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

ARTICLE IN PRESS Physica B 403 (2008) 4139–4150

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Influence of light on the Debye screening length in ultrathin films of 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, India Department of Metallurgical Engineering, National Institute of Technology, Agartala, Tripura (West) 799 055, India Department of Computer Science and Engineering, West Bengal University of Technology, B.F. 142, Sector 1, Salt Lake City, Kolkata 700 064, India d Department of Electronic Science, The University of Calcutta, 92, Acharyya Prafulla Road, Kolkata, West Bengal 700 009, India b c

a r t i c l e in fo

abstract

Article history: Received 2 July 2008 Received in revised form 25 August 2008 Accepted 26 August 2008

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-Hg1xCdxTe, 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.

PACS: 71.55.Eq Keywords: Ultra-thin films Debye screening length Optoelectronic Light waves Experimental determination

1. Introduction It is well known that the Debye screening length (DSL) of the carriers in semiconductors is a very important quantity characterizing the screening of the Coulomb field of the ionized impurity centers by the free carriers [1]. It affects many of the special features of modern nanodevices, the carrier mobilities under different mechanisms of scattering and the carrier plasmas in semiconductors [2]. The DSL is a very good approximation to the accurate selfconsistent screening in presence of band tails and is also used to illustrate the interaction between the colliding carriers in Auger effect in solids [1]. In the conventional form, the DSL decreases with increasing n2D at a constant temperature and this relation holds only under the condition of carrier non-degeneracy. Since, the performance of the electron devices at the device terminals and the speed of operation of modern switching transistors are significantly influenced by the degree of carrier degeneracy present in these devices [3], the simplest way of analyzing such devices taking into account of the degeneracy of the band is to use the appropriate DSL

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E-mail address: [email protected] (K.P. Ghatak). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.08.017

to express the performance at the device terminal and switching speed 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 of optoelectronic and related compounds having various band structures [2]. It is well known from the fundamental study of Landsberg [1] that the DSL for electronic materials having degenerate electron concentration is essentially determined by their respective energy band structures. It has, therefore, different values in different materials and varies with the electron concentration, with the magnitude of the reciprocal quantizing magnetic field under magnetic quantization, with the quantizing electric field as in inversion layers, with the nanothickness as in quantum wells, with superlattice period as in the quantum-confined superlattices of small-gap compounds with graded interfaces having various carrier energy spectra. The nature of these variations has been investigated by 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 and such variations are significantly influenced by constants of the energy band spectra;

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(b) the DSL decreases with the magnitude of the quantizing electric field as in inversion layers; (c) the DSL oscillates with the inverse quantizing magnetic field under magnetic quantization due to the SdH effect; and (d) the DSL exhibits composite oscillations with the various controlled parameters as in superlattices of non-parabolic compounds with graded interfaces.

The above information has been obtained through theoretical analyses and no experimental results are available to the knowledge of the authors in support of the predictions for ultrathin films having arbitrary dispersion relations in the presence of light waves. In this paper, we have studied the two-dimensional (2D) DSL for ultrathin films of optoelectronic compounds in the presence of light waves in addition to the suggestion of the experimental determination of the same. In this context, it may be noted that with the advent of fine lithographical methods [13], molecular beam epitaxy [14], organometallic vapor-phase epitaxy [15] and other experimental techniques, low-dimensional structures having quantum confinement of one (inversion layers, accumulation layers, nipi structures and ultrathin films), two (quantum well wires) and three dimensions (quantum dots, magneto inversion layers, magneto accumulation layers, magneto size quantization, quantum well superlattices under magnetic quantization, quantum dot superlattices and magneto nipi structures) have, in the last few years, attracted much attention not only for their potential in uncovering new phenomena in nanoscience but also for their interesting quantum device applications [16–18]. In ultrathin films, the restriction of the motion of the carriers in the direction normal to the film (say, the z-direction) may be viewed as carrier confinement in an infinitely deep one-dimensional (1D) square potential well, leading to quantization [known as quantum size effect (QSE)] of the wave vector of the carrier, allowing 2D electron transport parallel to the surface of the film representing new physical features not exhibited in bulk semiconductors [19]. The low-dimensional heterostructures based on various materials are widely investigated because of the enhancement of carrier mobility [20]. These properties make such structures suitable for applications in quantum well lasers [21], heterojunction FETs [22], high-speed digital networks [23], high-frequency microwave circuits [24], optical modulators [25], optical switching systems [26] and other devices. We shall use n-Hg1xCdxTe matched as an example of ternary materials. The ternary alloy n-Hg1xCdxTe is a classic small-gap material and is an important optoelectronic compound because its energy band gap can be varied to cover a spectral range from 0.8 mm to over 30 mm by varying the alloy composition [27]. The n-Hg1xCdxTe is being extensively used in infrared detector materials [28] and photovoltaic detector arrays [29] in the 8–12 mm wave bands. The aforementioned applications have spurred an Hg1xCdxTe technology for the generation of high mobility single crystals, with specially prepared surface layers and the same compound is ideally suitable for narrow sub-band physics because the relevant energy band constants are within easy experimental reach [30]. In Section 2.1 of the theoretical background, we have formulated the dispersion relation of the conduction electrons of optoelectronic materials in the presence of photo-excitation, whose unperturbed energy band structures are defined by the three- and two-band models of Kane together with the parabolic energy bands for the purpose of relative comparison. The expressions for the surface electron concentration and DSL for ultrathin films of the aforementioned materials in the presence and absence of photo-excitation have been formulated in Sections 2.2 and 2.3, respectively. In Section 2.4, we have suggested an

experimental method of determination of the DSL in semiconductor nanostructures having arbitrary carrier energy spectra in the present case. In Section 2.5, we have written two specific applications of the results of this paper in the fields of computational and theoretical nanostructures. The 2D DSL has been numerically investigated by taking n-Hg1xCdxTe as an example of ternary compounds in accordance with the three- and the two-band models of Kane together with parabolic energy bands, respectively, for the purpose of relative comparison both in the 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 ¯ ) of an electron in the presence of light The Hamiltonian (H wave characterized by the vector potential ~ A can be written in the following [31] form: H ¼ ½ðp¯ þ e~ AÞ2 =2m þ 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 ¯ ¼H ¯0 þH ¯0 H

