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Microelectronics Journal 39 (2008) 466–471

www.elsevier.com/locate/mejo

Polaron effects on the energy of a hydrogenic donor impurity inGaAs–(Ga,Al)As quantum-well wires

Pablo Villamila,�, Carlos Cabraa, N. Porras-Montenegrob

aDepartamento de Matematicas y Fısica, Universidad de Sucre, A.A. 406, Sincelejo, ColombiabDepartamento de Fısica, Universidad del Valle, A.A. 25360, Cali, Colombia

Available online 20 August 2007

Abstract

In this work using the effective-mass approximation within a variational approach we study the effect of the 3D longitudinal optical

phonons on the energy of a shallow donor impurity in a GaAs–(Ga,Al)As quantum-well wire with a magnetic field applied in the wire-

axis direction. We obtain the energy of the lower impurity states as a function of the wire radius, the donor position inside the wire, and

under the effect of the applied magnetic field. The polaron effects on the impurity energy are studied using second order perturbation

theory. We include non-parabolicity effects to explain the optical experiments under applied high magnetic fields.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Hydrogenic donor impurity; Polaron; Quantum-well wire

1. Introduction

Inside a crystal an electron interacts with the surround-ings. If the electron moves in a polar semiconductor itinteracts with phonons to constitute a quasi-particle calledpolaron, which is a long-range interaction. The electroneffective-mass increases due to this interaction. Theintensity of the electron–phonon interaction depends onthe non-dimensional coupling constant a, introduced byFrohlich et al. [1]. In weak polar materials, such as theIII–V compounds with a51 the polaron effects are veryweek for energies far from the resonance. Due to this smallvalue, the polaron problem may be treated in the lowcoupling limit, which means that it is enough to considerperturbed states with only one optical phonon.

Two effects modify the electron band structure in theGaAs: the electron–optical phonon interaction (polaron)and the non-parabolicity of the conduction band. As botheffects are small, they can be treated independently [2].

A polaron bounded to an impurity is called a boundpolaron. Most of the phonon modes in a crystal are thelongitudinal optical (LO) phonon modes and in the center

e front matter r 2007 Elsevier Ltd. All rights reserved.

ejo.2007.07.028

ing author.

ess: pabloe1954@yahoo.es (P. Villamil).

of the first Brillouin zone their energy is approximatelyindependent of the wave vector.The energy of the free polaron has been investigated with

and without applied magnetic fields [3–11]. Larsen [12]studied the polaron effect on the fundamental state of adonor impurity in InSb and found that this effect isnegligible; however Van Klarenbosch [13] and Cohn et al.[14] found that the polaron correction to shallow-impurityenergy levels is important. Later Cheng et al. [15] and VanKlarenbosch [16] observed experimentally the separation ofthe energy levels of donor impurity in GaAs under theaction of high magnetic fields up to 23.5 T. They observedresonant absorption with six branches due to the transitionenergies 1s-2p+. Also, Feddi et al. [17] calculated theeffects of both electric and magnetic fields on the energy ofthe ground state of a donor impurity confined in a polarmonocrystal of CdSe embedded in a non-polar matrix.They found that the polaron corrections on the bindingenergy, magnetic coefficient, and polarizability are veryimportant and cannot be neglected.This work is devoted to the study of polaron effects

on the binding energy of shallow donors impurities inGaAs–(Ga,Al)As quantum-well wires (QWWs). In Section 2we present the theory used to calculate variationally the 1s,2p7, 3p7, impurity-like states, in Section 3 we present the

ARTICLE IN PRESSP. Villamil et al. / Microelectronics Journal 39 (2008) 466–471 467

calculations for the polaron correction, results and discus-sions are presented in Section 4, and in Section 5 we presentthe conclusions.

