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A. Haji Malekkheili, N. Naghdiani, K. Moradi Ankoochaksaraee, G. H. Khorrami / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622

www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1010-1014

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Calculation of Electron Transport Properties in CdTe andHgCdTe at Low Electric Filed Application

1A. Haji Malekkheili, 2N. Naghdiani, 3K. Moradi Ankoochaksaraee and 4G.

H. Khorrami1Department of Physics, Robat Karim Branch, Islamic Azad University, Tehran, Iran2Department of Physics, Robat Karim Branch, Islamic Azad University, Tehran, Iran

3Department of Physics, Tehran Shomal Branch, Islamic Azad University, Tehran, Iran4Department of Physics (Materials and Electroceramics Laboratory), Ferdowsi University of Mashhad,

Mashhad, Iran

AbstractElectron mobility in CdTe and HgCdTe

are calculated, by solving Boltzmann equationusing iteration model, as a function of temperature for carrier concentrations of 10 16,10 17, and 10 18 cm -3. Both CdTe and HgCdTehave maximum mobility between 100 and 200 K,depending on the electron density. Thetheoretical maximum mobility in CdTe andHgCdTe at 300 K are about 1000 and 4400cm 2V -1s-1. We compared the results withexperimental data and find reasonablecorrelation.

Keywords -: Iteration Model; ionized impurityscattering; Electron mobilities .

I. INTRODUCTIONCdTe and HgCdTe semiconductors have

been intensively investigated for last two decadesfor their potential applications in electronic,optoelectronic, and microelectronic devicestructures. The recent findings show that III-nitridesemiconductors are also promising fornanostructure materials and nanodevices. Thefabrication of homo and heterojunction devicesmainly based on CdTe led to a rapidcommercialization of light emitting diodes andlaser diodes. The practical and commercialapplications of these materials can be used in trafficlights, back lighting of mobile phones, indoorlights, light indicators for electronic devices, opticalstorage and display and solid state lasers. Anotherimportant application of these semiconductors is infabrication of quantum infrared (IR) detectors.Photoconductors are the most common type of quantum IR detectors which can be fabricated byusing nitride semiconductors [1]. The low-fieldelectron mobility is one of the most importantparameters that determine the performance of afield-effect transistor. The purpose of the presentpaper is to calculate electron mobility for varioustemperatures and ionized-impurity concentrations.The formulation itself applies only to the centralvalley conduction band. We have also considerband non-parabolicity and the screening effects of free carriers on the scattering probabilities. All the

relevant scattering mechanisms, including polaroptic phonons, deformation potential, piezoelectric,acoustic phonons, and ionized impurity scattering.The Boltzmann equation is solved iteratively forour purpose, jointly incorporating the effects of allthe scattering mechanisms [2-4].

This paper is organized as follows. Detailsof the iteration model , the electron scatteringmechanism which have been used and the electronmobility calculations are presented in section IIand the results of iterative calculations carried outon CdTe and HgCdTe structures are interpreted insection III.

II. M ODEL DETAILS To calculate mobility, we have to solve the

Boltzmann equation to get the modified probabilitydistribution function under the action of a steadyelectric field. Here we have adopted the iterativetechnique for solving the Boltzmann transportequation. Under the action of a steady field, theBoltzmann equation for the distribution functioncan be written as,

collk r t f

f eF

f vt

f )(.. (1)

Where collt f ) / ( represents the change of

distribution function due to the electron scattering.In the steady-state and under application of auniform electric field the Boltzmann equation canbe written as,

collk t f

f eF

)(. (2)

Consider electrons in an isotropic, non-parabolic conduction band whose equilibriumFermi distribution function is f 0(k) in the absence of electric field. Note the equilibrium distribution f 0(k)is isotropic in k space but is perturbed when anelectric field is applied. If the electric field is small,we can treat the change from the equilibrium

distribution function as a perturbation which is firstorder in the electric field. The distribution in the

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presence of a sufficiently small field can be writtenquite generally as,

cos)()()( 10 k f k f k f (3)Where θ is the angle betwee n k and F and

f 1(k) is an isotropic function of k , which isproportional to the magnitude of the electric field .