(2)

where p2 ¯ 0 ¼ ¯ þ Vð~ rÞ H 2m and ¯0 ¼ H

e ~ A ~ p 2m

(3) 00

¯ can be written as The perturbed Hamiltonian H   i_e ~ ¯0 ¼ ðA  rÞ (4) H 2m pffiffiffiffiffiffiffi p ¼ i_r. where i ¼ 1, _ ¼ h/2p, h is Planck constant, ~ 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)

s is the polarization where A0 is the amplitude of the light wave, ~ vector, ~ s0 is the momentum vector of the incident photon, ~ r is the position vector, o is the angular frequency of light wave and t is ¯ 0nl between initial state, the time scale. The matrix element of H ~ ~ ~ ~ cl ðq; rÞ and final state cn ðk; rÞ in different bands can be written as ¯ 0nl ¼ H

e hn~ kj~ A ~ pjl~ qi 2m

(6)

Using Eqs. (4) and (5), we can re-write Eq. (6) as   i_eA0 ¯ ði~s0 ~rÞ rjl~ ¯ 0nl ¼ ~ s  ½fhnkje qieiot g þ fhn~ kjeði~s0 ~rÞ rl~ qieiot g H 4m (7) The first matrix element of Eq. (7) can be written as Z 3 hn~ kj expði~ s0  ~ rÞrjl~ qi ¼ expði½~ q þ~ s0  ~ k  ~ rÞi~ qun ð~ k; ~ rÞul ð~ q; ~ rÞ d r Z 3 þ expði½~ q þ~ s0  ~ k  ~ rÞun ð~ k; ~ rÞrul ð~ q; ~ rÞ d r (8) The functions un ul and un rul are periodic. The integral over all spaces can be separated into a sum over unit cells times an integral over a single unit cell. It is assumed that the wavelength

ARTICLE IN PRESS S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150

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of the electromagnetic wave is sufficiently large so that if ~ k and ~ q are within the Brillouin zone, ð~ q þ~ s0  ~ kÞ is not a reciprocal lattice vector. Therefore, we can write Eq. (8) as

and

hn~ kj expði~ s0  ~ rÞrjl~ qi " # Z 3  ð2pÞ q þ~ s0  ~ kÞdnl þ dð~ q þ~ s0  ~ kÞ iq¯ dð~ ¼

s is the s-type atomic orbital in both unprimed and primed coordinates, k0 indicates the spin down function in the primed h i 0 0 coordinates, ak  b Eg 0  ðg0k Þ2 ðEg 0  d Þ1=2 ðEg0 þ d Þ1=2 , b 

O

" ¼

ð2pÞ3

#

dð~ q þ~ s0  ~ kÞ

O

Z

cell

un ð~ k; ~ rÞr ul ð~ q; ~ rÞ d r 3

cell

3 un ð~ k; ~ rÞrul ð~ q; ~ rÞ d r





(9)

R 3 k; ~ rÞul ð~ q; ~ rÞ d r ¼ where O is the volume of the unit cell and un ð~ ~ 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 the dimension of a typical Brillouin zone and we set ~ q ¼~ k. From Eqs. (8) and (9), we can write ¯ 0nl H

eA0 ~ ¼ s  ~ pnl ð~ kÞdð~ q ~ kÞ cosðotÞ 2m

(10)

where ~ pnl ð~ kÞ ¼ i_

Z

3

un rul d r ¼

Z

eA0 0 ~ ¼  ~ p ð~ kÞ 2m s nl

3

w  ð6E2g0 þ 9Eg0 D þ 4D2 Þ, g0k 

½ð6ðEg0 þ 2D=3ÞðEg0 þ DÞÞ=w1=2 ,

0 ½ðx1k Eg0 Þ=2ðx1k þ d Þ1=2 , x1k ¼ Ec ð~ kÞ  Ev ð~ kÞ ¼ Eg0 ½1 þ 2ð1 þ ðmc =mv Þ

ðgðEÞ=Eg0 ÞÞ1=2 , d ¼ ðE2g0 DÞðwÞ1 , X0 , Y0 and Z0 are the p-type atomic 0

orbitals in the primed coordinates, m0 indicates the spin-up function in the primed coordinates, bk7rg0k7, r(4D2/3w)1/2, ck7tg0k7 and t  ½6ðEg0 þ 2D=3Þ2 =w1=2 . We can, therefore, write the expression for the OME as OME ¼ p¯ cv ð~ kÞ ¼ hu1 ð~ k; ~ rÞjpju k; ~ rÞi ¯ 2 ð~

(17)

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

hXjpjYi ¼ hYjpjZi ¼ hZjpjXi ¼0 ¯ ¯ ¯ (11)

0

There are finite interactions between the CB and the VB, and we can 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 "

where g(E) ¼ E(aE+1)(bE+1)/(cE+1), a ¼ 1=Eg0 ; Eg0 is the unperturbed band gap, b ¼ 1=ðEg0 þ DÞ, D is the spin–orbit splitting constant in the absence of any field, c ¼ 1=ðEg0 þ 2D=3Þ, mc is the effective electron mass at the edge of the CB in the absence of any field and hj~ 0s  ~ pcv ð~ kÞj2 iav represents the average of the square of the optical matrix element (OME). For the three-band model of Kane, we can write (14)

1 where mr is the reduced mass and is given by m1 ¼ m1 r c +mv , and mv is the effective mass of the heavy hole at the top of the valence band (VB) in the absence of any field. The doubly degenerate wave functions u1 ð~ k; ~ rÞ and u2 ð~ k; ~ rÞ can be expressed as [32]  0  X  iY 0 0 pffiffiffi " þ ckþ ½Z 0 #0  k; ~ rÞ ¼ akþ ½ðisÞ#0  þ bkþ (15) u1 ð~ 2

0

"

#

#0

(12)

With n ¼ c stands for CB and l ¼ v stands for valance band, the energy equation for the conduction electron can approximately be expressed as ! _2 k2 ðeA0 =2mÞ2 hj~ 0s  ~ pcv ð~ kÞj2 iav þ gðEÞ ¼ (13) 2mc Ec ð~ kÞ  Ev ð~ kÞ

2 2 x1k ¼ Ec ð~ kÞ  Ev ð~ kÞ ¼ ðE2g0 þ Eg0 _ k =mr Þ1=2

(16)

and

s ¼ s cos ot. where ~ When a photon interacts with a semiconductor, the carriers (i.e., electrons) are generated in the bands which are followed by the interband transitions. For example, when the carriers are generated in the valence band, the carriers then make interband transition to the conduction band (CB). The transition of the electrons within the same band i.e., H¯ 0 nn ¼ hn~ kjH0 jn~ k is neglected. Because, in such a case, i.e., when the carriers are generated within the same bands by photons, are lost by recombination within the aforementioned band resulting zero carriers. Therefore, hn~ kjH0 jn~ ki ¼ 0

