ARTICLE IN PRESS

Physica B 403 (2008) 4139–4150

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

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

Influence of light on the Debye screening length in ultrathin filmsof optoelectronic materials

S. Bhattacharya a, N.C. Paul b, D. De c, K.P. Ghatak d,�

a Nano Scale Device Research Laboratory, Centre for Electronics Design and Technology, Indian Institute of Science, Bangalore 560 012, Indiab Department of Metallurgical Engineering, National Institute of Technology, Agartala, Tripura (West) 799 055, Indiac Department of Computer Science and Engineering, West Bengal University of Technology, B.F. 142, Sector 1, Salt Lake City, Kolkata 700 064, Indiad Department of Electronic Science, The University of Calcutta, 92, Acharyya Prafulla Road, Kolkata, West Bengal 700 009, India

a r t i c l e i n f o

Article history:

Received 2 July 2008

Received in revised form

25 August 2008

Accepted 26 August 2008

PACS:

71.55.Eq

Keywords:

Ultra-thin films

Debye screening length

Optoelectronic

Light waves

Experimental determination

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.08.017

esponding author. Tel.: +91033 235 05213; fa

ail address: kamakhyaghatak@yahoo.co.in (K.

a b s t r a c t

In this paper, we study the Debye screening length (DSL) in ultrathin films of optoelectronic materials in

the presence of light waves. The solution of the Boltzmann transport equation on the basis of the newly

formulated electron dispersion laws will introduce new physical ideas and experimental findings under

different external conditions. It has been found, taking ultrathin films of n-Hg1�xCdxTe, as an example,

that the respective two-dimensional (2D) DSL in the aforementioned materials exhibits decreasing

quantum step dependence with the increasing film thickness, surface electron concentration, light

intensity and wavelength, respectively, with different numerical values. The nature of the variations is

totally band structure dependent which is influenced by the presence of the different energy band

constants. The strong dependence of the 2D DSL on both the light intensity and the wavelength reflects

the direct signature of the light waves. The well-known result for the 2D DSL for nondegenerate wide

gap materials in the absence of any field has been obtained as a special case of the present analysis

under certain limiting conditions and this compatibility is the indirect test of our generalized

formalism. Besides, we have suggested an experimental method of determining the 2D DSL in ultrathin

materials in the presence of light waves having arbitrary dispersion laws.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

It is well known that the Debye screening length (DSL) of thecarriers in semiconductors is a very important quantity characteriz-ing the screening of the Coulomb field of the ionized impuritycenters by the free carriers [1]. It affects many of the special featuresof modern nanodevices, the carrier mobilities under differentmechanisms of scattering and the carrier plasmas in semiconduc-tors [2]. The DSL is a very good approximation to the accurate self-consistent screening in presence of band tails and is also used toillustrate the interaction between the colliding carriers in Augereffect in solids [1]. In the conventional form, the DSL decreases withincreasing n2D at a constant temperature and this relation holdsonly under the condition of carrier non-degeneracy. Since, theperformance of the electron devices at the device terminals and thespeed of operation of modern switching transistors are significantlyinfluenced by the degree of carrier degeneracy present in thesedevices [3], the simplest way of analyzing such devices taking intoaccount of the degeneracy of the band is to use the appropriate DSL

ll rights reserved.

x: +91035 922 46112.

P. Ghatak).

to express the performance at the device terminal and switchingspeed in terms of the carrier concentration [3].

The DSL depends on the density-of-states (DOS) function which,in turn, significantly affects the different physical properties ofoptoelectronic and related compounds having various band struc-tures [2]. It is well known from the fundamental study of Landsberg[1] that the DSL for electronic materials having degenerate electronconcentration is essentially determined by their respective energyband structures. It has, therefore, different values in differentmaterials and varies with the electron concentration, with themagnitude of the reciprocal quantizing magnetic field undermagnetic quantization, with the quantizing electric field as ininversion layers, with the nanothickness as in quantum wells, withsuperlattice period as in the quantum-confined superlattices ofsmall-gap compounds with graded interfaces having various carrierenergy spectra. The nature of these variations has been investigatedby in the literature [1–12] and some of the significant features,which have emerged from these studies, are:

(a)

the DSL decreases with increasing electron concentration andsuch variations are significantly influenced by constants of theenergy band spectra;

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–41504140

(b)

the DSL decreases with the magnitude of the quantizingelectric field as in inversion layers;

(c)

the DSL oscillates with the inverse quantizing magnetic fieldunder magnetic quantization due to the SdH effect; and

(d)

the DSL exhibits composite oscillations with the variouscontrolled parameters as in superlattices of non-paraboliccompounds with graded interfaces.

The above information has been obtained through theoreticalanalyses and no experimental results are available to the knowl-edge of the authors in support of the predictions for ultrathinfilms having arbitrary dispersion relations in the presence of lightwaves. In this paper, we have studied the two-dimensional (2D)DSL for ultrathin films of optoelectronic compounds in thepresence of light waves in addition to the suggestion of theexperimental determination of the same.

In this context, it may be noted that with the advent of finelithographical methods [13], molecular beam epitaxy [14],organometallic vapor-phase epitaxy [15] and other experimentaltechniques, low-dimensional structures having quantum confine-ment of one (inversion layers, accumulation layers, nipi structuresand ultrathin films), two (quantum well wires) and threedimensions (quantum dots, magneto inversion layers, magnetoaccumulation layers, magneto size quantization, quantum wellsuperlattices under magnetic quantization, quantum dot super-lattices and magneto nipi structures) have, in the last few years,attracted much attention not only for their potential in uncoveringnew phenomena in nanoscience but also for their interestingquantum device applications [16–18]. In ultrathin films, therestriction of the motion of the carriers in the direction normalto the film (say, the z-direction) may be viewed as carrierconfinement in an infinitely deep one-dimensional (1D) squarepotential well, leading to quantization [known as quantum sizeeffect (QSE)] of the wave vector of the carrier, allowing 2D electrontransport parallel to the surface of the film representing newphysical features not exhibited in bulk semiconductors [19]. Thelow-dimensional heterostructures based on various materials arewidely investigated because of the enhancement of carriermobility [20]. These properties make such structures suitablefor applications in quantum well lasers [21], heterojunction FETs[22], high-speed digital networks [23], high-frequency microwavecircuits [24], optical modulators [25], optical switching systems[26] and other devices. We shall use n-Hg1�xCdxTe matched as anexample of ternary materials. The ternary alloy n-Hg1�xCdxTe is aclassic small-gap material and is an important optoelectroniccompound because its energy band gap can be varied to cover aspectral range from 0.8mm to over 30mm by varying the alloycomposition [27]. The n-Hg1�xCdxTe is being extensively used ininfrared detector materials [28] and photovoltaic detector arrays[29] in the 8–12mm wave bands. The aforementioned applicationshave spurred an Hg1�xCdxTe technology for the generation of highmobility single crystals, with specially prepared surface layers andthe same compound is ideally suitable for narrow sub-bandphysics because the relevant energy band constants are withineasy experimental reach [30].