2. Theory

The polaron hamiltonian in the effective-mass approx-imation is given by:

H ¼ He þHLO þHI, (1)

where He is the Hamiltonian of a shallow donor impurityin a cylindrical GaAs–Ga0.6Al0.4As QWW of radius R,with finite potential confinement, and under the action of amagnetic field applied in the wire-axial direction, written as

He ¼1

2m�Pþ

e

cA

h i2�

e2

�jr� r0jþ VðrÞ, (2)

where jr� r0j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðq� q0Þ

2þ z2

q, z is the relative separa-

tion between the electron and the impurity ion along thewire axis, e the static dielectric constant of GaAs, m* theelectron effective mass, and A(r) the potential vectorof the magnetic field, A(r) ¼ 1

2(B� r), with B ¼ BZ, which

in cylindrical coordinates is given by Ar ¼ Az ¼ 0,Aj ¼ (Br)/2. The finite confinement potential is describedby a square potential

V ðrÞ ¼0; 0prpR;

V0; r4R:

((3)

For the interface of GaAs–Ga0.6Al0.4As the height ofthe barrier, V0, corresponds to 60% of the differencebetween the forbidden bands of GaAs and Ga0.6Al0.4As,DEg ¼ (1.36x+0.22x2)meV [18]. The Hamiltonian of thesystem in cylindrical coordinates and in effective Rydbergs,R*, is written as,

He ¼ �r2 � ig

qqj

� �þ

g2r2

4�

2

rþ VðrÞ. (4)

g ¼ e_B/(2m*cR*) is the electron energy in the firstLandau level (n ¼ 0), due to the action of the appliedmagnetic field. For donor impurities in GaAs, m* ¼ 0.067,e ¼ 12.58, a0ffi100 A and R* ¼ 5.83meV.

Due to the inclusion of the impurity, the Hamiltonian inEq. (2) can not be solved exactly and in this work we usethe variational method to calculate the hamiltonianexpected values for the 1s-, 2p7- and 3p7 like impuritystates. We propose a trial wave function for every state asthe product of a hydrogenic part, Gnlm, and the ConfluentHypergeometric Functions. The trail wave functions forthe ground and some excited states of the impurity arewritten as

CnlmðrÞ ¼

Nnlm exp � x4

� �1F1ða01; 1; xÞ

�Gnlmðr; flnl ;bnlgÞ; rpR;

Nnlm1F1ða01;1;xRÞ

Uða001;1;xRÞ

exp � x4

� �Uða001; 1; xÞ

�Gnlmðr; flnl ;bnlgÞ; r4R:

8>>>><>>>>:

(5)

In Eq. (5), Nnlm are the normalization constants ofthe nlm sates, 1F1(a01,1,x), and U(a001, 1,x) are theConfluent Hypergeometric Functions for a finite confine-ment potential in the presence of a on-axis appliedmagnetic field, where x ¼ ðeBr2=2_cÞ ¼ ð1=2Þgr2, and xR

is x evaluated at r ¼ R; a01 and a001 are the parametersof the Confluent Hypergeometric Functions for theelectron ground state in and out of the wire, and arecalculated numerically by means of the boundary conditionðqCin=qrÞjr¼R ¼ ðqCout=qrÞjr¼R. Gnlm are the hydrogenicwave functions associated with the nlm states, as proposedby Latge et al. [19]. lnl, bnl are variational parameters usedby Chaudhury and Bajaj [20], which preserve the ortho-gonality between states with different nlm indexes.In the present work the energy of the impurity

unperturbed state is minimized with respect to thevariational parameter lnl:

E0nlm ¼ min

CnlmjHejCnlmh i

CnlmjCnlmh i, (6)

And the transition energies are calculated by means of

ET ðnlm! n0l0m0Þ ¼ jE0nlmðR;BÞ � E0

n0l0m0 ðR;BÞj, (7)

obeying the selection rules [25]

Dl ¼ l ¼ l0 ¼ �1

Dm ¼ m�m0 ¼ 0; �1. ð8Þ

In Eq. (1) HLO is the LO phonon described by

HLO ¼X~q

_w~q ay

~qa~q þ1

2

� �, (9)

where ay

~qya~q are the phonon creation and annihilationoperators with wave vector ~q and frequency wLO. ForGaAs, _wLO ¼ _w~q ¼ 36:75meV [21], is the energy valuefor the LO phonon at 4.2K [15,16].

HI is the Hamiltonian of the electron-phonon interactionin Eq. (1) and is given by

H1 ¼X~q

ðV~qa~qei~q�~r þ V�~qa

y

~qe�i~q�~rÞ, (10)

where V~q is the Fourier coefficient of the electron–phononinteraction, described by

jV~qj2 ¼

4paOð_wLOÞ

3=2

q2, (11)

where O is the crystal volume and a the Frohlich’s couplinggiven by

a ¼

ffiffiffiffiffiffiffiffiffiffiffiR�

_wLO

s�0�1� 1

� �, (12)

with eN the high-frequency dielectric constant of thematerial. In our calculations we have used a ¼ 0.068 forGaAs [22]. In the calculations we take into account theinteraction with 3D phonons. Due to the small value of a,the treatment of this problem is devoted to the weak

ARTICLE IN PRESSP. Villamil et al. / Microelectronics Journal 39 (2008) 466–471468

coupling limit, which means that it is enough to consideronly perturbed states with only one phonon present.