 0  X þ iY 0 0 pffiffiffi # þ ck ½Z 0 "0  2

hSjpjSi ¼ hXjpjXi ¼ hYjpjYi ¼ hZjpjZi ¼0 ¯ ¯ ¯ ¯

un ð~ k; ~ rÞ~ pul ð~ k; ~ rÞ d r

Therefore, we can write ¯ 0nl H

u2 ð~ k; ~ rÞ ¼ ak ½ðisÞ"0   bk

2

y

if=2 cos 6e 2 ¼6 4 y if=2 e sin 2

eif=2 sin if=2

e

3

y " #

" 27 7 y5 # cos 2

and 2

3 2 cos y cos f X0 6 0 7 6  sin f 4Y 5 ¼ 4 sin y cos y Z0

cos y sin f cos f sin y sin f

32

3 X 7 6 7 0 54 Y 5 cos y Z

 sin y

Besides, the spin vector can be written as ~ S¼

  _ ~ s 2

where     1 0 i ; sz ¼ ; sy ¼ 1 0 0 i 0 pffiffiffiffiffiffiffi and i ¼ 1. From above, we can write

sx ¼



0

1

0



1

kÞ ¼ hu1 ð~ k; ~ rÞjPju2 ð~ k; ~ rÞi PCV ð~      0 X  iY 0 0 0 pffiffiffi " þ ckþ ½Z 0 #0  ¼ akþ ½ðisÞ#  þ bkþ 2   0 

 X þ iY 0 0 0 pffiffiffi  jPj ak ½ðisÞ"   bk # þ ck ½Z 0 "0  2

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Also, we can write

Using above relations, we get bkþ ak ffiffiffi fhðX 0  iY 0 ÞjPjiSih"0 j"0 ig þ ckþ ak fhZ 0 jPjiSih#0 j"0 ig P CV ð~ kÞ ¼ p 2 akþ bk  pffiffiffi fhiSjPjðX 0 þ iY 0 Þih#0 j#0 ig þ akþ ck fhiSjPjZ 0 ih#0 j"0 ig 2 (18) From Eq. (18), we can write hðX 0  iY 0 jPjiSi ¼ hðX 0 ÞjPjiSi  hðiY 0 ÞjPjiSi ¼ ihX 0 jPjSi  hY 0 jPjSi

ckþ ak hZ 0 jPjiSih#0 j"0 i þ ck akþ hiSjPjZ 0 ih#0 j"0 i ¼ iPðckþ ak þ ck akþ Þr^ 3 ½h#0 j"0 i

(24)

Combining Eqs. (23) and (24), we find P kÞ ¼ pffiffiffi ðir^ 1  r^ 2 Þfðbkþ ak Þh"0 j"0 i  ðbk akþ Þh#0 j#0 i PCV ð~ 2 þ iPr^ 3 ðckþ ak þ ck akþ Þh#0 j"0 i

(25)

From the above relations, we obtain

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

"0 ¼ eif=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #

jX 0 i ¼ cos y cos fjXi þ cos y sin fjYi  sin yjZi

#0 ¼ eif=2 sin ðy=2Þ " þeif=2 cos ðy=2Þ #

Thus,

(26)

Therefore

hX 0 jPjSi ¼ cos y cos fhXjPjSi þ cos y sin fhYjPjSi  sin yhZjPjSi ¼ P r^ 1

h# j"0 i ¼ ðeif=2 sin ðy=2Þ " þeif=2 cos ðy=2Þ # Þ 0

 ðeif=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #Þ

where

¼ ðeif=2 sin ðy=2Þ" þ eif=2 cos ðy=2Þ# Þ

r^ 1 ¼ ^i cos y cos f þ ^j cos y sin f  k^ sin y

 ðeif=2 cos ðy=2Þ " þeif=2 sin ðy=2Þ #Þ

Similarly, we obtain

¼  sin ðy=2Þ cos ðy=2Þh" j "i þ eif cos2 ðy=2Þh# j "i

0

 eif sin2 ðy=2Þh" j #i þ sin ðy=2Þ cosðy=2Þh# j #i

jY i ¼  sin fjXi þ cos fjYi þ 0jZi Thus,

Therefore,

0

hY jPjSi ¼  sin fhXjPjSi þ cos fhYjPjSi þ 0hZjPjSi ¼ P r^ 2

h#0 j"0 ix ¼  sinðy=2Þ cosðy=2Þh" j " ix þ eif cos2 ðy=2Þh# j " ix  eif sin2 ðy=2Þh" j # ix þ sinðy=2Þ cosðy=2Þh# j # ix

where r^ 2 ¼ ^i sin f þ ^j cos f

(27)

But we know from above that h" j " ix ¼ 0; h# j " ix ¼ 12; h" j # ix ¼ 12 and h# j # ix ¼ 0

so that hðX 0  iY 0 ÞjPjSi ¼ Pðir^ 1  r^ 2 Þ

Thus, from Eq. (27), we get h#0 j"0 ix ¼ 12½eif cos2 ðy=2Þ  eif sin2 ðy=2Þ

Thus, ak bkþ a  bkþ pffiffiffi hðX 0  iY 0 ÞjPjSih"0 j"0 i ¼ kp ffiffiffi Pðir^ 1  r^ 2 Þh"0 j"0 i 2 2 Now since 0

0

¼ 12½ðcos f  i sin fÞcos2 ðy=2Þ  ðcos f þ i sin fÞsin2 ðy=2Þ

(19)

¼ 12½cos f cos y  i sin f

(28)