In Section 2.1 of the theoretical background, we haveformulated the dispersion relation of the conduction electronsof optoelectronic materials in the presence of photo-excitation,whose unperturbed energy band structures are defined by thethree- and two-band models of Kane together with the parabolicenergy bands for the purpose of relative comparison. Theexpressions for the surface electron concentration and DSL forultrathin films of the aforementioned materials in the presenceand absence of photo-excitation have been formulated in Sections2.2 and 2.3, respectively. In Section 2.4, we have suggested an

experimental method of determination of the DSL in semicon-ductor nanostructures having arbitrary carrier energy spectra inthe present case. In Section 2.5, we have written two specificapplications of the results of this paper in the fields ofcomputational and theoretical nanostructures. The 2D DSL hasbeen numerically investigated by taking n-Hg1�xCdxTe as anexample of ternary compounds in accordance with the three- andthe two-band models of Kane together with parabolic energybands, respectively, for the purpose of relative comparison both inthe presence and absence of photo-excitation.

2. Theoretical background

2.1. Formulation of the dispersion relation of the conduction

electrons of optoelectronic materials in the presence of light waves

The Hamiltonian (H) of an electron in the presence of lightwave characterized by the vector potential ~A can be written in thefollowing [31] form:

H ¼ ½ðpþ e~AÞ2=2mþ Vð~rÞ (1)

in which, p is the momentum operator, e is the electron energy,Vð~rÞ is the crystal potential and m is the free electron mass.

Eq. (1) can be expressed as

H ¼ H0 þ H0

(2)

where

H0 ¼p2

2mþ Vð~rÞ

and

H0¼

e

2m~A �~p (3)

The perturbed Hamiltonian H0 0 can be written as

H0¼�i_e

2m

� �ð~A � rÞ (4)

where i ¼ffiffiffiffiffiffiffi�1p

, _ ¼ h/2p, h is Planck constant, ~p ¼ �i_r.The vector potential ð~AÞ of the monochromatic light of plane

wave can be expressed as

~A ¼ A0~�s cosð~s0 �~r �otÞ (5)

where A0 is the amplitude of the light wave, ~�s is the polarizationvector,~s0 is the momentum vector of the incident photon,~r is theposition vector, o is the angular frequency of light wave and t isthe time scale. The matrix element of H

0

nl between initial state,clð~q;~rÞ and final state cnð

~k;~rÞ in different bands can be written as

H0

nl ¼e

2mhn~kj~A �~pjl~qi (6)

Using Eqs. (4) and (5), we can re-write Eq. (6) as

H0

nl ¼�i_eA0

4m

� �~�s � ½fhnkjeði~s0�~rÞrjl~qie�iotg þ fhn~kjeð�i~s0 �~rÞrl~qie�iotg�

(7)

The first matrix element of Eq. (7) can be written as

hn~kj expði~s0 �~rÞrjl~qi ¼

Zexpði½~qþ~s0 �

~k� �~rÞi~qu�nð~k;~rÞulð~q;~rÞd

3r

þ

Zexpði½~qþ~s0 �

~k� �~rÞu�nð~k;~rÞrulð~q;~rÞd

3r

(8)

The functions u�nul and u�nrul are periodic. The integral over allspaces can be separated into a sum over unit cells times anintegral over a single unit cell. It is assumed that the wavelength

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150 4141

of the electromagnetic wave is sufficiently large so that if ~k and ~qare within the Brillouin zone, ð~qþ~s0 �

~kÞ is not a reciprocal latticevector.

Therefore, we can write Eq. (8) as

hn~kj expði~s0 �~rÞrjl~qi

¼ð2pÞ3

O

" #iqdð~qþ~s0 �

~kÞdnl þ dð~qþ~s0 �~kÞ

Zcell

u�nð~k;~rÞrulð~q;~rÞd

3r

� �

¼ð2pÞ3

O

" #dð~qþ~s0 �

~kÞ

Zcell

u�nð~k;~rÞrulð~q;~rÞd

3r

� �(9)

where O is the volume of the unit cell andR

u�nð~k;~rÞulð~q;~rÞd

3r ¼

dð~q�~kÞdnl ¼ 0, since n6¼l.The delta function expresses the conservation of wave vector in

the absorption of light wave and ~s0 is small compared to thedimension of a typical Brillouin zone and we set ~q ¼ ~k.

From Eqs. (8) and (9), we can write

H0

nl ¼eA0

2m~�s �~pnlð

~kÞdð~q�~kÞ cosðotÞ (10)

where

~pnlð~kÞ ¼ �i_

Zu�nrul d3r ¼

Zu�nð~k;~rÞ~pulð

~k;~rÞd3r

Therefore, we can write

H0

nl ¼eA0

2m~�0s �~pnlð

~kÞ (11)

where ~�0s ¼ �s cosot.When a photon interacts with a semiconductor, the carriers

(i.e., electrons) are generated in the bands which are followed bythe interband transitions. For example, when the carriers aregenerated in the valence band, the carriers then make interbandtransition to the conduction band (CB). The transition of theelectrons within the same band i.e., H0nn ¼ hn~kjH

0jn~k is neglected.

Because, in such a case, i.e., when the carriers are generatedwithin the same bands by photons, are lost by recombinationwithin the aforementioned band resulting zero carriers.

Therefore,

hn~kjH0jn~ki ¼ 0 (12)

With n ¼ c stands for CB and l ¼ v stands for valance band, theenergy equation for the conduction electron can approximately beexpressed as

gðEÞ ¼ _2k2

2mc

!þðeA0=2mÞ2hj~�0s �~pcvð

~kÞj2iav

Ecð~kÞ � Evð

~kÞ(13)

where g(E) ¼ E(aE+1)(bE+1)/(cE+1), a ¼ 1=Eg0; Eg0

is the unper-turbed band gap, b ¼ 1=ðEg0

þ DÞ, D is the spin–orbit splittingconstant in the absence of any field, c ¼ 1=ðEg0

þ 2D=3Þ, mc is theeffective electron mass at the edge of the CB in the absence of anyfield and hj~�0s �~pcvð

~kÞj2iav represents the average of the square ofthe optical matrix element (OME).

For the three-band model of Kane, we can write

x1k ¼ Ecð~kÞ � Evð

~kÞ ¼ ðE2g0þ Eg0

_2k2=mrÞ1=2 (14)

where mr is the reduced mass and is given by mr�1¼ mc

�1+mv�1,

and mv is the effective mass of the heavy hole at the top of thevalence band (VB) in the absence of any field.

