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EFFECT OF ELECTRIC FIELD, TEMPERATUREAND CORE DIMENSIONS IN III–V COMPOUND

CORE–SHELL NANOWIRES

ASHWANI VERMA*, BAHNIMAN GHOSH*,†,‡

and AKSHAY KUMAR SALIMATH*,§

*Department of Electrical EngineeringIndian Institute of Technology, Kanpur

Kanpur 208016, India

†Microelectronics Research Center, 10100 Burnet RoadUniversity of Texas at Austin, Austin, TX, 78758, USA

‡[email protected]§[email protected]

Received 22 September 2013Accepted 10 February 2014

Published

In this paper, we have used semiclassical Monte Carlo method to show the dependence of spinrelaxation length in III–V compound semiconductor core–shell nanowires on di®erent parameterssuch as lateral electric ¯eld, temperature and core dimensions. We have reported the simulationresults for electric ¯eld in the range of 0.5–10 kV/cm, temperature in the range of 77–300K andcore length ranging from 2 nm to 8 nm. The spin relaxation mechanisms used in III–V compoundsemiconductor core–shell nanowire are D'yakonov–Perel (DP) relaxation and Elliott–Yafet (EY)relaxation. Depending upon the choice of materials for core and shell, nanowire forms two types ofband structures. We have used InSb–GaSb core–shell nanowire and InSb–GaAs core–shellnanowire and nanowire formed by swapping the core and shell materials to show all the results.

Keywords: Spintronics; Monte Carlo simulation; core-shell nanowire; compound semiconductor.

1. Introduction

In conventional electronic devices, electron charge ismainly used for information transfer whereas inspintronics the determining role is played by spin ofthe electron. Spin is a quantum mechanical propertyassociated with every elementary particle such aselectrons, neutrons, photons, neutrinos, etc. We canmeasure the spin of a particle and it has a quantizedvalue, including zero. Exploiting this quantum

mechanical property of electron in research and noveldevices would allow us to attain further miniatur-ization of devices which is the need and future ofelectronics. By using this intrinsic property of elec-tron we can store as well as transmit the information.The two states of electron i.e., spin \up" and \down"can be used to represent 1 and 0 so it can store in-formation in binary form.1 Further, the transport ofspin carriers or spin polarized electrons can be used

§Corresponding author.

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March 4, 2014 4:54:38pm WSPC/213-NANO 1450051 ISSN: 1793-2920

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NANO: Brief Reports and ReviewsVol. 9, No. 4 (2014) 1450051 (10 pages)© World Scienti¯c Publishing CompanyDOI: 10.1142/S1793292014500519

http://dx.doi.org/10.1142/S1793292014500519

for transmission of this information. In this paper, wemainly focus on the transport of the spin because inmost of the proposed application of spintronics likespin valves,2 spin-FET3 it is the essential factor forimproving the overall e±ciency of the device. Inorder to transfer spin-based information inside a spindevice it is important that the particle should retainits injected spin up to certain length. The criticallength over which the injected spin retains 1=e timesits initial injection value is referred to as SpinRelaxation Length. This irreversible process ofspin dephasing4–6 occurs because of spin–orbit in-teraction, momentum relaxation and spin °ip.

Most of the work has been done in past to studythe spin relaxation length in nanowires. In Ref. 7,spin dephasing in silicon and germanium nanowireshas been studied and in Ref. 8 spin transport in III–V compound semiconductors was discussed. In thispaper, we study the spin relaxation length in III–Vcompound core–shell nanowires using semiclassicalMonte Carlo approach. Structure of a core–shellnanowire is shown below which is of nanometerrange and having a core made up of III–V compoundsemiconductor and a shell made of another III–Vcompound semiconductor. In this paper, we haveinvestigated the dependence of the spin relaxationlength on variation of length of core, temperatureand electric ¯eld along the length of the core–shellnanowire. We have reported the simulation resultsfor electric ¯eld in the range of 0.5–10 kV/cm,temperature in the range of 77–300K and corelength ranging from 2nm to 8 nm. The spin relax-ation mechanisms used in III–V compound semi-conductor core–shell nanowire are D'yakonov-Perel(DP) relaxation and Elliott–Yafet (EY) relaxationand the scattering mechanisms considered areacoustic phonon scattering, optical phonon absorp-tion and emission, polar optical phonon absorptionand emission, surface roughness scattering and spin°ip due the EY relaxation mechanism. These scat-tering mechanisms are explained brie°y in Sec. 2 ofthis paper. The paper is organized as follows. In thenext section, i.e., Sec. 2, we discuss the model usedfor simulation in our work. In Sec. 3, we present ourresults along with the discussion. Finally in Sec. 4we present our conclusion.