Similarly, we obtain 0

0

hiSjPjðX þ iY Þi ¼ ihSjPjX i  hSjPjY i ¼ Pðir^ 1  r^ 2 Þ

h# j"0 iy ¼ 12½i cos f þ sin f cos y 0

We can write akþ bk akþ bk ffiffiffi fhiSjPjðX 0 þ iY 0 Þih#0 j#0 ig ¼  p ffiffiffi Pðir^ 1  r^ 2 Þh#0 j#0 i  p 2 2

and (20)

h#0 j"0 iz ¼ 12½ sin y Therefore,

Similarly, we get

^ 0 j"0 i h#0 j"0 i ¼ ^ih#0 j"0 ix þ ^jh#0 j"0 iy þ kh# z 1 ^ ¼ 2fðcos y cos f  i sin fÞ^i þ ði cos f þ sin f cos yÞ^j  sin ykg ^ ¼ 1½fðcos y cos fÞ^i þ ðsin f cos yÞ^j  sin ykg

jZ 0 i ¼ sin y cos fjXi þ sin y sin fjYi þ cos yjZi So that

2

hZ jPjiSi ¼ ihZ 0 jPjSi ¼ iPfsin y cos f^i þ sin y sin f^j þ cos yk^ ¼ iP r^ 3

þ if^i sin f þ ^j cos fg ¼ þ ir^ 2  ¼ 12i½ir^ 1  r^ 2 

0

1 ^ 2½r 1

where

Similarly, we can write

r^ 3 ¼ ^i sin y cos f þ ^j sin y sin f þ k^ cos y

h"0 j"0 i ¼ 12½^i sin y cos f þ ^j sin y sin f þ k^ cos y ¼ 12r^ 3

Thus, ckþ ak hZ 0 jPjiSih#0 j"0 i ¼ ckþ ak iP r^ 3 h#0 j"0 i

and (21)

Similarly, we can write ck akþ hiSjPjZ 0 ih#0 j"0 i ¼ ck akþ iP r^ 3 h#0 j"0 i

h#0 j#0 i ¼ 12r^ 3 Using the above results and following Eq. (25), we can write

(22)

Therefore, we obtain ak bkþ akþ bk pffiffiffi fhðX 0  iY 0 ÞjPjSih"0 j"0 ig  p ffiffiffi fhiSjPjðX 0 þ iY 0 Þih#0 j#0 ig 2 2 P 0 0 ¼ pffiffiffi ðakþ bk h# j# i þ ak bkþ h" j"0 iÞðir^ 1  r^ 2 Þ (23) 2

P kÞ ¼ pffiffiffi ðir^ 1  r^ 2 Þfðak bkþ Þh"0 j#0 i  ðbk akþ Þh#0 j#0 ig PCV ð~ 2 þ iPr^ 3 fðckþ ak  ck akþ Þh#0 j"0 ig   ak bkþ bk akþ P P pffiffiffi þ pffiffiffi ¼ r^ 3 ðir^ 1  r^ 2 Þ þ r^ 3 ðir^ 1  r^ 2 Þ 2 2 2 2 fðckþ ak  ck akþ Þg

ARTICLE IN PRESS S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150

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

Thus, 



b P þ c k kÞ ¼ r^ 3 ðir^ 1  r^ 2 Þ akþ pkffiffiffi P CV ð~ 2 2





bkþ þ c kþ þ ak pffiffiffi 2



4143

get (29)

We can write that

! !)1=2  ( 0 Eg0 Eg0  d r 2 2 2 g  g g Að~ kÞ ¼ b t þ pffiffiffi 0 0 0kþ 0kþ 0k Eg0 þ d Eg0 þ d 2 (35)

jr^ 1 j ¼ jr^ 2 j ¼ jr^ 3 j ¼ 1 ! !)1=2  ( 0 Eg0 Eg0  d r 2 2 2 ~ BðkÞ ¼ b t þ pffiffiffi 0 g0k  g0kþ g0k 0 Eg0 þ d Eg0 þ d 2

also P r^ 3 ¼ Px sin y cos f^i þ P y sin y sin f^j þ P z cos yk^

(36)

where in which

P ¼ hSjPjXi ¼ hSjPjYi ¼ hSjPjZi Z hSjPjXi ¼

  0   Eg0 Eg0 þ d 1 1 0 ¼ 2 2 x1k þ d x1k þ d0

x

g20kþ ¼ 1k

3

rÞPuVx ð0; ~ rÞ d r ¼ PCVx ð0Þ uC ð0; ~

and hSjPjYi ¼ PCVy ð0Þ

g20k ¼

and

Thus, P ¼ PCVx ð0Þ ¼ P CVy ð0Þ ¼ P CVz ð0Þ ¼ P CV ð0Þ where Z





Substituting x ¼ x1k+d0 in g20k , we can write !   ( 0 Eg0 Eg0 þ d r 1 1  Að~ kÞ ¼ b t þ pffiffiffi 0 x Eg0 þ d 2 2 ) !  0 0  0 Eg0 þ d Eg0  d 1 Eg0  d  1 þ 1  4 Eg0 þ d0 x x

hSjPjZi ¼ PCVz ð0Þ

P CV ð0Þ ¼



0 x1k þ Eg0 Eg0  d 1 1þ 0 ¼ 2ðx1k þ d Þ 2 x1k þ d0

3

uC ð0; ~ rÞPuV ð0; ~ rÞ d r ¼ P

Thus

For a plane polarized light wave, we have the polarization vector ^ when the light wave vector is traveling along the z-axis. ~ s ¼ k, Therefore, for a plane polarized light wave, we have considered ^ ~ s ¼ k. Then, from Eq. (29), we can write P ð~ 0s  ~ PCV ð~ kÞÞ ¼ ~ k  r^ 3 ðir^ 1  r^ 2 Þ½Að~ kÞ þ Bð~ kÞ cos ot 2

(30)

and   bkþ þ c kþ Að~ kÞ ¼ ak pffiffiffi 2   b Bð~ kÞ ¼ akþ pkffiffiffi þ c k 2

(31)

Thus,    P 2 j~ 0s  ~ pcv ð~ kÞj2 ¼ k^  r^ 3  jir^ 1  r^ 2 j2 ½Að~ kÞ þ Bð~ kÞ2 2 1 2 P cos2 y½Að~ kÞ þ Bð~ kÞ2 cos2 ot (32) 4 Z 2 ¯ for a plane polarized light So, the average value of j¯s  p¯ cv ðkÞj wave is given by !  Z 2p Z p 2 1 hj~ s  ~ pcv ð~ kÞj2 iav ¼ P 2z ½Að~ kÞ þ Bð~ kÞ2 df cos2 y sin y dy 4 2 0 0

   b r 2a0 a1 1=2 Að~ kÞ ¼ t þ pffiffiffi þ 2 1 2 x x 2 02

where a0 ¼ ðE2g0 þ d ÞðEg0 þ d Þ1 and a1 ¼ ðEg0  d Þ2 . After tedious algebra, one can show that " #1=2   b r 1 1 0 t þ pffiffiffi ðEg0  d Þ  Að~ kÞ ¼ 2 x1k þ d0 Eg0 þ d0 2 " #1=2 0 ðEg0 þ d Þ 1   x1k þ d0 ðEg0  d0 Þ2 0