The doubly degenerate wave functions u1ð~k;~rÞ and u2ð

~k;~rÞ canbe expressed as [32]

u1ð~k;~rÞ ¼ akþ½ðisÞ#

0� þ bkþX0 � iY 0ffiffiffi

2p "0

� �þ ckþ½Z

0#0� (15)

and

u2ð~k;~rÞ ¼ ak�½ðisÞ"

0� � bk�X0 þ iY 0ffiffiffi

2p #0

� �þ ck�½Z

0"0� (16)

s is the s-type atomic orbital in both unprimed and primedcoordinates, k0 indicates the spin down function in the primed

coordinates, ak� � b Eg0� ðg0k�Þ

2ðEg0� d0Þ�1=2ðEg0

þ d0Þ�1=2h i

, b �

½ð6ðEg0þ2D=3ÞðEg0

þDÞÞ=w�1=2, w � ð6E2g0þ 9Eg0

Dþ 4D2Þ, g0k� �

½ðx1kEg0Þ=2ðx1kþd

0Þ�1=2, x1k ¼ Ecð

~kÞ � Evð~kÞ ¼ Eg0

½1þ 2ð1þ ðmc=mvÞ

ðgðEÞ=Eg0ÞÞ�1=2, d0 ¼ ðE2

g0DÞðwÞ�1, X0, Y0 and Z0 are the p-type atomic

orbitals in the primed coordinates, m0 indicates the spin-upfunction in the primed coordinates, bk7�rg0k7, r�(4D2/3w)1/2,

ck7�tg0k7 and t � ½6ðEg0þ 2D=3Þ2=w�1=2.

We can, therefore, write the expression for the OME as

OME ¼ pcvð~kÞ ¼ hu1ð

~k;~rÞjpju2ð~k;~rÞi (17)

Since the photon vector has no interaction in the same band forthe study of interband optical transition, we can therefore write

hSjpjSi ¼ hXjpjXi ¼ hY jpjYi ¼ hZjpjZi ¼ 0

and

hXjpjYi ¼ hY jpjZi ¼ hZjpjXi ¼ 0

There are finite interactions between the CB and the VB, and wecan obtain

hSjPjXi ¼ i �~P ¼ i �~Px

hSjPjYi ¼ j �~P ¼ j �~Py

hSjPjZi ¼ k �~P ¼ k �~Pz

where i; j and k are the unit vectors along x, y and z axes,respectively.

It is well known that

"0

#0

" #¼

e�if=2 cosy2

eif=2 siny2

�e�if=2 siny2

eif=2 cosy2

2664

3775 "#" #

and

X0

Y 0

Z0

264

375 ¼

cos y cos f cos y sin f � sin y� sin f cos f 0

sin y cos y sin y sin f cos y

264

375

X

Y

Z

264

375

Besides, the spin vector can be written as

~S ¼_2

� �~s

where

sx ¼0 1

1 0

� �; sy ¼

0 �i

i 0

� �; sz ¼

1 0

0 �1

� �

and i ¼ffiffiffiffiffiffiffi�1p

.From above, we can write

PCVð~kÞ ¼ hu1ð

~k;~rÞjPju2ð~k;~rÞi

¼ akþ ½ðisÞ#0� þ bkþ

X0 � iY 0ffiffiffi2p

� �"0

� �þ ckþ ½Z

0#0�

� �

� jPj ak� ½ðisÞ"0� � bk�

X0 þ iY 0ffiffiffi2p

� �#0 þ ck� ½Z

0"0�

� �� �

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–41504142

Using above relations, we get

PCVð~kÞ ¼

bkþak�ffiffiffi2p fhðX0 � iY 0ÞjPjiSih"0j"0ig þ ckþak� fhZ

0jPjiSih#0j"0ig

�akþbk�ffiffiffi

2p fhiSjPjðX0 þ iY 0Þih#0j#0ig þ akþck� fhiSjPjZ

0ih#0j"0ig

(18)

From Eq. (18), we can write

hðX0 � iY 0jPjiSi ¼ hðX0ÞjPjiSi � hðiY 0ÞjPjiSi ¼ ihX0jPjSi � hY 0jPjSi

From the above relations, for X0, Y0 and Z0, we get

jX0i ¼ cos y cos fjXi þ cos y sin fjYi � sin yjZi

Thus,

hX0jPjSi ¼ cos y cos fhXjPjSi þ cos y sin fhYjPjSi � sin yhZjPjSi ¼ Pr1

where

r1 ¼ i cos y cos fþ j cos y sin f� k sin y

Similarly, we obtain

jY 0i ¼ � sin fjXi þ cos fjYi þ 0jZi

Thus,

hY 0jPjSi ¼ � sin fhXjPjSi þ cos fhY jPjSi þ 0hZjPjSi ¼ Pr2

where

r2 ¼ �i sinfþ j cosf

so that

hðX0 � iY 0ÞjPjSi ¼ Pðir1 � r2Þ

Thus,

ak�bkþffiffiffi2p hðX0 � iY 0ÞjPjSih"0j"0i ¼

ak�bkþffiffiffi2p Pðir1 � r2Þh"

0j"0i (19)

Now since

hiSjPjðX0 þ iY 0Þi ¼ ihSjPjX0i � hSjPjY 0i ¼ Pðir1 � r2Þ

We can write

�akþbk�ffiffiffi

2p fhiSjPjðX0 þ iY 0Þih#0j#0ig ¼ �

akþbk�ffiffiffi2p Pðir1 � r2Þh#

0j#0i (20)

Similarly, we get

jZ0i ¼ sin y cos fjXi þ sin y sin fjYi þ cos yjZi

So that

hZ0jPjiSi ¼ ihZ0jPjSi ¼ iPfsin y cos fiþ sin y sin fjþ cos yk ¼ iPr3

where

r3 ¼ i sin y cos fþ j sin y sin fþ k cos y

Thus,

ckþak� hZ0jPjiSih#0j"0i ¼ ckþak� iPr3h#

0j"0i (21)

Similarly, we can write

ck�akþ hiSjPjZ0ih#0j"0i ¼ ck�akþ iPr3h#

0j"0i (22)

Therefore, we obtain

ak�bkþffiffiffi2p fhðX0 � iY 0ÞjPjSih"0j"0ig �

akþbk�ffiffiffi2p fhiSjPjðX0 þ iY 0Þih#0j#0ig

¼Pffiffiffi2p ð�akþbk� h#

0j#0i þ ak�bkþ h" j"0iÞðir1 � r2Þ (23)

Also, we can write

ckþak� hZ0jPjiSih#0j"0i þ ck�akþ hiSjPjZ

0ih#0j"0i

¼ iPðckþak� þ ck�akþ Þr3½h#0j"0i� (24)

Combining Eqs. (23) and (24), we find

PCVð~kÞ ¼

Pffiffiffi2p ðir1 � r2Þfðbkþak� Þh"

0j"0i � ðbk�akþ Þh#0j#0i

þ iPr3ðckþak� þ ck�akþ Þh#0j"0i (25)

From the above relations, we obtain

"0 ¼ e�if=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #

#0 ¼ �e�if=2 sin ðy=2Þ " þeif=2 cos ðy=2Þ # (26)