2. Model

The model used is similar to the Datta–Das tran-sistor which provides a two-dimensional channel for

electron transport in between two ferromagneticelectrodes. The two electrodes are similar to thesource and drain in a ¯eld e®ect transistor. Oneelectrode (emitter) is used to inject the electron intothe channel having the spin in the direction of itsmagnetization (in this paper, we assumed injectedelectron has spin polarization in z-direction), whilethe other electrode (collector) acts as a spin ¯lter andonly accepts electrons with the same spin. Channelbetween the electrodes is modeled as core–shellnanowire shown in Fig. 1. The outer dimensions ofthis core–shell nanowire are 10 nm� 10 nm� 10 umin z-, y- and x-directions, while core has the samelength in x-direction i.e., of 10 um and variablelengths in y- and z-directions. Core part is made upof one III–V compoundmaterial and shell is made upof another III–V compound material.

We have applied a potential di®erence across thesource and drain which is responsible for the °ow ofcurrent in x-direction only. Along with this a ver-tical ¯eld of 100 kV/cm from the gate is also appliedwhich is responsible for Rashba Spin–Orbitalcoupling.9

Depending on the bandgap of the materials usedfor core and shell we can get two types of bandstructures in a core–shell nanowire as shown inFig. 2. If the bandgap of the shell material is greaterthan the core material, we get band structure shown

Fig. 1. Core–Shell nanowire structure.

(a) (b)

Fig. 2. Band structure formed by core–shell nanowires.

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A. Verma, B. Ghosh & A. K. Salimath

in Fig. 2(a) and for the reverse case we get bandstructure shown in Fig. 2(b). Core and shell aremade up of two di®erent materials, so they havedi®erent masses and hence we have used mass-dependent Schrodinger equation to ¯nd out thedi®erent energy levels (E) and the wave function(�) corresponding to these energy levels. Mass-dependent Schrodinger equation is shown below[Eq. (1)]

�h2

2

@�

@x

1

mðxÞ@�

@x

� �þ V ðxÞ� ¼ E�; ð1Þ

where h is reduced Planck's constant, mðxÞ ise®ective mass depending upon the position in thecore–shell nanowire, � is the wave function, V ðxÞ ispotential in the well and E is the total energy.

We have used the weighted average constantsbased on the probability of ¯nding the electron inthe core or shell. Probability of ¯nding the electronin core or shell is calculated with the help of wavefunction (�) given by as followsZ x2

x1

��dx; ð2Þ

where �� represents the conjugate of �, x1 and x2are the points in between which the probability of¯nding the electron has to be calculated.

After ¯nding out all the constants we usedsemiclassical Monte Carlo method. To ensure sys-tem stability, the simulations are run for 1000 000iterations and last 50 000 iterations are recorded fordata. The iterations corresponded to a step durationof dt ¼ 0:05 fs. The °ight calculations and spinupdating are done after every iteration.

III–V compound semiconductors have bulkinversion asymmetry which results in Dresselhausspin–orbital interaction.10 The transverse electric¯eld is responsible for Rashba spin–orbit coupling11

as it breaks the structural inversion asymmetry ofthe compound. Now the electron spin is in°uencedby spin–orbit Hamiltonian which also comprises ofthe Dresselhaus interaction and Rashba interaction.For nanowire Dresselhaus spin–orbit Hamiltonian10

is given by

HD ¼ ��ðhkyi2 � hkzi2Þkx�x ð3ÞAnd Rashba spin–orbit interaction is given as

HR ¼ � � ðð¾xpÞÞ; ð4Þ

where ¾ is the Pauli matrix, p is the momentumvector, k is the wave vector, � and � are Rashba andDresselhaus components which depend on the ma-terial. � also depends on the external transverseelectric ¯eld. The expression of �12 is given below,

� ¼ }

2m��

Eg

2Eg þ�

ðEg þ�Þð3Eg þ 2�ÞeE; ð5Þ

where m� is the e®ective mass of electron, � is thespin–orbit splitting of valence band, Eg is bandgapand E is the transverse electric ¯eld.