0

(37)

Similarly, from Eq. (36), we can write !   ( 0 Eg0 Eg  d r 1 ~ 1þ 0 BðkÞ ¼ b t þ pffiffiffi 0 x Eg0 þ d 2 2 ! )    0 0 0  1=2 Eg0 þ d Eg0  d 1 Eg0  d  1 þ 1  4 Eg0 þ d0 x x

 cos2 ot ¼

¼

p 3

kÞ þ Bð~ kÞ2 P 2z ½Að~

(33)

where P 2z ¼ j~ k ~ pcv ð0Þj2 and m2 Eg0 ðEg0 þ DÞ j~ k ~ pcv ð0Þj2 ¼ 4mr ðEg0 þ ð2=3ÞDÞ

(34)

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

So that, finally we get   0 Eg  d b r 1þ 0 t þ pffiffiffi Bð~ kÞ ¼ 2 x1k þ d0 2

(38)

Using Eqs. (33), (34), (37) and (38) we can write   eA0 2 hj~ s  ~ pcv ð~ kÞj2 iav ~ 2m Ec ðkÞ  Ev ð~ kÞ     eA0 2 p b2 r 2 j~ t þ pffiffiffi s  ~ pcv ð0Þj2 ¼ 2m 3 4 2 8 " #1=2  0 < Eg  d 1 1 1 0  1þ 0  d Þ  þ ðE g 0 x1k : x1k þ d0 x1k þ d0 Eg0 þ d0 9 " #1=2 2 0 = Eg0 þ d 1  (39) 0  0 2 ; x1k þ d ðEg  d Þ 0

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

Following Nag [33], it can be shown that

2 2

2

A20 ¼

Il pffiffiffiffiffiffiffiffiffiffiffi 2p2 c3 sc 0

(40)

where I is the light intensity of wavelength l, c is the velocity of light and e0 is the permittivity of vacuum. Thus, the simplified electron energy spectrum in III–V, ternary and quaternary materials up to the second order in the presence of light waves can approximately be written as

_2 k2 2mc

(41)

¼ b0 ðE; lÞ

where

  e Il Eg0 ðEg0 þ DÞ b r 2 1

p ffiffiffi y0 ðE; lÞ ¼ p ffiffiffiffiffiffiffiffiffiffi ffi t þ 96mr pc3 sc 0 Eg0 þ 23D 4 2 f0 ðEÞ 8 " #1=2   0 < Eg0  d 1 1 0  1þ þ ðEg0  d Þ 0 0  0 : f0 ðEÞ þ d f0 ðEÞ þ d Eg0 þ d 9 " #1=2 2 0 = Eg0 þ d 1   0 0 2 ; f0 ðEÞ þ d ðEg0  d Þ 2

2

and 



f0 ðEÞ ¼ Eg0 1 þ 2 1 þ

  mc gðEÞ 1=2 mv Eg0

2.1.1. Special cases (1) For the two-band model of Kane, we have D-0. Under this condition, 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 write the energy spectrum of III–V, ternary and quaternary materials in the presence of external photo-excitation whose unperturbed conduction electrons obey the two-band model of Kane as

2mc

¼ o0 ðE; lÞ

(42)

where

o0 ðE; lÞ ¼ Eð1 þ aEÞ  B0 ðE; lÞ 2

B0 ðE; lÞ ¼

e 2 I l E g0 1 pffiffiffiffiffiffiffiffiffiffiffi 384pc3 mr sc 0 f1 ðEÞ



f1 ðEÞ ¼ Eg0 1 þ

¼ r0 ðE; lÞ

r0 ðE; lÞ ¼ E 

   2  E g0 1 1 1þ þ Eg0  f1 ðEÞ f1 ðEÞ Eg0

2mc Eð1 þ aEÞ Eg0 mr

1=2

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

(43)

    2 e2 Il 2mc E 3=2 1 þ p ffiffiffiffiffiffiffiffiffiffi ffi mr Eg0 96pc3 mr sc 0

(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 

 1 e2 qn2D 2sc qE¯ F

(45)

Where esc is the semiconductor permittivity, E¯ F is the Fermi energy. It appears then that the formulation of the 2D DSL requires an expression of electron statistics, which, in turn, is determined by the DOS function: (i) The 2D electron energy spectrum in ultrathin films of III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the three-band model of Kane, in the presence of light waves can be expressed from Eq. (41) as   _2 k2s _2 nz p 2 þ ¼ b0 ðE; lÞ (46) 2mc 2mc dz 2

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

_2 k2

2mc

L2D ¼

b0 ðE; lÞ ¼ ½gðEÞ  y0 ðE; lÞ 2

_ k

2

2

where ks ¼ kx þ ky , nz ( ¼ 1, 2, 3, y) is the size quantum number along z-direction and dz is the film thickness along the z-direction. The sub-band energies ðEnz Þcan be written as

_2

ðnz p=dz Þ2

(47)

The DOS function is given by   nzmax mc X 0 N 2D ðE; lÞ ¼ ½b0 ðE; lÞHðE  Enz Þ 2

(48)

b0 ðEnz ; lÞ ¼

2mc

p_

nz ¼1

where the prime indicates the differentiation of the differentiable functions with respect to E and H is the Heaviside step function. Combining Eq. (48) with the Fermi-Dirac occupation probability factor, the surface electron concentration can thus be written as n2D ¼

nzmax mc X

p_2 nz ¼1

½T 1 ðEF ; nz ; lÞ þ T 2 ðEF ; nz ; lÞ

(49)

where T1(EF, nz, l)[b0(EF, l)((_2)/2mc)((nzp)/dz)2], T 2 P0 ðEF ; nz ; lÞ  sr¼1 Z r T 1 ðEF ; nz ; lÞ, Zr2(kBT)2r(1212r)z(2r)((q2r)/ (qE2r )), z (2r) is the zeta function of order 2r and r is the set of F positive integers whose upper limit is S0 [35] and EF is the Fermi energy in the presence of light waves as measured from the edge of the CB in the absence of any field in the vertically upward direction. The use of Eqs. (49) and (45) leads to the expression of the 2D DSL in this case as ( )1 !n zmax X e2 mc 0 0 L2D ¼ ½T ðE ; n ; l Þ þ T ðE ; n ; l Þ (50) z z 1 F 2 F 2 2sc p_ nz ¼1 where the primes denote the first-order differentiation of the differentiable 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 surface electron concentration for ultrathin films of III–V, ternary and