3. Polaron correction

Due to the fact that the GaAs semiconductor is a weakpolar material, second orden perturbation theory is used tocalculate the polaron correction to the energy of the nlm

donor impurity state:

DEnlm ¼ �Xn0m0l0

X~q

n0l0m0;~qjHIjnlm;~0D E��� ���2

_w~q þ E0n0l0m0 � E0

nlm � Dnlm

. (13)

Dnlm ¼ DEnlm � DE1s. In the perturbation theory ofRayleigh–Schrodinger (RSPT), valid in the non-resonantpolaron region, Dnlm ¼ 0, for all the sates. Dnlm 6¼0 for allthe states in the perturbation theory of Wigner–Brillouin(WBPT) [2,23,24]. nlm;~q

�� �represents a vector tensor

product compound by a donor electron with unperturbedenergy E0

nlm and a phonon with momentum _~q ¼ _ðqr; qzÞ

and energy _w~q. We have assumed that the LO phononenergy, _w~q ¼ _wLO, does not depend on its wave vector.

All the impurity states must be included in Sn0l0m0 in orderto take into account the polaron correction of every state.In the present work the calculations are limited to 1s, 2p7and 3p7 -like impurity states.

The matrix elements of the electron–phonon interac-

tion, Hn0l0m0;nlmI ¼

P~q n0l0m0;~qjHIjnlm;~0D E��� ���2, in units of

ð_wLOÞ3=2, are given by

Hn0l0m0;nlmI ¼

aa0

2p2ðR�Þ3=2

Zn0l0m0je�i~q�~rjnlm ��� ��2

q2d3

q. (14)

In this work we have also considered the non-parabolicityeffect of the conduction band. A mixing between differentenergy bands, especially valence and conduction bands, causethe non-parabolicity of conduction band. In this situationEqs. (2) and (4) are not correct. By means of a three-levelmodel and neglecting all the energy bands except the valenceand conduction band, Kane [26] obtained the expression

EðE þ EgÞðE þ Eg þ DÞ � K2P2 E þ Eg þ2

3D

� �¼ 0,

(15)

where Eg is the energy gap, D the spin–orbit splitting fromthe valence band, K the absolute value of the electron wavevector, and P the momentum matrix element between statesof s symmetry (in the conduction band) and z symmetry ( inthe valence band).

In the presence of a magnetic field B, the latter equationsmodified to [25–27]

EðE þ EgÞðE þ Eg þ DÞ � P2 eB

_ð2nþ 1Þ þ K2

z

�

� E þ Eg þ2

3D

� ��

1

3_P2DeB ¼ 0. ð16Þ

Neglecting the small term in E3, the small spin contribu-tion, and introducing the free electron contribution

E ¼ E0 �_2k2

2m, (17)

Vrehen [28] obtained for the borders of the bands of theLandau levels En

0(Kz ¼ 0)

E0n ¼ _wc nþ1

2

� �� 1�

m�0m

�23Eg þ 4Dþ 2D2=Eg

ðEg þ DÞð3Eg þ 2DÞ

�

� nþ1

2

� �2

ð_wcÞ2, ð170Þ

where the effective mass, m0*, in the conduction-bandminimum is given by

1

m�0¼

1

mþ

2P2

3_22

Egþ

1

Eg þ D

�. (18)

The first term describes the non-perturbed Landau levelswhich correspond to the parabolic case and the secondcorresponds to the non-parabolic correction that dependsquadratically on Ruf and Cardona [29] found an expressionfor the bulk dispersion at Uc

6,

Enp ¼ �E�g

2þ

E�g

2

� �2

þmb

me

me

mb� 1� C�

� �E�gEp

" #1=2

þmb

me

ð1þ C�ÞEp, ð19Þ

where Enp and Ep are donor energies with and without thenon-parabolicity effect, respectively, E�g ¼ 1631meV is thespin–orbit energy of GaAs, and C* is an adjustableparameter, which has been experimentally determined as�2.3. If C* is taken as �1, and the spin–orbit interaction isneglected, Eq. (19) reduces to

Enp ¼Eg

2�1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

4Ep

Eg

s !, (20)

which is the Kane’s result, a good approximation forelectron energies lower than 100meV. We have used thisexpression to take into account the non-parabolicity effects.