During the free °ight, the evolution of the spinvector is done by following equation13,14

dS

dt¼ �� S; ð6Þ

where S is the spin vector, � is the precession vectorand it consists of two components, one from Rashbainteraction and another from Dresselhaus inter-action and is given by13,14

�D ¼� 2�effkx�

}; ð7Þ

�R ¼� 2�kx�

}; ð8Þ

where �eff ¼ ��ðhkyi2 � hkzi2Þ.Representing S ¼ Sx� þ Sy�þ Szk and using

Eqs. (7) and (8) in Eq. (6), we get each componentof spin as

dSx

dt¼� 2

}�kxSz; ð9Þ

dSy

dt¼� 2

}�effkxSz; ð10Þ

dSx

dt¼� 2

}kxð�Sx � �effSyÞSz: ð11Þ

We have also used EY15 relaxation mechanism,which is the cause of spin °ip scattering and givenby16

1

� EYs

¼ AkBT

Eg

� �2

�2 1� �=2

1� �=3

� �2 1

�p; ð12Þ

� EYs here represents the spin relaxation time due to

spin °ip scattering, �p represents the total momen-tum relaxation time, Eg is the bandgap and � isgiven by the following expression

� ¼ �

Eg þ�: ð13Þ

Here � is the spin–orbit splitting.

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E®ect of Electric Field, Temperature and Core Dimensions in III–V Compound Core–Shell Nanowires

In our model, the energy wave vector relation16,18

chosen is

"ð1þ �"Þ ¼ }2�2

2m; ð14Þ

where " is the energy and � is the nonparabolicityparameter given by

� ¼ 1

Eg

1� m

m0

� �2

: ð15Þ

The ensemble spin vector11 when calculated com-ponent-wise under conditions of steady state, isgiven by the expression

hSiiðx; tÞ ¼P t¼T

t¼t1

Pnxðx;tÞn¼1 sn;iðtÞP t¼T

t¼t1 nxðx; tÞ: ð16Þ

Here nxðx; tÞ is the number of electrons at time t andposition x within an accuracy of �xsn;iðtÞ is the nthelectron's spin at time \t" and t1, T are start and¯nish times, respectively.

In our simulation, the scattering rates consideredare optical phonon scattering,17,18 acoustic phononscattering,17,18 polar optical phonon scattering18,19

and surface roughness scattering.17,20

Parameters that have been used to calculatedi®erent scattering rates (mentioned above) areshown in Table 1.

3. Results

3.1. Electric ¯eld variation

As mentioned above, we have applied a voltagebetween the two ends of the nanowire (across sourceand drain) which allows electron to °ow in x-direc-tion. We vary this voltage from 0.5V to 10V which

results in an electric ¯eld inside the nanowire of theorder of 0.5 kV/cm to 10 kV/cm (length of nanowireis chosen as 10 um). Firstly, we used the bandgapshown in Fig. 2(a) i.e., for the combination in whichshell has higher bandgap as compared to core(shown under heading 1.A below) and after that weused bandgap shown in Fig. 2(b) i.e., reverse offormer one (shown below under heading 1.B).

(A) Shell has higher bandgap than core[Fig. 2(a)].

(B) When core has higher bandgap than shell[Fig. 2(b)].

Figures 3 and 4 show the dependence of spinrelaxation length on lateral electric ¯eld in GaAs–InSb and GaSb–InSb core–shell nanowires andFigs. 5 and 6 show the same dependence afterinterchanging core and shell materials. By observingthe above ¯gures we found that for both the com-binations, spin relaxation length increases with theincrease in the lateral electric ¯eld. This can beexplained on the basis of two opposing features thatare scattering rates and ensemble averaged driftvelocity. With the increase in the voltage across thetwo ends of nanowire, the driving electric ¯eld alsoincreases and hence the drift velocity increases.However, the inherent scattering rates do not showthe same drastic increase as with case of ¯eld andthey tend to saturate and thus they fail to counter-balance the e®ect. From above ¯gures we have alsoobserved that at smaller electric ¯eld (0.5–4 kV/cm), there is large change in spin relaxation lengthas compared to higher electric ¯elds (6–10 kV/cm)which is because at higher electric ¯elds drift vel-ocity starts saturating and hence the e®ect of elec-tric ¯eld experienced by the electron is less andhence we get less change in spin relaxation length.