ARTICLE IN PRESS S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150

quaternary materials, whose unperturbed band structure is defined by the two-band model of Kane, can respectively be written in the presence of photo-excitation as   _2 k2s _2 nz p 2 þ ¼ o0 ðE; lÞ (51) 2mc 2mc dz

4145

photo-excitation as   _2 k2s _2 nz p 2 þ ¼ gðEÞ 2mc 2mc dz

gðEnz Þ ¼

_2 2mc

(61)

ðnz p=dz Þ2

(62)

2

o0 ðEnz ; lÞ ¼



N2D ðE; lÞ ¼

n2D ¼

_

2mc

mc

p_

nzmax mc X

p_2 nz ¼1

ðnz p=dz Þ2

(52) N 2D ðEÞ ¼

 nX zmax

2

½o00 ðE; lÞHðE  Enz Þ

(53) n2D ¼

nz ¼1

½T 3 ðEF ; nz ; lÞ þ T 4 ðEF ; nz ; lÞ

(54)

where T3(EF, nz, l)[o0(EF, l)(_2/(2mc))((nzp)/dz))2], P0 T 4 ðEF ; nz ; lÞ  sr¼1 Z r T 3 ðEF ; nz ; lÞ. The use of Eqs. (54) and (45) leads to the expression of the 2D DSL in this case as ( )1 !n zmax X e2 mc 0 0 L2D ¼ ½T ðE ; n ; l Þ þ T ðE ; n ; l Þ (55) z z F F 3 4 2 2sc p_ nz ¼1 (iii) Using Eq. (43), the expressions for the 2D dispersion relation, the sub-band energies, the DOS function and the electron concentration for ultrathin films of III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the parabolic energy bands, can respectively be written in the presence of photo-excitation as

_2 k2s 2mc

þ

  nz p 2 ¼ r0 ðE; lÞ 2mc dz

_2

r0 ðEnz ; lÞ ¼

N2D ðE; lÞ ¼

n2D ¼

_2

ðnz p=dz Þ2

2mc 

mc

 nX zmax

p_2

nz ¼1

nzmax mc X

p_2 nz ¼1

(56)

(57)

½r00 ðE; lÞHðE  Enz Þ

(58)

½T 5 ðEF ; nz ; lÞ þ T 6 ðEF ; nz ; lÞ

( L2D ¼

!n zmax X

e2 mc 2

2sc p_

½T 05 ðEF ; nz ; lÞ þ T 06 ðEF ; nz ; lÞ

(60)

nz ¼1

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-band energies, the DOS function and the surface electron concentration for ultrathin films of optoelectronic materials, whose unperturbed band structure is defined by the three-band model of Kane, can respectively be written in the absence of

½g0 ðEÞHðE  Enz Þ

(63)

nz ¼1

½T 7 ðEF0 ; nz Þ þ T 8 ðEF 0 ; nz Þ

(64)

where EF0 is the Fermi energy in the presence of size quantization as measured from the edge of the CB in the vertically upward direction in the absence of external photoexcitation, "   # _ 2 nz p 2 T 7 ðEF 0 ; nz Þ  gðEF 0 Þ  2mc dz and T 8 ðEF 0 ; nz Þ 

s0 X

Z r T 7 ðEF 0 ; nz Þ

r¼1

The use of Eqs. (64) and (45) leads to the expression of the DMR in this case as !n ( )1 zmax X e2 mc 0 0 ½T ðE ; n Þ þ T ðE ; n Þ (65) L2D ¼ z z 7 F0 8 F0 2 2sc p_ nz ¼1 (ii) The expressions for the 2D dispersion relation, the sub-band energies, the DOS function and the surface electron concentration for ultrathin films of optoelectronic materials, whose unperturbed band structure is defined by the two-band model of Kane, can respectively be written in the absence of photo-excitation as Eð1 þ aEÞ 

_2 2mc

Enz ð1 þ aEnz Þ 

2

)1

p_2

p_2 nz ¼1

N 2D ðEÞ ¼

where T5(EF, nz, l)[r0(EF, l)(_ /(2mc))((nzp)/dz)) ], T 6 ðEF ; nz ; lÞ  Ps0 r¼1 Z r T 5 ðEF ; nz ; lÞ. The use of Eqs. (54) and (45) leads to the expression of the 2D DSL in this case as

 nX zmax

mc

nzmax mc X

(59)

2



n2D ¼

ðnz p=dz Þ2 þ

_2 2mc

nzmax mc X

p_2

2mc

ðnz p=dz Þ2

(66)

(67)

½1 þ 2aEHðE  Enz Þ

(68)

½ð1 þ 2aEnz ÞF 0 ðZn Þ þ 2akB TF 1 ðZn Þ

(69)

p_2 nz ¼1

nzmax mc kB T X

_2 k2s

nz ¼1

where Fj(Zn) is the Fermi-Dirac integral of order j, Zn  ðEF 0  Enz Þ=kB T is the one parameter Fermi–Dirac integral of order t0 which can be written as [36]  Z 1 1 F t0 ðZÞ ¼ yt0 ð1 þ expðy  ZÞÞ1 dy; y4  1 Gðt0 þ 1Þ 0 (70) where G(t0+1) is the complete gamma function or for all t, analytically continued as a complex contour integral around the negative axis Z ð0þÞ F t0 ðZÞ ¼ At0 yt0 ð1 þ expðy  ZÞÞ1 dy (71) 1

pffiffiffiffiffiffiffi in which At0 ¼ ðGðt 0 ÞÞ=ð2p 1Þ.

