ARTICLE IN PRESS

Physica E 42 (2010) 1623–1626

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physe

The hydrostatic pressure and temperature effects on donor impurities incylindrical quantum wire under the magnetic field

E. Kasapoglu a,n, F. Ungan a, H. Sari a, I. Sokmen b

a Cumhuriyet University, Department of Physics, 58140 Sivas, Turkeyb Dokuz Eylul University, Department of Physics, 35160 _Izmir, Turkey

a r t i c l e i n f o

Article history:

Received 9 October 2009

Accepted 5 January 2010Available online 13 January 2010

Keywords:

Donor impurities

Cylindrical quantum wire

Hydrostatic pressure

Temperature

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2010.01.009

esponding author. Tel.: +90 346 2191010/19

ail address: ekasap@cumhuriyet.edu.tr (E. Kas

a b s t r a c t

The combined effects of hydrostatic pressure and temperature on donor impurity binding energy in

cylindrical GaAs/Ga0.7Al0.3As quantum wire in the presence of the magnetic field have been studied by

using a variational technique within the effective-mass approximation. The results show that an

increment in temperature results in a decrement in donor impurity binding energy while an increment

in the pressure for the same temperature enhances the binding energy and the pressure effects on

donor binding energy are lower than those due to the magnetic field.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Semiconductor quantum wires (QWRs) have been studiedintensively worldwide for a wide spectrum of materials. Suchone-dimensional (1D) nanostructures are not only interesting forfundamental research due to their unique structural and physicalproperties relative to their bulk counterparts, but also offerfascinating potential for future technological applications. Theunderstanding of the electronic and optical properties of impuritiesin such systems is important because the optical and transportproperties of devices made from these materials are stronglyaffected by the presence of shallow impurities. And also, magneticand electric fields, intense laser field and hydrostatic pressure areeffective tools for studying the properties of impurities in hetero-structures and thus a number of studies have been performed todiscuss the hydrogenic-shallow impurities in QWWs [1–12].

In this letter, the combined effects of hydrostatic pressure andtemperature on donor impurity binding energy in cylindricalGaAs/Ga0.7Al0.3 quantum wire in the presence of the magneticfield are investigated for the pressure values where there is noG�X crossover by using a variational technique within theeffective-mass approximation. As known, for pressures Pr10kbar a G-like electron is confined in GaAs layer by the G barriersof constant height while for 10oPr30 kbar the X-minima of thebarrier layers drop below the G-minimum of these layers and passthrough the energies of the confined electron states. For P430kbar the X-minima of the Ga0.7Al0.3As layers become theminimum of the conduction-band states of the system andelectrons are no longer confined to the GaAs layer. For PZ40 kbar

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apoglu).

the X-minima are the lowest energy states in the GaAs layer andboth the barrier and well materials become indirect.

2. Theory

In the effective mass approximation the Hamiltonian for ahydrogenic on-center shallow donor impurity in GaAs/Ga1�xAlxAsquantum wire under the magnetic field is given by

H¼�1

2meðP; TÞp!

eþe

cA!ð r!Þ

� �2

þVðr; P; TÞ� e2

eðP; TÞr ð1Þ

where p!

e is the momentum operator, A!ð r!Þ¼ 1

2ð B!

x r!Þ¼

ð�ðB=2Þy; ðB=2Þx;0Þ the vector potential associated with the

magnetic field ( B!¼ ð0;0;BÞ), P the hydrostatic pressure in units

of kbar, T is the temperature in units of Kelvin,

r¼ ð9r!�r!i92þz2Þ

1=2 the distance between the electron and the

donor impurity site, V(r,P,T) the pressure and temperaturedependent confinement potential, e(P,T) the pressure and tem-perature dependent dielectric constant and me(P,T) the pressureand temperature dependent effective mass. By introducing the

effective Rydberg ðRyd¼ ðmeðP; TÞe4=2e2ðP; TÞ_2ÞÞ as the unit of

energy and the effective Bohr radius ðaB ¼ ðeðP; TÞ_2=meðP; TÞe2ÞÞ asthe unit of length, Hamiltonian in cylindrical coordinates and inthe reduced units can be written as