Table 1. Parameters used for the di®erent core–shell III–V compounds.

S. no Parameters InP InSb GaN GaAs GaSb

1 E®ective electron mass 0.08mo 0.014mo 0.13mo 0.063mo 0.041mo

2 Bandgap (eV) 1.344 0.17 3.26 1.424 0.7263 Density (g/cm3) 4.79 5.79 6.15 5.32 5.61

4 Speed of sound (105cm/s) 5.13 3.7 6.9 5.5 6.07

5 Static dielectric constant ("s) 12.35 17.9 9.7 12.9 15.76 Optical Dielectric constant ("1) 9.52 15.7 5.35 10.89 11.057 Nonparabolicity factor (eV�1) 0.635 4.05 0.232 0.616 1.266

8 Acoustic phonon deformation potential (eV) 7.0 30.0 8.4 8.8 8.99 Polar optical phonon energy (eV) 0.043 0.025 0.087 0.035 0.029710 Lande-g-factor 1.20 �50 1.97 0.0148 �8.5911 Spin–Orbit splitting (eV) 0.11 0.80 0.02 0.34 0.80

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0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Variation of spin relaxation length with lateral electric ¯eld for InSb–GaAs core–shell nanowire (color online).

Fig. 4. Variation of spin relaxation length with lateral electric ¯eld for InSb–GaSb core–shell nanowire (color online).

Fig. 5. Variation of spin relaxation length with lateral electric ¯eld for GaAs–InSb core–shell nanowire (color online).

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3.2. Temperature variation

In this section, we show the dependence of spinrelaxation length on the variation of tempera-ture in a III–V core–shell nanowire. We vary the

temperature from 77K to 300K for both types of

bands as mentioned above.(A) When shell has higher bandgap than core

[Fig. 2(a)].(B) When core has higher bandgap than shell

[Fig. 2(b)].Figures 7–10 show the dependence of spin relax-

ation length on temperature. It is observed from the¯gures that when temperature increases the spinrelaxation length decreases for both types of bandstructures mentioned above. This can be concludedas both acoustic phonon scattering and polar optical

phonon scattering increases with increase in thetemperature and hence reduces spin relaxationlength. Temperature dependence can be clearly seenfrom the scattering expressions shown below

Acoustic phonon scattering rate17,18 is given by

�acn;mðkxÞ ¼

�2ackbT

ffiffiffiffiffiffiffiffiffi2m�

p

}2�v2Dnm

ð1þ 2�"fÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"fð1þ �"fÞ

p �ð"fÞ;

ð17ÞWhere �ac is the acoustic deformation potential, v isthe velocity of sound and �ð"fÞ is the heavysidestep function.Dnm is the overlap integral pertainingto electron–phonon interaction. The acousticphonon scattering rate increases with increase intemperature.

The polar optical phonon scattering rate is givenby

½W e=af;b �POðE;�Þ ¼ e2!

ffiffiffiffiffiffiffiffiffi2m�

p

}

1

e"1� 1

"es

� �nþ 1

2þ1

2

� �D Eþ}!;�ð Þ

xX1p¼1

X1r¼1

RLx

0

RLy

0 j�0ðx; yÞj2 sin p�xLx

� �sin r�y

Ly

� �dxdy

�� 2

qe=af;b

� �2 þ p�

Lx

� �2 þ r�

Ly

� �2

; ð18Þ

where

qe=af ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2m�E

p

}�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�ðEþ}!Þp

};

qe=ab ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2m�E

p

}þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�ðEþ}!Þp

}

and

n ¼ 1

e}!kbT

� �� 1

;

where ! is the angular frequency of a polar opticalphonon, e"1 and e"s are the optical dielectric con-stant and static permittivity of the compoundsemiconductor, n is the phonon occupation number.The superscripts e=a stand for emission andabsorption of phonon and subscript bðfÞ refers tobackward (forward) scattering.