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ARTICLE IN PRESS S. Bhattacharya et al. / Physica B 403 (2008) 4139–4150

The use of Eqs. (69) and (45) leads to the expression of the 2D DSL in this case as ( L2D ¼

!n zmax X

e2 mc 2

2sc p_ kB T

)1 ½ð1 þ 2aEnz F 1 ðZn Þ þ 2akB TF 0 ðZn ÞÞ

nz ¼1

(72) (iii) Under the condition a-0 as for ultrathin films of parabolic energy bands, the expressions for the 2D dispersion relation, the sub-band energies, the DOS function, the surface electron concentration and the 2D DSL can be written as E

_2

2

2mc

Enz 

ðnz p=dz Þ þ

_2 2mc

N2D ðEÞ ¼

n2D ¼

(73)

2mc

ðnz p=dz Þ2 nzmax mc X

p_2 nz ¼1

nzmax mc kB T X

p_2 (

L2D ¼

_2 k2s

(74)

HðE  Enz Þ

(75)

½F 0 ðZn Þ

(76)

nz ¼1

e2 mc

!n zmax X

2

2sc p_ kB T

)1 ½F 0 ðZn Þ

(77a)

nz ¼1

Under the condition of non-degeneracy, Eqs. (76) and (77a) assume the forms n2D ¼

nzmax mc kB T X

p_2 (

L2D ¼

½expðZn Þ

(77b)

nz ¼1

e2 mc

!n zmax X

2

2sc p_ kB T

)1 ½expðZn Þ

(77c)

nz ¼1

power G in the present case can be expressed as [37]   Z 1 1 qf G¼ ðE  EF ÞRðEÞ  0 dE eTn0 1 qE

4147

(78)

where R(E) is the total number of states. Following Tsidilkovski [38], Eq. (78) can be written under the condition of carrier degeneracy as !  p2 k2B T qn0 G¼ (79) 3n0 qEF The use of Eqs. (79) and (45) leads to the result ! 2 2p2 kB T sc L2D ¼ 3e3 n2D G

(80)

Thus, the 2D DSL for degenerate materials can be determined by knowing the experimental values of G. From the suggestion for the experimental determination of the DSL for degenerate materials having arbitrary dispersion laws as given by Eq. (80), we observe that for a constant T, the DSL varies inversely with Gn2D. Only the experimental values of G for any material as a function of electron concentration will generate the experimental values of the 2D DSL for that range of n0 for that material. Since (Gn2D)1 decreases with increasing n2D for constant T, from Eq. (80) we can conclude that the 2D DSL will decrease with increasing n2D. This statement provides a compatibility test of our theoretical analysis. Eq. (80) provides an experimental check of the 2D DSL and also a technique for probing the band structures of the materials having arbitrary band structures. Thus, the 2D DSL for degenerate materials can be determined by knowing the experimental values of G. The suggestion for the experimental determination of the 2D DSL for degenerate materials having arbitrary dispersion laws as given by Eq. (80) does not contain any band parameters. This statement provides 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 the materials having arbitrary band structures.

Using Eqs. (77b) and (77c), we get L2D ¼ ½2sc kB T=e2 n2D 

(77d)

The classical 2D DSL equation of wide gap non-degenerate materials has been obtained in Eq. (77d) as a special case of our generalized analysis under certain limiting conditions from all the results. This indirect test not only exhibits the mathematical compatibility of our formulation but also shows the fact that our simple analysis is a more generalized one, since one can obtain the corresponding results for relatively wide gap 2D materials having parabolic energy bands under certain limiting conditions from 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 electrons in materials in the presence of a classically large magnetic field is independent of the scattering mechanism and is determined only by the dispersion law [37]. The magnitude of the thermoelectric

2.5. Two applications of the results of this paper in the field of nanoelectronics in general (i) It is well known that the Einstein relation for the DMR is an important quantity for studying the transport properties of modern semiconductor devices since the diffusion constant (a quantity very useful for device analysis but whose exact experimental determination is rather difficult) can be derived from this ratio if one knows the experimental values of the mobility [39]. In addition, it is more accurate than any of the individual relations for the diffusivity or the mobility, which are the two widely used features of electron transport. Besides, the performance of the electronic devices at device terminal and the speed of operation of modern switching transistors are significantly influenced by the degree of carrier degeneracy present in these devices. The simplest way of analyzing such devices taking into account the degeneracy of bands is to use the appropriate Einstein relation to express the performance at the device terminals and switching speed in terms of carrier concentration [40]. The 2D Einstein relation

Fig. 1. (a) Plot of the normalized 2D DSL as a function of film thickness for ultrathin films of n-Hg1xCdxTe in the presence of light waves in which the curves (a–c) represent the three and two band models of Kane with that of the parabolic energy bands, respectively. (b) Plot of the normalized 2D DSL as a function of surface electron concentration per unit area for ultrathin films of n-Hg1xCdxTe in the presence of light waves in which the curves (a–c) represent the three and two band models of Kane and that of the parabolic energy bands, respectively. (c) Plot of the normalized 2D DSL as a function of light intensity for ultrathin films of n-Hg1xCdxTe in which the curves (a–c) represent the three and two band models of Kane and that of parabolic energy bands, respectively. (d) Plot of the normalized 2D DSL as a function of wavelength for ultrathin films of n-Hg1xCdxTe in which the curves (a–c) represent the three and two band models of Kane and that of parabolic energy bands, respectively.

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can, in general, be written as [41] D

ðn2D =eÞ ¼ m ðqn2D =qE¯ F Þ

(81)

find two specific applications namely the Einstein relation and the carrier contribution to the elastic constants and in the realm of nanodevices.

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

m

3. Result and discussions ¼ p2 kB T=ð3e2 GÞ

(82)

Therefore, we can experimentally determine the 2D DMR for any materials having arbitrary dispersion laws by knowing the experimental values of G. Since G decreases with increasing n2D, from Eq. (82), we can infer that the DMR will increase with increase in n2D and this compatibility is the direct test of our suggestion for experimental determination of the 2D DMR. (ii) The knowledge of the carrier contribution to the elastic constants is very important in studying the mechanical properties of the nanomaterials and has been relatively been less investigated in the literature [42]. The electronic contribution to the second- and third-order elastic constants can be written as [42,43]

DC 44 ¼ 

a20 qn2D

(83)

9dz qE¯ F

and

DC 456 ¼

a30 q2 n2D

(84)

27dz qE¯ 2 F

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

DC 44 ¼ ½n2D a20 eG=ð3dz p2 k2B TÞ

(85)

and 

  n2D qG 1þ G qn2D

DC 456 ¼ n2D ea30 G2 =ð3p4 k3B Tdz Þ

(86)

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

Thus, we can summarize the whole mathematical background in the following way: in this paper, we have investigated the 2D DSL in ultrathin films of optoelectronic materials in the presence of photo-excitation on the basis of a newly formulated electron dispersion law whose unperturbed conduction electrons obey the three- and two-band models of Kane together with parabolic energy bands. Under certain special conditions, we have also obtained the results for materials whose unperturbed electron energy spectra are defined by the two-band model of Kane and that of the parabolic energy bands. We have also investigated the 2D DSL both in the presence and absence of photo-excitation in bulk specimens of the aforementioned materials. The classical 2D DSL equation of wide gap non-degenerate materials has been obtained in Eq. (77d) as a special case of our generalized analysis under certain limiting conditions from all the results. This indirect test not only exhibits the mathematical compatibility of our formulation but also shows the fact that our simple analysis is a more generalized one, since one can obtain the corresponding results for relatively wide gap 2D materials having parabolic energy bands under certain limiting conditions from our present derivation. In addition, we have suggested an experimental method for the determination of 2D DSL, for nanomaterials having arbitrary dispersion laws in the presence of light together with two new suggestions for the measurement of band gap of semiconductors in this context. Besides, the results of this paper

