holes accumulate in LH1S through resonant capture of holes in HH1 with kinetic energy ␧共k兲=Er. As the field further in-creases, the fraction ␭ starts to increase because the field pumping overwhelms resonant capture and acoustic phonon scattering. In that case a large fraction of holes in HH1 ac-quire kinetic energy larger than Er. The temperature effect diminishes in this regime as shown by the coincidence of the two curves in Fig.8. Eventually the growth of␭ in the high-field regime saturates when optical phonon scattering sets in. Next we consider the subsystem consisting of HH1 and the lower localized states. At low temperature and

equilib-rium, most of the holes are bound by the acceptors and oc-cupy the lowest HH1S. There are very few holes in HH1 and even fewer holes with enough kinetic energy to inelastically collide with the localized holes. Therefore the process of impact ionization is negligible and the so is the Auger re-combination because in such a dilute case the average dis-tance between the free holes is so large that the probability of collision is extremely small. Hence the populations of these levels are dominated by thermal processes and the statistics obey the Boltzmann distribution. When the electric field is turned on, holes can acquire more kinetic energy and impact ionization of the low-lying localized state is possible through inelastic collisions with energetic holes. The subsequent dis-tribution of holes is balanced by those upward and down-ward transitions, as illustrated by Eq.共50兲. In order to have a quantitative understanding of how an electric field change the steady-state distribution of holes as the impact ionization rates increase, it is easier to consider the subsystem as HH1 and one single localized state, which is below HH1 mini-mum by eg. The rate equation can be written in a similar manner—that is,

A

˜ na␭2␰3+ B˜ ␭␰2+

冉

C˜ − B˜ +

D˜

␭

冊

␭␰− D = 0. 共52兲 The variable ␰= 1 − f˜ and f˜ stands for the population in the localized state. The capital letters with tildes represent the effective coefficients for the corresponding processes. Now we first focus on the limit of low temperature and low field. In such a case the occupation of lower localized levels is close to unity共␰Ⰶ1兲 and the impact ionization coefficient B˜ is nearly zero. So it is a good approximation to neglect the term of highest power in ␰ in the rate equation 共52兲. The solution is given by

␰=共B˜ − C˜兲␭ +

冑

共B˜ − C˜兲

2_␭2_{+ 4␭B˜D}˜

2␭B˜ , 共53兲

where the term for thermal excitation D˜ /␭ is dropped in the parentheses of Eq.共52兲 because the thermal excitation pro-cess is much weaker than the thermal recombination propro-cess

共

C˜ ⰇD˜_␭

兲

at low temperature. Note␭⯝1 at low field. In order to illustrate how an electric field affects the solution ␰ through the impact ionization coefficient B˜ , we set ␭=1 and define the relative coefficients for impact ionization, bc⬅

B ˜ C ˜, and thermal excitation, dc⬅

D ˜ C

˜, to the thermal recombination coefficient C. The solution can be rewritten as

␰=共bc− 1兲 +

冑

共bc− 1兲

2_{+ 4b}

cdc 2bc

. 共54兲

dc⯝e−␤eg is a temperature-dependent parameter in the ex-pression as suggested by Eq. 共51兲. For bc= 1, ␰ is

冑

dcⰆ1, justifying the omission of the ␰3 _{term in Eq. 共52兲 in the}

regime of discussion. The relation between␰and the relative impact ionization coefficient bcis plotted in Fig.9 for tem-perature from 1 K to 4 K. In the limit of small bcthe solu-tion can be approximated as␰= dcwhich is nothing but the

FIG. 7. Normalized subband hole distribution f˜共␧兲 versus hole energy␧ for electric field strength of F=1000 V/cm and T=1 K. The vertical line denotes the resonance energy Er. The distribution

is concentrated more in the␧⬍Erregion as the acceptor density na

increases.