Therefore

h#0j"0i ¼ ð�e�if=2 sin ðy=2Þ " þeif=2 cos ðy=2Þ # Þ�

� ðe�if=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #Þ

¼ ð�eif=2 sin ðy=2Þ"� þ e�if=2 cos ðy=2Þ#�Þ

� ðe�if=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #Þ

¼ � sin ðy=2Þ cos ðy=2Þh" j "i þ e�if cos2ðy=2Þh# j "i

� eif sin2ðy=2Þh" j #i þ sin ðy=2Þ cosðy=2Þh# j #i

Therefore,

h#0j"0ix ¼ � sinðy=2Þ cosðy=2Þh" j " ix þ e�if cos2ðy=2Þh# j " ix

� eif sin2ðy=2Þh" j # ix þ sinðy=2Þ cosðy=2Þh# j # ix (27)

But we know from above that

h" j " ix ¼ 0; h# j " ix ¼12; h" j # ix ¼

12 and h# j # ix ¼ 0

Thus, from Eq. (27), we get

h#0j"0ix ¼12½e�if cos2ðy=2Þ � eif sin2

ðy=2Þ�

¼ 12½ðcos f� i sin fÞcos2ðy=2Þ � ðcos fþ i sin fÞsin2

ðy=2Þ�

¼ 12½cos f cos y� i sin f� (28)

Similarly, we obtain

h#0j"0iy ¼12½i cos fþ sin f cos y�

and

h#0j"0iz ¼12½� sin y�

Therefore,

h#0j"0i ¼ ih#0j"0ix þ jh#0j"0iy þ kh#0j"0iz

¼ 12fðcos y cos f� i sin fÞiþ ði cos fþ sin f cos yÞj� sin ykg

¼ 12½fðcos y cos fÞiþ ðsin f cos yÞj� sin ykg

þ if�i sin fþ j cos fg�¼ 1

2½r1 þ ir2� ¼ �12i½ir1 � r2�

Similarly, we can write

h"0j"0i ¼ 12½i sin y cos fþ j sin y sin fþ k cos y� ¼ 1

2r3

and

h#0j#0i ¼ �12r3

Using the above results and following Eq. (25), we can write

PCVð~kÞ ¼

Pffiffiffi2p ðir1 � r2Þfðak�bkþ Þh"

0j#0i � ðbk�akþ Þh#0j#0ig

þ iPr3fðckþak� � ck�akþ Þh#0j"0ig

¼P

2r3ðir1 � r2Þ

ak�bkþffiffiffi2p þ

bk�akþffiffiffi2p

� �� �þ

P

2r3ðir1 � r2Þ

�fðckþak� � ck�akþ Þg

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150 4143

Thus,

PCVð~kÞ ¼

P

2r3ðir1 � r2Þ akþ

bk�ffiffiffi2p þ ck�

� �þ ak�

bkþffiffiffi2p þ ckþ

� �� �(29)

We can write that

jr1j ¼ jr2j ¼ jr3j ¼ 1

also

Pr3 ¼ Px sin y cos fiþ Py sin y sin fjþ Pz cos yk

where

P ¼ hSjPjXi ¼ hSjPjYi ¼ hSjPjZi

hSjPjXi ¼

Zu�Cð0;~rÞPuVxð0;~rÞd

3r ¼ PCVxð0Þ

hSjPjYi ¼ PCVyð0Þ

and

hSjPjZi ¼ PCVzð0Þ

Thus,

P ¼ PCVxð0Þ ¼ PCVyð0Þ ¼ PCVzð0Þ ¼ PCVð0Þ

where

PCVð0Þ ¼

Zu�Cð0;~rÞPuVð0;~rÞd

3r ¼ P

For a plane polarized light wave, we have the polarization vector~�s ¼ k, when the light wave vector is traveling along the z-axis.Therefore, for a plane polarized light wave, we have considered~�s ¼ k.

Then, from Eq. (29), we can write

ð~�0s �~PCVð~kÞÞ ¼ ~k �

P

2r3ðir1 � r2Þ½Að~kÞ þ Bð~kÞ� cos ot (30)

and

Að~kÞ ¼ ak�

bkþffiffiffi2p þ ckþ

� �

Bð~kÞ ¼ akþ

bk�ffiffiffi2p þ ck�

� �(31)

Thus,

j~�0s �~pcvð~kÞj2 ¼ k �

P

2r3

��������2

jir1 � r2j2½Að~kÞ þ Bð~kÞ�2

� cos2 ot ¼1

4P2

Z cos2 y½Að~kÞ þ Bð~kÞ�2 cos2 ot (32)

So, the average value of j�s � pcvðkÞj2 for a plane polarized light

wave is given by

hj~�s �~pcvð~kÞj2iav ¼

2

4P2

z ½Að~kÞ þ Bð~kÞ�2

Z 2p

0dfZ p

0cos2 y sin ydy

!1

2

� �

¼p3

P2z ½Að

~kÞ þ Bð~kÞ�2 (33)

where

P2z ¼ j

~k �~pcvð0Þj2

and

j~k �~pcvð0Þj2 ¼

m2

4mr

Eg0ðEg0þDÞ

ðEg0þ ð2=3ÞDÞ

(34)

We shall express Að~kÞ and Bð~kÞ in terms of constants of theenergy spectra in the following way.

Substituting ak� , bk� , ck� and g0k� in Að~kÞ and Bð~kÞ in Eq. (31), weget

Að~kÞ ¼ b t þrffiffiffi2p

� �Eg0

Eg0þ d0

!g2

0kþ� g2

0kþg2

0k�

Eg0� d0

Eg0þ d0

!( )1=2

(35)

Bð~kÞ ¼ b t þrffiffiffi2p

� �Eg0

Eg0þ d0

!g2

0k�� g2

0kþg2

0k�

Eg0� d0

Eg0þ d0

!( )1=2

(36)

in which

g20kþ¼

x1k � Eg0

2 x1k þ d0� ¼ 1

21�

Eg0þ d0

x1k þ d0

� �� �

and

g20k�¼

x1k þ Eg0

2ðx1k þ d0Þ¼

1

21þ

Eg0� d0

x1k þ d0

� �� �

Substituting x ¼ x1k+d0 in g20k�

, we can write

Að~kÞ ¼ b t þrffiffiffi2p

� �Eg0

Eg0þ d0

!1

21�

Eg0þ d0

x

� �(

�1

4

Eg0� d0

Eg0þ d0

!1�

Eg0þ d0

x

� �1þ

Eg0� d0

x

� �)

Thus

Að~kÞ ¼b2

t þrffiffiffi2p

� �1�

2a0

xþ

a1

x2

� �1=2

where a0 ¼ ðE2g0þ d02ÞðEg0

þ d0Þ�1 and a1 ¼ ðEg0� d0Þ2.