4. Results and discussion

In Fig. 1 we present the energy of some excited states of ashallow donor impurity on-axis located in a GaAs–(Ga,AlAs)QWW as a function of a magnetic field applied on thewire-axis direction. It is observed that for B ¼ 0 the energyfor all the impurity states is positive due to the geometricconfinement and while the energy of the states 1s, 2p+- and3p+-like increases, that of the 2p� and 3p�-like decreaseswith the magnetic field. The reason for this behavior is thatfor all the states the kinetic energy is higher than the potentialenergy and their absolute value increases with the magneticfield, and while the diamagnetic energy is positive for aloneentire range of magnetic field, the paramagnetic energy ispositive for the np+ and negative for the np� excited states.

ARTICLE IN PRESSP. Villamil et al. / Microelectronics Journal 39 (2008) 466–471 469

Also it is observed that the effect of the non-parabolicity is todecrease the energy in each excited state.

The width Dr and DZ of the wave functions for theground and some excited states of a donor impurity on-axislocated in a GaAs–(Ga,Al)As QWW as a function of themagnetic field are presented in Figs. 2(a) and (b). Resultsfor 2p+(3p+)-like are the same as for the 2p�(3p�)-likesates taking into account that the wave functions for thesestates differ only by a phase factor. As expected the wavefunctions for these states are more localized in the radialthan in the z-direction and as much extended as they are,appear more sensible to the action of the magnetic field.Also, it is observed that in the entire range of the magneticfield the z-localization of the wave function is greater forthe lower principal quantum numbers.

In Fig. 3 we present the energy of the 1s-, 2p7- and 3p7-like inpurity states taking into account the non-paraboli-city of conduction band with and without the polaroneffects. Also, these energies modified by the presence of aphonon are presented and the energy 1sþ _wLO is

0 5 10 15 20 25

3

6

9

12

15

18ρi = 0

R = 1 a0

1S

2p_

3p_

2p+

3p+

Parabolic Band - - - -

MAGNETIC FIELD ( T )

30

IMPU

RIT

Y E

NE

RG

Y (

R* )

Non-parabolic Band

Fig. 1. Some energy levels of a shallow donor impurity on-axis located in

a GaAs–Ga0.6Al0.4As QWW as a function of a magnetic field applied in

the wire-axis direction. R* ¼ 5.83meV.

0 5 10 15 20 25 30

3p±

1s

2p±

R = 100Åρi = 0

Non-parabolic Band

0.8

0.6

0.4

0.2

MAGNETIC FIELD ( T )

Δρ (a

0)

Fig. 2. Width of the wave function for the ground state and some excited state

(a) is presented the radial, while in (b) the z dependence, as a function of the

presented with its polaron correction. It is observed thatthe polaron correction diminishes the energy of each of theconsidered states and the shift is approximately constant inthe entire range of the magnetic field. The polaroncorrection is, in absolute value, greater for the 2p7- and3p7-like sates due to the greater geometric confinement ofthese states.In Fig. 4. we present the polaron energy correction for

the 1s-, 2p7- and 3p7-like states of a onor impurity on-center located in a GaAs–Ga0.6Al0.4As QWW as a functionof an applied magnetic field. As observed, the polaroncorrection, in absolute value, is greater for the less localizedstates and it is constant over the entire range of themagnetic field for the 2p7-like state. Due to the symmetry,the polaron correction is the same for the states withm ¼ �1 and m ¼ 1.The results for matrix elements, Hnlm;n0l0m0

I , are presentedin Fig. 5 in units of (R*)2 as a function of an appliedmagnetic field. Three main aspects are found: (1) most ofthe Hnlm;n0l0m0

I increase with the applied magnetic field due to

0 5 10 15 20 25 30

1

2

3

4

5

6

7

8

R = 100Å ρi = 0

Non-parabolic BandΔ Z

(a 0

)

MAGNETIC FIELD ( T )

3p±

2p±1s

s of a donor impurity on-center located in a GaAs–Ga0.6Al0.4As QWW. In

applied magnetic field.