FIG. 8. The subband hole fractions ␭共F,T兲 as a function of electric field F for T = 1 K共solid line兲 and T=4 K 共dashed line兲 are shown. The coincidence for different temperatures at higher electric fields suggests that k_BT becomes irrelevant compared with the scale of Erand optical phonon energyប␻0.

thermal equilibrium. Such a case corresponds to the low-field situation in which impact ionization is not yet activated. As the electric field increases, bcgrows towards unity because more holes in HH1 acquire enough kinetic energy from the field. In the crossover regime where the term共bc− 1兲 in Eq. 共54兲 turns positive from negative,␰grows rapidly as both the population and average kinetic energy of holes in HH1 in-crease. As bc gets larger and larger than 1 the solution ap-proaches 1 −_b1

c. In the crossover there is competition between

the two terms 共bc− 1兲2 and bcdc in the square root of Eq. 共54兲. Consequently the size of the crossover is determined by

冑

dc. Since the impact ionization parameter bc is strongly field dependent, this crossover corresponds to the variation of field␦F as

␦F⬃ e−␤eg/2

冉

⳵bc

⳵F

冊

−1

. 共55兲

This␦F characterizes how sensitive pumping is to electric

field. The dramatic jump of␰ at bc⯝1 is due to the domi-nance of upward impact ionization over the downward ther-mal recombination. The depletion of the lower localized lev-els when bc⬎1 is critical for the realization of the hole population inversion.

After combining the two subsystems, we are able to ob-tain the occupation of each level in the system. The occupa-tion probabilities fg, f1, and f2 for the strain-split acceptor

levels and the ratio f₂/ f₁ at 4 K are shown in Fig. 10. By definition a population inversion is established if f₂/ f₁⬎1. There is a threshold acceptor density na of about 10−3nm−2 when the applied field is 100 V / cm. The threshold acceptor density reflects the fact that the resonance scattering is nec-essary for building the population inversion. As naincreases further, it becomes harder for HH1 holes to acquire higher energy, which is shown in Fig.7, and this effect leads to a suppression of the impact ionization processes from the lower levels. Even though the upward transitions get sup-pressed due to more resonant scattering, the population f₂

remains at fixed values due to the increase of␭ with increas-ing na. However, this effect leads to the fact that the popu-lation ratio fg/ f1 is getting closer to its equilibrium value.

For T⬍4 K the result is the same because acoustic phonon scattering is irrelevant for low temperature and higher field. The behaviors of the system differ for low-temperature 共kBT⬍␦兲 and high-temperature 共kBT⬎␦兲 regimes. At low temperature共T⬍10 K兲 population inversion can be realized for only a moderate electric field共100 V/cm兲 because there is almost no acoustic phonon scattering and the hole distri-bution in HH1 can be easily distorted by the field. At high temperature, the distribution is stabilized by the strong acoustic phonon scattering. Therefore population inversion is impossible even for a stronger field.

In Fig.11the populations of localized levels versus field strength F for T = 10 K are shown. As the field is turning on and increasing toward 20 V / cm, holes in HH1 become more and more energetic. Consequently more and more free holes are generated due to the increase of the coefficients B1sand B2p. Note that presently the resulting upward transition is

mainly from HH1S to HH1 because the upper level HH2P± is empty and the population f1 mainly results from the

com-bined processes, impact ionization HH1S to HH1 plus the thermal recombination from HH1 to HH2P± is empty and the population f₁. As F continues to increase, the populations

f₁ and f₂grow significantly and the lowest HH1S begins to be depleted due to the fact that the intracenter recombination from HH2Pñ to HH1S is quite slow. Now the upward tran-sition is contributed more by HH2P± is empty and the popu-lation f₁ than HH1S. When the field exceeds the threshold field, 20 V / cm in our case, the lowest HH1S is almost empty and the pumping process is mainly controlled by the transi-tions between HH1 and HH2P± is empty and the population

f1. The abruptness of the growth of f2 is inversely

propor-tional to the temperature according to Eq.共55兲. However, the population f2 comes to a fixed value for the field F FIG. 9. The solution␰ of Eq. 共52兲 at low field and low

tempera-ture is shown as a function of the relative impact ionization coeffi-cient bc. The abrupt jump around bc= 1 is due to the depletion of the

lower localized levels by impact ionization.

FIG. 10. Occupation probabilities f₁共HH2P⫾兲, f₂共LH1S兲, and fg共HH1S兲 for T=4 K are plotted as functions of the acceptor

den-sity na at fixed electric field F = 100 V / cm. The population ratio

f2/ f1 is shown as a solid line. The horizontal line at f2/ f1= 1

de-notes that population inversion is built when na exceeds some