After tedious algebra, one can show that

Að~kÞ ¼b2

t þrffiffiffi2p

� �ðEg0� d0Þ

1

x1k þ d0�

1

Eg0þ d0

" #1=2

�1

x1k þ d0�ðEg0þ d0Þ

ðEg0� d0Þ2

" #1=2

(37)

Similarly, from Eq. (36), we can write

Bð~kÞ ¼ b t þrffiffiffi2p

� �Eg0

Eg0þ d0

!1

21þ

Eg0� d0

x

� �(

�1

4

Eg0� d0

Eg0þ d0

!1�

Eg0þ d0

x

� �1þ

Eg0� d0

x

� �)1=2

So that, finally we get

Bð~kÞ ¼b2

t þrffiffiffi2p

� �1þ

Eg0� d0

x1k þ d0

� �(38)

Using Eqs. (33), (34), (37) and (38) we can write

eA0

2m

� �2hj~�s �~pcvð

~kÞj2iav

Ecð~kÞ � Evð

~kÞ

¼eA0

2m

� �2 p3j~�s �~pcvð0Þj

2 b2

4t þ

rffiffiffi2p

� �2

�1

x1k1þ

Eg0� d0

x1k þ d0

� �þ ðEg0

� d0Þ1

x1k þ d0�

1

Eg0þ d0

" #1=28<:

�1

x1k þ d0�

Eg0þ d0

ðEg0� d0Þ2

" #1=29=;

2

(39)

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–41504144

Following Nag [33], it can be shown that

A20 ¼

Il2

2p2c3ffiffiffiffiffiffiffiffiffiffiffi�sc�0p (40)

where I is the light intensity of wavelength l, c is the velocity oflight and e0 is the permittivity of vacuum. Thus, the simplifiedelectron energy spectrum in III–V, ternary and quaternarymaterials up to the second order in the presence of light wavescan approximately be written as

_2k2

2mc¼ b0 E; lð Þ (41)

where

b0ðE; lÞ ¼ ½gðEÞ � y0ðE;lÞ�

y0ðE; lÞ ¼e2

96mrpc3

Il2ffiffiffiffiffiffiffiffiffiffiffi�sc�0p

Eg0ðEg0þ DÞ

Eg0þ 2

3D� b2

4t þ

rffiffiffi2p

� �2 1

f0ðEÞ

� 1þEg0� d0

f0ðEÞ þ d0

� �þ ðEg0

� d0Þ1

f0ðEÞ þ d0�

1

Eg0þ d0

" #1=28<:

�1

f0ðEÞ þ d0�

Eg0þ d0

ðEg0� d0Þ2

" #1=29=;

2

and

f0ðEÞ ¼ Eg01þ 2 1þ

mc

mv

� �gðEÞEg0

� �1=2

Thus, under the limiting condition ~k! 0, from Eq. (41), weobserve that E6¼0 and is positive. Therefore, in the presence ofexternal light waves, the energy of the electron does not tend tozero when ~k! 0, whereas for the unperturbed three-band modelof Kane, g(E) ¼ (_2k2)/2mc in which E-0 for ~k! 0. As the CB istaken as the reference level of energy, therefore the lowestpositive value of E for ~k! 0 provides the increased band gap(DEg) of the semiconductor due to photon excitation. The values ofthe increased band gap can be obtained by computer iterationprocesses for various values of I and l, respectively.

2.1.1. Special cases

(1)

For the two-band model of Kane, we have D-0. Under thiscondition, g(E)-E(1+aE) ¼ (_2k2)/2mc with a ¼ 1=Eg0

. Since,b-1, t-1, r-0, d0-0 for D-0, from Eq. (41), we can writethe energy spectrum of III–V, ternary and quaternarymaterials in the presence of external photo-excitation whoseunperturbed conduction electrons obey the two-band modelof Kane as

_2k2

2mc¼ o0ðE; lÞ (42)

where

o0ðE;lÞ ¼ Eð1þ aEÞ � B0ðE; lÞ

B0ðE; lÞ ¼e2Il2Eg0

384pc3mrffiffiffiffiffiffiffiffiffiffiffi�sc�0p

1

f1ðEÞ1þ

Eg0

f1ðEÞ

� �þ Eg0

1

f1ðEÞ�

1

Eg0

� �� �2

f1ðEÞ ¼ Eg01þ

2mc

mr

Eð1þ aEÞ

Eg0

� �1=2

(2)

In the case of relatively wide band gap semiconductor, we canwrite, a-0, b-0, c-0 and g(E)-E.

Thus, from Eq. (42), we can find

_2k2

2mc¼ r0ðE; lÞ (43)

r0ðE; lÞ ¼ E�e2Il2

96pc3mrffiffiffiffiffiffiffiffiffiffiffi�sc�0p 1þ

2mc

mr

� �E

Eg0

� ��3=2

(44)

2.2. Formulation of the DSL in the presence of light waves in

ultrathin films of optoelectronic materials

The 2D DSL (L2D) can, in general, Refs. [11,34], be written as

L2D ¼e2

2�sc

qn2D

qEF

� �� ��1

(45)

Where esc is the semiconductor permittivity, EF is the Fermienergy. It appears then that the formulation of the 2D DSLrequires an expression of electron statistics, which, in turn, isdetermined by the DOS function:

(i)

The 2D electron energy spectrum in ultrathin films of III–V,ternary and quaternary materials, whose unperturbed bandstructure is defined by the three-band model of Kane, in thepresence of light waves can be expressed from Eq. (41) as

_2k2s

2mcþ

_2

2mc

nzpdz

� �2

¼ b0ðE;lÞ (46)

where k2s ¼ k2

x þ k2y , nz ( ¼ 1, 2, 3, y) is the size quantum

number along z-direction and dz is the film thickness alongthe z-direction.The sub-band energies ðEnz Þcan be written as

b0ðEnz ; lÞ ¼_2

2mcðnzp=dzÞ

2 (47)

The DOS function is given by

N2DðE; lÞ ¼mc

p_2

� �Xnzmax

nz¼1

½b00ðE;lÞ�HðE� Enz Þ (48)

where the prime indicates the differentiation of the differ-entiable functions with respect to E and H is the Heavisidestep function.Combining Eq. (48) with the Fermi-Dirac occupation prob-ability factor, the surface electron concentration can thus bewritten as

n2D ¼mc

p_2

Xnzmax

nz¼1

½T1ðEF;nz; lÞ þ T2ðEF;nz; lÞ� (49)

where T1(EF, nz, l)�[b0(EF, l)�((_2)/2mc)((nzp)/dz)2], T2

ðEF;nz; lÞ �Ps0

r¼1ZrT1ðEF;nz; lÞ, Zr�2(kBT)2r(1�21�2r)z(2r)((q2r)/(qEF

2r)), z(2r) is the zeta function of order 2r and r is the set ofpositive integers whose upper limit is S0 [35] and EF is theFermi energy in the presence of light waves as measured fromthe edge of the CB in the absence of any field in the verticallyupward direction.The use of Eqs. (49) and (45) leads to the expression of the 2DDSL in this case as

L2D ¼e2mc

2�scp_2

!Xnzmax

nz¼1

½T 01ðEF;nz;lÞ þ T 02ðEF;nz; lÞ�

( )�1

(50)

where the primes denote the first-order differentiation of thedifferentiable functions with respect to EF.