0 5 10 15 20 25 300

5

10

15

20

25

30

2p ++ wLO

3p ++ w LO

3p_ + w

LO2p

_ + wLO

IMPU

RIT

Y E

NE

RG

Y (

R*

)

MAGNETIC FIELD ( T )

1S

without polaron correction

⎯with polaron correction - - - -- -

2p_

3p_2p+

3p+

1s + wLO

wL O

+ γ

Fig. 3. Energy of the 1s-, 2p7- and 3p7-like impurity states of an on-axis

donor impurity in a GaAs–Ga0.6Al0.4As QWW, taking into account the

non-parabolicity of the conduction band with (dashed curves) and without

polaron effects (solid curves).

ARTICLE IN PRESSP. Villamil et al. / Microelectronics Journal 39 (2008) 466–471470

the compression which the magnetic field causes to thewave function, closing the electron to the impurity ion.(2) the diagonal matrix elements are larger than the non-diagonals, and (3) the matrix elements are smaller for stateswith a large difference in energy, as expected.

The transition energies 1s-2p7 and 1s-2p7 arepresented in Fig. 6. The polaron effect weakly diminishesthe transition energy and the shift is approximatelyconstant due to the fact that the polaron energy remainsapproximately constant for each energy considered in thetransition.

0 5 10 15 20 25 30

-0.28

-0.24

-0.20

-0.16

3P±

POL

AR

ON

CO

RR

EC

TIO

N (

R* )

MAGNETIC FIELD (T)

1S

2P±

non-parabolic Band

ρi = 0

R = a0

Fig. 4. Polaron energy correction for the 1s-, 2p7- and 3p7-like donor

impurity states of an impurity on-center located in a GaAs–Ga0.6Al0.4As

QWW, as a function of the applied magnetic field.

0 5 10 15 20 25 30

0 5 10

0.90

0.95

1.00

1.05

MAGNETIC FIELD (T)

1s.1s

Σ q |

<j;q

| H

I | i;

0> |

2 /R

* 2

Σ q |

<j;q

| H

I | i;

0> |

2 /R

* 2

3x10-4

6x10-4

9x10-4

3

MAGN

Fig. 5. Some transition matrix elements of the electron–phonon interaction

GaAs–Ga0.6Al0.4As QWW.

5. Conclusions

In this work using the effective-mass approximationand within the variational approach we have studied theeffect of the 3D LO phonons on the binding energy of adonor impurity in a GaAs–(Ga,Al)As QWW with amagnetic field applied parallel to the axis of the QWW.We have found that the non-parabolicity diminishes theenergy of the impurity excited states and the polaroneffect weakly diminishes the transition energy between the1s and the first impurity excited states. We believe that

0 5 10 15 20 25 30

15 20 25 30

5.0x10-2

1.0x10-1

1.5x10-1

2.0x10-1

2.5x10-1

2P±. 3P±

3P± . 3P±

Σ q |

<j;q

| H

I | i;

0> |

2 /R

* 2

MAGNETIC FIELD (T)

2P±. 2P±

P+ - 2P

+

ETIC FIELD (T)

3P- - 2P

-

(LO) in units of (R*)2 as a function of the applied magnetic fielding a

0 5 10 15 20 25 30

4

6

8

10

12

14

16

1S -> 3P-

1S -> 3P +

TR

AN

SIT

ION

EN

ER

GY

( R

* )

MAGNETIC FIELD ( T )

1S -> 2P +

1S -> 2P-

without polaron correction ____

with polaron correction - - - -

Fig. 6. Transition energies 1s-2p7 and 1s-3p7 with (dotted lines) and

without (solid lines) polaron correction of an on-axis donor impurity in a

GaAs–Ga0.6Al0.4As QWW 100 A in radius, as a function of the applied

magnetic field.

ARTICLE IN PRESSP. Villamil et al. / Microelectronics Journal 39 (2008) 466–471 471

these results are of importance for futures interpretationsof experimental results.

Acknowledgments

Authors want to thank Dr. Ivan D. Nunez for theconstruction of some Fortran routines used in variouscalculations of this work. Equally, authors want to thankthe Colombian Scientific Agency (Colciencias, Grant no.1106-05-13828), and the CENM (Excellence Center forNovel Materials) for financial support.

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