~H ¼�@2

@r2þ

1

r@

@r þ1

r2

@2

@j2

!�@2

@z2þgLzþ

g2r2

4

þ ~Vðr;P;TÞ� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9r!�r!i9

2þz2

q ; ð2Þ

ARTICLE IN PRESS

00

10

20

30

40

2 P = 13.5 kbar

B = 0

1 P = 0

B = 15 T

T = 50 K

Eb

(meV

)

do ( )

1

2

1

2

- - - - - - T = 250 K

30025020015010050

Fig. 1. The variation of the ground state binding energy of donor impurity located

in the wire center in cylindrical quantum wire versus the wire radius for different

hydrostatic pressure, magnetic field and temperature values. Solid (dashed) curve

is corresponding to T=50 K (T=250 K).

E. Kasapoglu et al. / Physica E 42 (2010) 1623–16261624

where Lz is the z-component of the angular momentum operator

in units of _ and gð ¼ e_B=2meðP; TÞcRydðP; TÞÞ is the dimensionlessmeasure of the magnetic field.

The pressure and temperature dependent effective mass forthe electron is given by Refs. [13,14]

meðP; TÞ ¼mo

1þEGp ½ð2=EGg ðP; TÞÞþð1=EGg ðP; TÞþDoÞ�ð3Þ

where mo is the free electron mass, EGp ¼ 7:51 eV, Do=0.341 eV and

EGg ðP; TÞ the pressure and temperature dependent energy gap forthe GaAs quantum well at the G-point in units of eV is given byRef. [13]

EGg ðP; TÞ ¼ EGg ð0; TÞþ1:26� 10�2 P�3:77� 10�5 P2 ð4aÞ

EGg ð0; TÞ ¼ 1:519�ð5:405� 10�4T2Þ=ðTþ204Þ eV ð4bÞ

The pressure and temperature dependent static dielectricconstant is given by Refs. [15,16]

eðP; TÞ ¼12:74exp½�1:73� 10�3P�exp½9:4� 10�5

ðT�75:6Þ� for To200K

13:18exp½�1:73� 10�3P�exp½20:4� 10�5ðT�300Þ� for TZ200K

(

ð5Þ

for the electron which is given by

Vðr;P; TÞ ¼0; 0rrrdðP; TÞ

VðP; TÞ; r4dðP; TÞ

(ð6Þ

where d(P,T) is the pressure and temperature dependent radius ofthe wire and V(P,T) the barrier height of the confinementpotential. The barrier height is given by Refs. [17–19]

VðP; TÞ ¼ QcDEGg ðx; P; TÞ: ð7Þ

where Qc(=0.6) is the conduction band offset parameter, x themole fraction of aluminum in Ga1�xAlxAs layer, DEGg ðx;P; TÞ is theband gap difference between quantum well and the barrier matrixat the G-point as a function of pressure and temperature and it isgiven by Refs. [14,20]

DEGg ðx; P; TÞ ¼DEGg ðxÞþDðxÞPþGðxÞT ð8Þ

where DEGg ðxÞ ¼ ð1:155xþ0:37x2Þ eV, D(x)=[�(1.3�10�3)x]eV/

kbar and G(x)=[(�1.11�10�4)x]eV/K. The pressure dependenceof the wire radius obtained from the fractional change in volume

associated with the hydrostatic pressure (DV/Vo=�3P(S11+S12))[21] is given by

dðPÞ ¼ do½1�3PðS11þ2S12Þ�1=2; ð9Þ

where do is the original radius of the wire and S11=1.16�10�3

and S12=3.7�10�4 kbar�1 are the elastic constants of the GaAs[22]