(ii)

Using Eq. (42), the expressions for the 2D dispersion relation,the sub-band energies, the DOS function and the surfaceelectron concentration for ultrathin films of III–V, ternary and

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150 4145

quaternary materials, whose unperturbed band structure isdefined by the two-band model of Kane, can respectively bewritten in the presence of photo-excitation as

_2k2s

2mcþ

_2

2mc

nzpdz

� �2

¼ o0ðE; lÞ (51)

o0ðEnz ; lÞ ¼_2

2mcðnzp=dzÞ

2 (52)

N2DðE;lÞ ¼mc

p_2

� �Xnzmax

nz¼1

½o00ðE; lÞ�HðE� Enz Þ (53)

n2D ¼mc

p_2

Xnzmax

nz¼1

½T3ðEF;nz; lÞ þ T4ðEF;nz; lÞ� (54)

where T3(EF, nz, l)�[o0(EF, l)�(_2/(2mc))((nzp)/dz))2],

T4ðEF;nz; lÞ �Ps0

r¼1ZrT3ðEF;nz; lÞ.The use of Eqs. (54) and (45) leads to the expression of the 2DDSL in this case as

L2D ¼e2mc

2�scp_2

!Xnzmax

nz¼1

½T 03ðEF;nz; lÞ þ T 04ðEF;nz; lÞ�

( )�1

(55)

(iii)

Using Eq. (43), the expressions for the 2D dispersion relation,the sub-band energies, the DOS function and the electronconcentration for ultrathin films of III–V, ternary andquaternary materials, whose unperturbed band structure isdefined by the parabolic energy bands, can respectively bewritten in the presence of photo-excitation as

_2k2s

2mcþ

_2

2mc

nzpdz

� �2

¼ r0ðE; lÞ (56)

r0ðEnz ; lÞ ¼_2

2mcðnzp=dzÞ

2 (57)

N2DðE; lÞ ¼mc

p_2

� �Xnzmax

nz¼1

½r00ðE; lÞ�HðE� Enz Þ (58)

n2D ¼mc

p_2

Xnzmax

nz¼1

½T5ðEF;nz; lÞ þ T6ðEF;nz; lÞ� (59)

where T5(EF, nz, l)�[r0(EF, l)�(_2/(2mc))((nzp)/dz))2], T6ðEF;nz; lÞ �Ps0

r¼1ZrT5ðEF;nz; lÞ.The use of Eqs. (54) and (45) leads to the expression of the 2D

DSL in this case as

L2D ¼e2mc

2�scp_2

!Xnzmax

nz¼1

½T 05ðEF;nz; lÞ þ T 06ðEF;nz;lÞ�

( )�1

(60)

2.3. Formulation of the 2D DSL in the absence of light waves in

ultrathin films of optoelectronic materials

(i)

The expressions for the 2D dispersion relation, the sub-bandenergies, the DOS function and the surface electron concen-tration for ultrathin films of optoelectronic materials, whoseunperturbed band structure is defined by the three-bandmodel of Kane, can respectively be written in the absence of

photo-excitation as

_2k2s

2mcþ

_2

2mc

nzpdz

� �2

¼ gðEÞ (61)

gðEnz Þ ¼_2

2mcðnzp=dzÞ

2 (62)

N2DðEÞ ¼mc

p_2

� �Xnzmax

nz¼1

½g0ðEÞ�HðE� Enz Þ (63)

n2D ¼mc

p_2

Xnzmax

nz¼1

½T7ðEF0;nzÞ þ T8ðEF0

;nz� (64)

where EF0is the Fermi energy in the presence of size

quantization as measured from the edge of the CB in thevertically upward direction in the absence of external photo-excitation,

T7ðEF0;nzÞ � gðEF0

Þ �_2

2mc

nzpdz

� �2" #

and

T8ðEF0;nzÞ �

Xs0

r¼1

ZrT7ðEF0;nzÞ

The use of Eqs. (64) and (45) leads to the expression of theDMR in this case as

L2D ¼e2mc

2�scp_2

!Xnzmax

nz¼1

½T 07ðEF0;nzÞ þ T 08ðEF0

;nz�

( )�1

(65)

(ii)

The expressions for the 2D dispersion relation, the sub-bandenergies, the DOS function and the surface electron concen-tration for ultrathin films of optoelectronic materials, whoseunperturbed band structure is defined by the two-bandmodel of Kane, can respectively be written in the absence ofphoto-excitation as

Eð1þ aEÞ �_2

2mcðnzp=dzÞ

2þ_2k2

s

2mc(66)

Enz ð1þ aEnz Þ �_2

2mcðnzp=dzÞ

2 (67)

N2DðEÞ ¼mc

p_2

Xnzmax

nz¼1

½1þ 2aE�HðE� Enz Þ (68)

n2D ¼mckBT

p_2

Xnzmax

nz¼1

½ð1þ 2aEnz ÞF0ðZnÞ þ 2akBTF1ðZnÞ� (69)

where Fj(Zn) is the Fermi-Dirac integral of order j, Zn �

ðEF0� Enz Þ=kBT is the one parameter Fermi–Dirac integral of

order t0 which can be written as [36]

Ft0ðZÞ ¼ 1

Gðt0 þ 1Þ

� �Z 10

yt0 ð1þ expðy� ZÞÞ�1dy; y4� 1

(70)

where G(t0+1) is the complete gamma function or for all t,analytically continued as a complex contour integral aroundthe negative axis

Ft0ðZÞ ¼ At0

Z ð0þÞ�1

yt0 ð1þ expð�y� ZÞÞ�1 dy (71)

in which At0¼ ðGð�t0ÞÞ=ð2p

ffiffiffiffiffiffiffi�1p

Þ.

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–41504146

ARTICLE IN PRESS

Fig. 1the th

conce

and t

(a–c)

ultrat

S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150 4147

The use of Eqs. (69) and (45) leads to the expression of the 2DDSL in this case as

L2D ¼e2mc

2�scp_2kBT

!Xnzmax

nz¼1

½ð1þ 2aEnz F�1ðZnÞ þ 2akBTF0ðZnÞÞ�

( )�1

(72)

(iii)

Under the condition a-0 as for ultrathin films of parabolicenergy bands, the expressions for the 2D dispersion relation,the sub-band energies, the DOS function, the surface electronconcentration and the 2D DSL can be written as

E �_2

2mcðnzp=dzÞ

2þ_2k2

s

2mc(73)

Enz �_2

2mcðnzp=dzÞ

2 (74)

N2DðEÞ ¼mc

p_2

Xnzmax

nz¼1

HðE� Enz Þ (75)

n2D ¼mckBT

p_2

Xnzmax

nz¼1

½F0ðZnÞ� (76)

L2D ¼e2mc

2�scp_2kBT

!Xnzmax

nz¼1

½F0ðZnÞ�

( )�1

(77a)

Under the condition of non-degeneracy, Eqs. (76) and (77a)assume the forms

n2D ¼mckBT

p_2

Xnzmax

nz¼1

½expðZnÞ� (77b)

L2D ¼e2mc

2�scp_2kBT

!Xnzmax

nz¼1

½expðZnÞ�

( )�1

(77c)

Using Eqs. (77b) and (77c), we get

L2D ¼ ½2�sckBT=e2n2D� (77d)

The classical 2D DSL equation of wide gap non-degeneratematerials has been obtained in Eq. (77d) as a special case of ourgeneralized analysis under certain limiting conditions from all theresults. This indirect test not only exhibits the mathematicalcompatibility of our formulation but also shows the fact that oursimple analysis is a more generalized one, since one can obtainthe corresponding results for relatively wide gap 2D materialshaving parabolic energy bands under certain limiting conditionsfrom our present derivation.