f(k) satisfies the Boltzmann equation 2 and itfollows that,

i

k d f i

S f i

S f k d f i

S f i

S f t

f eF 30

)0

1(1

30

)0

1(1

cos0

(4)

In general there will be both elastic andinelastic scattering processes. For exampleimpurity scattering is elastic and acoustic andpiezoelectric scattering are elastic to a goodapproximation at room temperature. However,polar and non-polar optical phonon scattering areinelastic. Labeling the elastic and inelasticscattering rates with subscripts el and inelrespectively and recognizing that, for any process i,seli(k ’, k ) = s eli(k , k ’) equation 4 can be written as,

k d f S f S k d S

k d f S f S f k

f eF

k f inelinelel

inelinel

300

3

3001

0

1])1([)cos1(

])1([cos)(

(5)

Note the first term in the denominator is simply themomentum relaxation rate for elastic scattering.

Equation 5 may be solved iteratively by therelation,

k d f S f S k d S

k d f S f S n f k

f eF

k f inelinelel

inelinel

n 300

3

3001

0

1])1([)cos1(

])1(][1[cos)(

(6)

where f 1n (k ) is the perturbation to thedistribution function after the n-th iteration. It isinteresting to note that if the initial distribution ischosen to be the equilibrium distribution, for which

f 1(k ) is equal to zero, we get the relaxation timeapproximation result after the first iteration. Wehave found that convergence can normally beachieved after only a few iterations for smallelectric fields. Once f 1 (k ) has been evaluated to therequired accuracy, it is possible to calculatequantities such as the drift mobility , which isgiven in terms of spherical coordinates by,

(7)

Here, we have calculated low field driftmobility in GAN and InN structures using theiterative technique. In the following sections

electron-phonon and electron-impurity scatteringmechanisms will be discussed.

Deformation potential scatteringThe acoustic modes modulate the inter

atomic spacing. Consequently, the position of theconduction and valence band edges and the energyband gap will vary with position because of thesensitivity of the band structure to the latticespacing. The energy change of a band edge due tothis mechanism is defined by a deformationpotential and the resultant scattering of carriers iscalled deformation potential scattering. The energyrange involved in the case of scattering by acousticphonons is from zero to 2 vk , where v is thevelocity of sound, since momentum conservationrestricts the change of phonon wave vector tobetween zero and 2 k , where k is the electron wavevector. Typically, the average value of k is of theorder of 10 7 cm -1 and the velocity of sound in themedium is of order 10 5 cms -1. Hence, 2 vk ~ 1meV, which is small compared to the thermalenergy at room temperature. Therefore, thedeformation potential scattering by acoustic modescan be considered as an elastic process except atvery low temperature. The deformation potentialscattering rate with either phonon emission orabsorption for an electron of energy E in a non-parabolic band is given by Fermi's golden rule as[2-5],

22

42

2 / 1*2*2

)(3 / 1)1(

)21(

)1()(2

E E

E E

E E

v

T K mm D R

Blt ac

de

(8)

Where Dac is the acoustic deformationpotential, is the material density and is thenon-parabolicity coefficient. The formula clearlyshows that the acoustic scattering increases withtemperature.

Piezoelectric scatteringThe second type of electron scattering by

acoustic modes occurs when the displacements of the atoms create an electric field through thepiezoelectric effect. The piezoelectric scatteringrate for an electron of energy E in an isotropic,parabolic band has been discussed by Ridley [4].The expression for the scattering rate of an electronin a non-parabolic band structure retaining only theimportant terms can be written as [2-6]:

(9)0

30

2

0

31

3

*

)21 / (

3k d f k

k d f F k

F m

22

21

2

*22

)(31

1

2122

)(

s

av BPZ

mTK K ek R

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Where s is the relative dielectric constant of the

material and K av is the dimensionless so calledaverage electromechanical coupling constant.