It is clear from the expression of n that itincreases with increase in temperature and so doesthe polar optical phonon scattering rate which

Fig. 6. Variation of spin relaxation length with lateral electric ¯eld for GaSb–InSb core–shell nanowire (color online).

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Fig. 7. Variation of spin relaxation length with temperature for InSb–GaAs core–shell nanowire (color online).

Fig. 8. Variation of spin relaxation length with temperature for InSb–GaN core–shell nanowire (color online).

Fig. 9. Variation of spin relaxation length with temperature for GaAs–InSb core–shell nanowire (color online).

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reduces the spin relaxation length with increase intemperature.

3.3. Core diameter variation

In this section, we show the dependence of spinrelaxation length on the variation of dimensions ofcore. We have ¯xed the height and width of shell to10 nm and varied the core dimensions as 2 nm, 4 nm,6 nm and 8 nm.

(A) When shell has higher bandgap than core[Fig. 2(a)].

(B) When core has higher bandgap than shell[Fig. 2(b)].

Figures 11–14 show the variation of spin relax-ation length in III–V compound core–shell nano-wires with the variation of core dimensions in theinterval of 2 nm for both types of con¯gurationsmentioned in Fig. 2 above. From these ¯gures, weobserved that for the band diagram 2(a) (i.e., whenshell bandgap is greater than core bandgap), there isa continuous drop in the spin relaxation length as

dimensions (width and height) of core is increasedprogressively while for other type of band diagram(i.e., 2(b), when core has higher bandgap thanshell), spin relaxation length increases with increasein core dimensions. This change in the trend afterswapping core–shell materials can be explained asfollows. When core and shell materials are chosensuch that they form a potential well as in the case ofInSb–GaAs core–shell nanowire and InSb–GaSbcore–shell nanowire, core has lower energy thanshell and it tries to trap carriers inside the well. Butwhen we reduce the core dimensions, the quantumcon¯nement e®ect comes in to picture whichincreases the overall subband energy of core abovethe interface potential barrier and electron or carriercomes out of core trap and gets con¯ned to shell.Increase in core dimension decreases this quantumcon¯nement e®ect due to which large trapping ofcarriers in core occurs.

Whereas when core and shell materials arechosen such that they form a barrier [Fig. 2(b)] asin the case of GaAs–InSb core–shell nanowire and

Fig. 11. Variation of spin relaxation length with core dimension for InSb–GaAs core–shell nanowire (color online).

Fig. 10. Variation of spin relaxation length with temperature for InP–InSb core–shell nanowire (color online).

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GaSb–InSb core–shell nanowire, core has higherenergy then shell. This forms the same well inshell this time which reduces with the increase inthe core dimensions and quantum con¯nemente®ect comes into picture this time in the shell,and hence more carriers come out of shell trap

and get con¯ned to core. Hence, spin relaxationlength increases with the increase in the coredimensions. For smaller core dimensions, shell haslower energy than core, so it tries to trap carrierinside shell well and hence spin relaxation lengthdecreases.

Fig. 13. Variation of spin relaxation length with core dimension for GaAs–InSb core–shell nanowire (color online).

Fig. 12. Variation of spin relaxation length with core dimension for InSb–GaSb core–shell nanowire (color online).

Fig. 14. Variation of spin relaxation length with core dimension for GaSb–InSb core–shell nanowire (color online).

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4. Conclusion

In this paper, we have used semiclassical MonteCarlo method to show the dependence of spinrelaxation length on di®erent parameters, such as,lateral electric ¯eld, temperature and core dimensionfor III–V compound semiconductor core–shellnanowire. Depending upon the choice of materialsfor core and shell, nanowire forms two types of bandstructures and we have shown results for both typesof band structures. We found that with the increasein the lateral electric ¯eld, spin relaxation lengthincreases for both type of band structures. Whilewith the increase in the temperature, we found thatspin relaxation length decreases for both type ofband structures. Variation of core dimension givesinteresting results, for band structure 2(a) we foundthat spin relaxation length decreases with theincrease in core dimensions and for band structure2(b) reverse trend occurs. We mainly used InSb–GaSb core–shell nanowire and InSb–GaAs core–shellnanowire for band structure 2(a), and after swap-ping core–shell materials in both of the nanowires weformed nanowire having band structure 2(b).

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