2.4. Suggestion for the experimental determination of the DSL in

materials having arbitrary dispersion laws

It is well known that the thermoelectric power of the electronsin materials in the presence of a classically large magnetic field isindependent of the scattering mechanism and is determined onlyby the dispersion law [37]. The magnitude of the thermoelectric

. (a) Plot of the normalized 2D DSL as a function of film thickness for ultrathin films

ree and two band models of Kane with that of the parabolic energy bands, re

ntration per unit area for ultrathin films of n-Hg1�xCdxTe in the presence of light

hat of the parabolic energy bands, respectively. (c) Plot of the normalized 2D DSL as

represent the three and two band models of Kane and that of parabolic energy ba

hin films of n-Hg1�xCdxTe in which the curves (a–c) represent the three and two

power G in the present case can be expressed as [37]

G ¼1

eTn0

Z 1�1

ðE� EFÞRðEÞ �qf 0

qE

� �dE (78)

where R(E) is the total number of states. Following Tsidilkovski[38], Eq. (78) can be written under the condition of carrierdegeneracy as

G ¼p2k2

BT

3n0

!qn0

qEF

� �(79)

The use of Eqs. (79) and (45) leads to the result

L2D ¼2p2k2

BT�sc

3e3n2DG

!(80)

Thus, the 2D DSL for degenerate materials can be determined byknowing the experimental values of G.

From the suggestion for the experimental determination of theDSL for degenerate materials having arbitrary dispersion laws asgiven by Eq. (80), we observe that for a constant T, the DSL variesinversely with Gn2D. Only the experimental values of G for anymaterial as a function of electron concentration will generate theexperimental values of the 2D DSL for that range of n0 for thatmaterial. Since (Gn2D)�1 decreases with increasing n2D forconstant T, from Eq. (80) we can conclude that the 2D DSL willdecrease with increasing n2D. This statement provides a compat-ibility test of our theoretical analysis. Eq. (80) provides anexperimental check of the 2D DSL and also a technique forprobing the band structures of the materials having arbitrary bandstructures. Thus, the 2D DSL for degenerate materials can bedetermined by knowing the experimental values of G. Thesuggestion for the experimental determination of the 2D DSL fordegenerate materials having arbitrary dispersion laws as given byEq. (80) does not contain any band parameters. This statementprovides a compatibility test of our theoretical analysis. Eq. (80)provides an experimental check of the diffusivity–mobility ratio(DMR) and also a technique for probing the band structures of thematerials having arbitrary band structures.

2.5. Two applications of the results of this paper in the field of

nanoelectronics in general

(i)

of n

spec

wave

a fun

nds, r

band

It is well known that the Einstein relation for the DMR is animportant quantity for studying the transport properties ofmodern semiconductor devices since the diffusion constant(a quantity very useful for device analysis but whose exactexperimental determination is rather difficult) can be derivedfrom this ratio if one knows the experimental values of themobility [39]. In addition, it is more accurate than any of theindividual relations for the diffusivity or the mobility, whichare the two widely used features of electron transport.Besides, the performance of the electronic devices at deviceterminal and the speed of operation of modern switchingtransistors are significantly influenced by the degree of carrierdegeneracy present in these devices. The simplest way ofanalyzing such devices taking into account the degeneracy ofbands is to use the appropriate Einstein relation to express theperformance at the device terminals and switching speed interms of carrier concentration [40]. The 2D Einstein relation

-Hg1�xCdxTe in the presence of light waves in which the curves (a–c) represent

tively. (b) Plot of the normalized 2D DSL as a function of surface electron

s in which the curves (a–c) represent the three and two band models of Kane

ction of light intensity for ultrathin films of n-Hg1�xCdxTe in which the curves

espectively. (d) Plot of the normalized 2D DSL as a function of wavelength for

models of Kane and that of parabolic energy bands, respectively.

ARTICLE IN PRESS

S. Bhattacharya et al. / Physica B 403 (2008) 4139–41504148

can, in general, be written as [41]

D

m¼ðn2D=eÞ

ðqn2D=qEFÞ(81)

Using Eqs. (81) and (45), we get

D

m ¼ p2kBT=ð3e2GÞ (82)

Therefore, we can experimentally determine the 2D DMRfor any materials having arbitrary dispersion laws by knowingthe experimental values of G. Since G decreases withincreasing n2D, from Eq. (82), we can infer that the DMR willincrease with increase in n2D and this compatibility is thedirect test of our suggestion for experimental determinationof the 2D DMR.

(ii)

The knowledge of the carrier contribution to the elasticconstants is very important in studying the mechanicalproperties of the nanomaterials and has been relatively beenless investigated in the literature [42]. The electroniccontribution to the second- and third-order elastic constantscan be written as [42,43]

DC44 ¼ �a2

0

9dz

qn2D

qEF

(83)

and

DC456 ¼a3

0

27dz

q2n2D

qE2F

(84)

where a0 is the deformation potential constant. Thus, usingEqs. (83), (84) and (79), we can write

DC44 ¼ ½�n2Da20eG=ð3dzp2k2

BT� (85)

and

DC456 ¼ n2Dea30G2=ð3p4k3

BTdzÞ

� �1þ

n2D

G

qG

qn2D

� �(86)

Thus, the experimental graph of G versus n2D allows us todetermine the electronic contribution to the elastic constantsfor materials having arbitrary dispersion laws.

Thus, we can summarize the whole mathematical backgroundin the following way: in this paper, we have investigated the 2DDSL in ultrathin films of optoelectronic materials in the presenceof photo-excitation on the basis of a newly formulated electrondispersion law whose unperturbed conduction electrons obey thethree- and two-band models of Kane together with parabolicenergy bands. Under certain special conditions, we have alsoobtained the results for materials whose unperturbed electronenergy spectra are defined by the two-band model of Kane andthat of the parabolic energy bands. We have also investigated the2D DSL both in the presence and absence of photo-excitation inbulk specimens of the aforementioned materials. The classical 2DDSL equation of wide gap non-degenerate materials has beenobtained in Eq. (77d) as a special case of our generalized analysisunder certain limiting conditions from all the results. This indirecttest not only exhibits the mathematical compatibility of ourformulation but also shows the fact that our simple analysis is amore generalized one, since one can obtain the correspondingresults for relatively wide gap 2D materials having parabolicenergy bands under certain limiting conditions from our presentderivation. In addition, we have suggested an experimentalmethod for the determination of 2D DSL, for nanomaterialshaving arbitrary dispersion laws in the presence of light togetherwith two new suggestions for the measurement of band gap ofsemiconductors in this context. Besides, the results of this paper

find two specific applications namely the Einstein relation and thecarrier contribution to the elastic constants and in the realm ofnanodevices.