The eigenfunctions of the Hamiltonian in the absence of theimpurity are given by

Fðr; zÞ ¼N

J0ðr10rÞe�gr2=4; rod

J0ðr10rÞK0ðb10dÞ

K0ðb10rÞe�gr2=4; r4d

8><>: ð10Þ

where N is the normalization constant, r10 ¼ffiffiffiffiffiEo

p,

b10 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðP; TÞ�Eo

p, J0 and K0 are zero-order and modified Bessel

functions, respectively.By considering the Coulombic interaction between the electron

and impurity ion, the wave function of Eq. (2) can be written as

cðr; zÞ ¼N1Fðr; zÞe�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9r!�r!

i92þ z2

q=l ð11Þ

where N1 is the normalization constant of the trial wave functionand l the variational parameter. The ground state impurity energy

is evaluated by minimizing the expectation value of theHamiltonian in Eq. (2) with respect to l.

The ground state donor binding energy is calculated as

EbðP; TÞ ¼ EoðP; TÞþg�minl

/ ~Hðr;j; zÞS: ð12Þ

where Eo(P,T) is the ground-state energy of electron and g is thefirst Landau level.

3. Results and discussion

Fig. 1 shows the variation of the ground state binding energy ofdonor impurity located in the wire center in cylindrical quantumwire versus the wire radius for different hydrostatic pressure,magnetic field and temperature values. Binding energy increaseswith wire radius until it reaches a maximum value and thenbegins to decrease for further large wire radii due to the leakageof the wave function into the barriers. Also the binding energyincreases with the magnetic field since the magnetic field gives anadditional confinement of the electronic wave function. As thehydrostatic pressure increases dielectric constant, the potentialheight, the first subband energy, wire radius decrease theeffective mass increases for electron, leading to moreconfinement of the electron and thus the donor binding energyincreases for all wire radii. This increment in donor bindingenergy due to the hydrostatic pressure is lower than that due tothe magnetic field. As the temperature increases dielectricconstant and first subband energy increase while effective massand potential height of electron decrease and so donor bindingenergy decreases for all wire radii. The dependency of thephysical parameters(dielectric constant, the potential height, thefirst subband energy, wire radius and the effective mass)mentioned above upon the pressure and temperature are givenin Fig. 2(a)–(e).

ARTICLE IN PRESS

012.00

12.40

12.80

13.20

13.60

14.00

T = 4 K

T = 50 K

T = 250 K

T = 500 K

P (kbar)0 5 10 15

212

216

220

224

228

P (kbar)

V (

meV

)

T = 4 K

T = 50 K

T = 250 K

T = 500 K

0 5 10 15

24

28

32

36

40

44

P (kbar)

Eo

(meV

)

T = 4 K

T = 50 K

T = 250 K

T = 500 K

510150

151050

13.00

13.50

14.00

14.50

15.00

P (kbar)

d (

)

0.06

0.06

0.07

0.07

0.08

T = 4 K

T = 50 K

T = 250 K

T = 500 K

P (kbar)

m /

mo

*

15105

Fig. 2. The dependency of the pressure and temperature upon the physical parameters (dielectric constant, the potential height, the first subband energy, wire radius and

the effective mass).

E. Kasapoglu et al. / Physica E 42 (2010) 1623–1626 1625

ARTICLE IN PRESS

E. Kasapoglu et al. / Physica E 42 (2010) 1623–16261626

4. Conclusions

As a result, the effects of hydrostatic pressure and temperatureon donor impurity binding energy in cylindrical GaAs/Ga0.7Al0.3Asquantum wire in the presence of the magnetic field by using avariational technique within the effective-mass approximationhave been investigated. For simplicity, in this letter, we use thepressure values where there is no G�X crossover. The resultsshow that an increment in temperature results in a decrement indonor impurity binding energy while an increment in thepressure for the same temperature enhances the binding energyand the effects of the magnetic field on donor binding energy arehigher than that of the pressure effects. The obtained results arequalitatively consistent with theoretical studies on this subject. Itis hoped that the present work would stimulate further experi-mental activities in semiconductor heterostructures.

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