Polar optical phonon scattering

The dipolar electric field arising from theopposite displacement of the negatively andpositively charged atoms provides a couplingbetween the electrons and the lattice which resultsin electron scattering. This type of scattering iscalled polar optical phonon scattering and at roomtemperature is generally the most importantscattering mechanism for electrons in III-Vsemiconductors, and this is also the case in CdTeand HgCdTe despite the fact that the opticalphonon energy is particularly high at ~ 93 meVwhich suppresses the phonon population and alsoelectrons must reach that energy before phonon

emission is possible. The scattering rate due to thisprocess for an electron of energy E in an isotropic,non-parabolic band is [2-6],

(11)

Where E = E '±hw po is the final stateenergy phonon absorption (upper case) andemission (lower case) and Nop is the phononoccupation number and the upper and lower cases

refer to absorption and emission, respectively. Forsmall electric fields, the phonon population will bevery close to equilibrium so that the averagenumber of phonons is given by the Bose- Einsteindistribution.

Impurity scattering This scattering process arises as a result of

the presence of impurities in a semiconductor. Thesubstitution of an impurity atom on a lattice sitewill perturb the periodic crystal potential and resultin scattering of an electron. Since the mass of theimpurity greatly exceeds that of an electron and the

impurity is bonded to neighboring atoms, thisscattering is very close to being elastic. Ionizedimpurity scattering is dominant at low temperaturesbecause, as the thermal velocity of the electronsdecreases, the effect of long-range Coulombicinteractions on their motion is increased. Theelectron scattering by ionized impurity centres hasbeen discussed by Brooks Herring [5] who includedthe modification of the Coulomb potential due tofree carrier screening. The screened Coulombpotential is written as,

r r qe

r V s

)exp(4

)( 0

0

2

(12)

Where s is the relative dielectric

constant of the material and q0 is the inversescreening length, which under non-degenerateconditions is given by

T K

neq

Bs 0

22

0(13)

Where n is the electron density. The scattering ratefor an isotropic, non-parabolic band structure isgiven by [2-6],

bb

b Ln E m

E e N R

s

iim 1

)1())((232

)21(2 / 32*

4

(14)

20

2

* )(8q

E mb

(15)

Where N i is the impurity concentration.

III. R ESULTS The electron mobility is a function of

temperature and electron concentration .our resultsshow that the electron mobility depends on totalscattering in bulk CdTe and HgCdTe materials.Figures 1and 2 show total scattering depends onEnergy , also increasing Temperature causesincreasing total scattering in bulk CdTe andHgCdTe materials.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

2

3

4

5

6

7

8

9

10

11T = 450 K

T = 300 K

T o t a l s c a t t e r i n g ( s - 1 ) * ( 1 0

1 3 )

Energy (ev)

Fig 1. Changes total scattering electron Function

in terms of energy in bulk CdTe at the differenttemperature.

1,,

211182 *2

opopPO

POPO

N N E E F

E

E mek R

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Fig 2. Changes total scattering electron Function interms of energy in bulk HgCdTe at the differenttemperature.

Figure 3 shows the comparison totals scattering of CdTe and HgCdTe at the different temperature .ourcalculation results show that the totals scatteringCdTe is more than HgCdTe .

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2

4

6

8

10T=450 k CdTe

T=300 k CdTe

T=450 k HgCdTe

T=300 k HgCdTe

T

o t a l s c a t t e r i n g ( s - 1 ) * ( 1 0

1 3 )

Energy (ev)

Fig 3. Changes total scattering electron Function interms of energy in bulk CdTe and HgCdTe at thedifferent temperature.

Figures 4 and 5 shows the electron mobilitydepends on the Temperature at the differentelectron concentration in bulk CdTe and HgCdTematerials.

Fig 4. Changes the electron mobility Function interms of temperature in bulk CdTe at thedifferent electron concentration.