3. Result and discussions

Using the appropriate equations, we have plotted the 2D DSL asa function of film thickness at T ¼ 4.2 K in the presence of photo-excitation for ultrathin films of n-Hg1�xCdxTe;(x ¼ 0.3, mv ¼ 0.4,m0 ¼ 12.642 (D ¼ 0.063+0.24x+0.27x2) eV, Eg0

¼ ½�0:302þ 1:93xþ

5:25�10�4Tð1� 2xÞ � 0:810x2+0:832x3� eV) [43] whose unper-turbed electron dispersion laws are defined by the three- andtwo-band model of Kane together with the parabolic energy bandsas shown by curves (a–c) of Fig. 1a, respectively. Fig. 1b–d exhibitsthe dependencies of the 2D DSL on surface electron concentration,light intensity and wavelength, respectively. All the plots havebeen normalized to the value of the DSL, as given in theintroduction.

It should be noted that the 2D DSL decreases from the light-offcase to the light-on case, since the value of the Fermi energy in thepresence of light waves becomes larger due to the increase in thecarrier concentration as compared with the same in the absenceof photo-excitation. Therefore, the numerical magnitude of the 2DDSL in the presence of light is smaller as compared with the samein the light-off case in the whole range of the appropriatevariables as considered, although the 2D DSL decreases withincrease in said variables.

The combined influence of the energy band constants on the2D DSL for all the said compounds can easily be assessed from allthe figures. From Fig. 1c, we observe that the 2D DSL decreaseswith increasing light intensity whereas in the absence of externalphoto-excitation, the same is independent of intensity. Fig. 1dexhibits the fact that the 2D DSL decreases as the wavelengthshifts from red to violet color. For the ternary material, we havetaken x ¼ 0.3, since for xo0.17, the band gap becomes negative inn-Hg1�xCdxTe leading to semi-metallic state.

The influence of quantum confinement on the aforementionedmaterials is immediately apparent from all the figures, since the2D DSL depends strongly on the thickness of the size-quantizedmaterials, which is in direct contrast with their respective bulkspecimens. Moreover, the 2D DSL for ultrathin films can becomeseveral orders of magnitude larger than of their bulk specimens,which is also a direct signature of quantum confinement.It appears from the said figures that the 2D DSL decreaseswith the increasing film thickness in a step-like manner, both inthe presence and absence of photo-excitation for all types ofmaterials as considered here, although, the numerical valuesvary widely and determined by the constants of the energyspectra. The oscillatory dependence is due to the crossing over ofthe Fermi level by the size-quantized levels. For each coincidenceof a size-quantized level with the Fermi level, there would be adiscontinuity in the DOS function resulting in a peak ofoscillations. With large values of film thickness, the height ofthe steps decreases and the 2D DSL will decrease with increasingfilm thickness in non-oscillatory manner and exhibit monotonicdecreasing dependence. The height of step size and the rate ofdecrement are totally dependent on the band structure.The influence of light is immediately apparent from the plots inFig. 1c and d, since the 2D DSL depends strongly on I and l,which is in direct contrast as compared with the correspondingcases for ultrathin films in the absence of external photo-excitation, respectively. The variations of the 2D DSL in allthe figures reflect the direct signature of the light waves on theelectronic, optic and the other band structure-dependent proper-ties of semiconducting materials in the presence of light waves

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and the photon-assisted transport for the corresponding semi-conductor devices, since the incident photons drastically changethe electron dispersion law. From the figures, we observe that the2D DSL decreases with increasing film thickness, intensity,wavelength and surface electron concentration, together withthe fact that the rate of variation is totally band structuredependent.

It appears from Fig. 1b that the 2D DSL decreases withincreasing carrier degeneracy, which exhibits the signaturesof the 1D confinement through the step-like dependence.This oscillatory dependence will be less and less prominentwith increasing film thickness and carrier concentration, respec-tively. Ultimately, for bulk specimens of the same material, theDSL will be found to decrease continuously with increasingelectron concentration in a non-oscillatory manner. The appear-ance of the humps of the respective figures is due to theredistribution of the electrons among the quantized energy levelswhen the size quantum number corresponding to the highestoccupied level changes from one fixed value to the others. Withvarying electron concentration, a change is reflected in the 2D DSLthrough the redistribution of the electrons among the size-quantized levels.

In this paper, we have suggested the experimental determina-tions of the 2D DSL, the Einstein relation for the diffusivity-to-mobility ratio and the carrier contribution to the elasticconstants and our suggestions are valid for ultrathin films havingarbitrary dispersion relations. Since the experimental curves ofn2D versus G are not available in the literature to the best of ourknowledge for the present generalized systems, we cannotcompare our theoretical formulation with the proposed experi-ment although, the generalized analysis as presented in thiscontext can be checked when the experimental investigations of G

would appear in the literature.Thus, we can conclude that the influence of the presence

of an external photo-excitation is to change radically the originalband structure of the material. Our method is not at all relatedto the DOS technique as used in the literature [44]. From theE–k dispersion relation, we can obtain the DOS, but the DOStechnique as used in the literature [44] cannot provide the E–kdispersion relation. Therefore, our study is more fundamentalthan those of the existing literature because the Boltzmanntransport equation, which controls the study of the chargetransport properties of semiconductor devices, can be solvedif and only if the E–k dispersion relation is known. It maybe remarked that, in recent years, the carrier statistics has beenextensively been studied [45], but the screening length of the non-parabolic materials has relatively been less investigated. Wewish to note that we have not considered the hot electron andmany-body effects in this simplified theoretical formalism dueto the lack of availability in the literature of proper analyticaltechniques for including them for the generalized systemsas considered in this paper. Our simplified approach will beuseful for the purpose of comparison when methods of tack-ling the formidable problem after inclusion of the many-bodyand the hot electron effects for the present generalized systemsappear. The inclusion of the said effects would certainly increasethe accuracy of the results, although the qualitative features ofthe 2D DSL, as discussed in this paper, would not change in thepresence of the aforementioned effects. Our suggestions forthe experimental determinations of the 2D DMR and the elasticconstants are independent of the inclusion of the said effects.We have not considered other types of optoelectronic materialsand other external variables in order to keep the presentationbrief.

The numerical results presented in this paper would bedifferent for other materials but the nature of variation would

be unaltered. The theoretical results as given here would be usefulin analyzing various other experimental data related to thisphenomenon. Finally, we can write that this theory can be used toinvestigate the Burstien Moss shift, the effective electron mass,the specific heat and other different transport coefficients ofmodern ultrathin film semiconductor devices operated under theinfluence of external photon field.

Acknowledgement

The authors K. P. Ghatak and D. De are grateful to All IndianCouncil for Technical Education for granting the project having thereference number 8023/BOR/RID/RPS-95/2007-08 under researchpromotion scheme 2008 under which this research paper hasbeen completed.

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