Using the appropriate equations, we have plotted the 2D DSL as a function of film thickness at T ¼ 4.2 K in the presence of photoexcitation for ultrathin films of n-Hg1xCdxTe;(x ¼ 0.3, mv ¼ 0.4, m0 ¼ 12.642 (D ¼ 0.063+0.24x+0.27x2) eV, Eg0 ¼ ½0:302 þ 1:93xþ 5:25  104 Tð1  2xÞ  0:810x2 +0:832x3  eV) [43] whose unperturbed electron dispersion laws are defined by the three- and two-band model of Kane together with the parabolic energy bands as shown by curves (a–c) of Fig. 1a, respectively. Fig. 1b–d exhibits the dependencies of the 2D DSL on surface electron concentration, light intensity and wavelength, respectively. All the plots have been normalized to the value of the DSL, as given in the introduction. It should be noted that the 2D DSL decreases from the light-off case to the light-on case, since the value of the Fermi energy in the presence of light waves becomes larger due to the increase in the carrier concentration as compared with the same in the absence of photo-excitation. Therefore, the numerical magnitude of the 2D DSL in the presence of light is smaller as compared with the same in the light-off case in the whole range of the appropriate variables as considered, although the 2D DSL decreases with increase in said variables. The combined influence of the energy band constants on the 2D DSL for all the said compounds can easily be assessed from all the figures. From Fig. 1c, we observe that the 2D DSL decreases with increasing light intensity whereas in the absence of external photo-excitation, the same is independent of intensity. Fig. 1d exhibits the fact that the 2D DSL decreases as the wavelength shifts from red to violet color. For the ternary material, we have taken x ¼ 0.3, since for xo0.17, the band gap becomes negative in n-Hg1xCdxTe leading to semi-metallic state. The influence of quantum confinement on the aforementioned materials is immediately apparent from all the figures, since the 2D DSL depends strongly on the thickness of the size-quantized materials, which is in direct contrast with their respective bulk specimens. Moreover, the 2D DSL for ultrathin films can become several 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 decreases with the increasing film thickness in a step-like manner, both in the presence and absence of photo-excitation for all types of materials as considered here, although, the numerical values vary widely and determined by the constants of the energy spectra. The oscillatory dependence is due to the crossing over of the Fermi level by the size-quantized levels. For each coincidence of a size-quantized level with the Fermi level, there would be a discontinuity in the DOS function resulting in a peak of oscillations. With large values of film thickness, the height of the steps decreases and the 2D DSL will decrease with increasing film thickness in non-oscillatory manner and exhibit monotonic decreasing dependence. The height of step size and the rate of decrement are totally dependent on the band structure. The influence of light is immediately apparent from the plots in Fig. 1c and d, since the 2D DSL depends strongly on I and l, which is in direct contrast as compared with the corresponding cases for ultrathin films in the absence of external photoexcitation, respectively. The variations of the 2D DSL in all the figures reflect the direct signature of the light waves on the electronic, optic and the other band structure-dependent properties of semiconducting materials in the presence of light waves

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and the photon-assisted transport for the corresponding semiconductor devices, since the incident photons drastically change the electron dispersion law. From the figures, we observe that the 2D DSL decreases with increasing film thickness, intensity, wavelength and surface electron concentration, together with the fact that the rate of variation is totally band structure dependent. It appears from Fig. 1b that the 2D DSL decreases with increasing carrier degeneracy, which exhibits the signatures of the 1D confinement through the step-like dependence. This oscillatory dependence will be less and less prominent with increasing film thickness and carrier concentration, respectively. Ultimately, for bulk specimens of the same material, the DSL will be found to decrease continuously with increasing electron concentration in a non-oscillatory manner. The appearance of the humps of the respective figures is due to the redistribution of the electrons among the quantized energy levels when the size quantum number corresponding to the highest occupied level changes from one fixed value to the others. With varying electron concentration, a change is reflected in the 2D DSL through the redistribution of the electrons among the sizequantized levels. In this paper, we have suggested the experimental determinations of the 2D DSL, the Einstein relation for the diffusivityto-mobility ratio and the carrier contribution to the elastic constants and our suggestions are valid for ultrathin films having arbitrary dispersion relations. Since the experimental curves of n2D versus G are not available in the literature to the best of our knowledge for the present generalized systems, we cannot compare our theoretical formulation with the proposed experiment although, the generalized analysis as presented in this context 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 original band structure of the material. Our method is not at all related to the DOS technique as used in the literature [44]. From the E–k dispersion relation, we can obtain the DOS, but the DOS technique as used in the literature [44] cannot provide the E–k dispersion relation. Therefore, our study is more fundamental than those of the existing literature because the Boltzmann transport equation, which controls the study of the charge transport properties of semiconductor devices, can be solved if and only if the E–k dispersion relation is known. It may be remarked that, in recent years, the carrier statistics has been extensively been studied [45], but the screening length of the nonparabolic materials has relatively been less investigated. We wish to note that we have not considered the hot electron and many-body effects in this simplified theoretical formalism due to the lack of availability in the literature of proper analytical techniques for including them for the generalized systems as considered in this paper. Our simplified approach will be useful for the purpose of comparison when methods of tackling the formidable problem after inclusion of the many-body and the hot electron effects for the present generalized systems appear. The inclusion of the said effects would certainly increase the accuracy of the results, although the qualitative features of the 2D DSL, as discussed in this paper, would not change in the presence of the aforementioned effects. Our suggestions for the experimental determinations of the 2D DMR and the elastic constants are independent of the inclusion of the said effects. We have not considered other types of optoelectronic materials and other external variables in order to keep the presentation brief. The numerical results presented in this paper would be different for other materials but the nature of variation would

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

Acknowledgement The authors K. P. Ghatak and D. De are grateful to All Indian Council for Technical Education for granting the project having the reference number 8023/BOR/RID/RPS-95/2007-08 under research promotion scheme 2008 under which this research paper has been completed. References [1] [2] [3] [4]

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