100 200 300 400 500 600

4000

8000

12000

16000

Ionized impurity densiy =10 18[cm -3]

Ionized impurity densiy =10 16 [cm -3]

m o

b i l i t y [ c m

2 / V s ]

Temperature (k)

Fig 5. Changes the electron mobility Function interms of temperature in bulk HgCdTe at thedifferent electron concentration. figures 4 and 5show that electrons mobility at the definitetemperature 300 k for the CdTe semiconductors isgained about 1100cm 2v-1s-1 and for HgCdTe about4400 cm 2v-1s-1.also electrons mobility decreasequickly by temperature increasing from 100k to600 k for all the different electron concentrationsbecause temperature increasing causes increase of ponons energy too.so it causes a strong interactionbetween electrons and these ponons that its result isincrease of electrons scattering rate and finallydecrease of electrons mobility. Figure 6 showscomparison the electron mobility of CdTe andHgCdTe at the different electron concentration.ourcalculation results show that the electron mobilityInN is more than GaN this increasing is because of small eff mass.

100 200 300 400 500 6000

4000

8000

12000

16000

HgCdTe

CdTe

m o b i l i t y [ c m

2 / V s ]

Temperature (k)

Fig 6. Changes the electron mobility Function interms of temperature in bulk CdTe and HgCdTeat the electron concentration 10 16 (cm -3).

Figures 7 and 8 show the electron mobilitydepends on the electron concentration at thedifferent Temperature in bulk CdTe and HgCdTematerials

0.0 0.4 0.82.0

2.5

3.03.5

4.0

4.5

5.0

5.5 T = 450 K

T = 300 K

Energy (ev)

100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

Ionized impurity densiy =10 18 [cm -3 ]

Ionized impurity densiy =10 16 [cm -3 ]

m o

b i l i t y [ c m

2 / V s ]

Temperature (k)

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Fig 7. Changes the electron mobility Function interms of electron concentration in bulk CdTe atthe different Temperature.

Fig 8. Changes the electron mobility Function interms of electron concentration in bulk HgCdTeat the different Temperature

Figures 7 and 8 show that semiconductorsmobility decrease by electrons concentrationsincreasing because electrons increasing causesincrease of Ionized impurity centers in crystals thatit causes times more electrons under the influenceof the Coulomb potential of impurity centers locatedthat its result is increase of electrons scatteringrate and finally decrease of electrons mobilityFigure 9 shows comparison the electron mobility of CdTe and HgCdTe at the different Temperature.Our calculation results show that the electronmobility HgCdTe is more than CdTe

1E16 1E17 1E18

1000

2000

3000

4000

T =400 K GaN

T =300 K GaN

T =400 K InN

T =300 K InN

L o w - f i l d m o

b i l i t y [ c m

2 / ( V s ) ]

Ionized impurity densiy [cm -3]

Fig 9. Changes the electron mobility Function interms of electron concentration in bulk HgCdTeand at the different temperature.

IV. CONCLUSION1. HgCdTe semiconductor having high mobility of

CdTe because the effective mass is smallcompared with CdTe.

2. The ionized impurity scattering in both thesemiconductor CdTe and HgCdTe at alltemperatures is an important factor in reducingthe mobility.

R EFERENCES [1] Bhatta, Rudra Prasad, "Electron

Spectroscopic Study of Indium NitrideLayers" (2008). Physics & Astronomy

Dissertations. Paper 23.[2] Jacoboni C, Lugli P (1989). The Monte

Carlo Method for semiconductor andDevice Simulation, Springer-Verlag.

[3] Moglestue C (1993) . Monte CarloSimulation of SemiconductorDevices,Chapman and Hall

[4] Ridley BK (1997) . Electrons and phononsin semiconductor multilayers,CambridgeUniversity Press.

[5] Chattopadhyay D, Queisser HJ (1981).Review of Modern Physics, 53,part1

[6]H. Arabshahi, "Comparison of SiC and ZnO

Field Effect Transistors for High PowerApplications", Modern Physics Letters B,20, (2006), pp.787-793.

1E16 1E17 1E18

800

1000

1200

1400

1600

T= 400 K

T=300 k

T =200 k

L o w - f i l d

m o b i l i t y [ c m

2 / ( V s ) ]

Ionized impurity densiy [cm -3 ]

1E16 1E17 1E18

3000

3600

4200

4800

5400

6000

T =400 K

T =300 k

T =200 K

L o w - f i l d

m o

b i l i t y [ c m

2 / ( V s ) ]

Ionized impurity densiy [cm -3 ]

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