Band to band tunneling in III-V semiconductors: Implications of complex bandstructure, strain, orientation, and off-zone center contributionKausik Majumdar Citation: Journal of Applied Physics 115, 174503 (2014); doi: 10.1063/1.4874917 View online: http://dx.doi.org/10.1063/1.4874917 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin splitting in bulk wurtzite AlN under biaxial strain J. Appl. Phys. 111, 103716 (2012); 10.1063/1.4720469 Theory of hole mobility in strained Ge and III-V p-channel inversion layers with high- κ insulators J. Appl. Phys. 108, 123713 (2010); 10.1063/1.3524569 Semiempirical Tight Binding Modeling of Electronic Band Structure of IIIV Nitride Heterostructures AIP Conf. Proc. 893, 171 (2007); 10.1063/1.2729824 Strain-induced changes in the gate tunneling currents in p -channel metal–oxide–semiconductor field-effecttransistors Appl. Phys. Lett. 88, 052108 (2006); 10.1063/1.2168671 Full-band-structure calculation of Shockley–Read–Hall recombination rates in InAs J. Appl. Phys. 90, 848 (2001); 10.1063/1.1381051

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

Band to band tunneling in III-V semiconductors: Implications of complexband structure, strain, orientation, and off-zone center contribution

Kausik Majumdara)

SEMATECH, 257 Fuller Road, STE 2200, Albany, New York 12203, USA

(Received 14 March 2014; accepted 22 April 2014; published online 2 May 2014)

In this paper, we use a tight binding Hamiltonian with spin orbit coupling to study the real and

complex band structures of relaxed and strained GaAs. A simple d orbital on-site energy shift coupled

with appropriate scaling of the off-diagonal terms is found to correctly reproduce the band-edge shifts

with strain. Four different h100i strain combinations, namely, uniaxial compressive, uniaxial tensile,

biaxial compressive, and biaxial tensile strain are studied, revealing rich valence band structure and

strong relative orientation dependent tunneling. It is found that complex bands are unable to provide

unambiguous tunneling paths away from the Brillouin zone center. Tunneling current density

distribution over the Brillouin zone is computed using non-equilibrium Green’s function approach

elucidating a physical picture of band to band tunneling. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4874917]

I. INTRODUCTION

Band to band tunneling (BTBT)1 is a technologically

important phenomenon2,3 which, on one hand, may result in

degradation of device performance by increasing the leakage

current in metal-oxide-semiconductor field effect transistors

(MOSFET), and on the other hand, can be tailored to obtain

ultra-low power devices, like the tunnel FET.4,5 BTBT prob-

ability in group III–V semiconductors2,3 is enhanced due to

the presence of a direct bandgap and low carrier tunneling

mass. This resulted in a tremendous amount of research

effort in the implementation of III–V based TFET in the

recent past6–8 helping to boost the drive current—one of the

important roadblocks in Si TFETs.9

Under the application of strain,10 III–V semiconductors11–16

show rich valence band structure including relative move-

ments of the different bands, possible spin splitting and band

warping.17–19 In addition, strain is expected to warp the con-

duction band as well. Such band warping and bandgap

changes affect the BTBT probability significantly, helping to

further boost the tunneling current. However, a systematic

study of such strain induced band structure effect on BTBT

is lacking in the literature. Such an investigation would

require (a) the calculation of full band structure of the

strained semiconductor and (b) solution of the proper tunnel-

ing transport equation using the calculated band structure. In

this paper, we focus on both these aspects to gain insights

into the tunneling picture in both relaxed and strained III–V

semiconductors.

To address the full band structure problem, we use an

sp3d5s* tight binding Hamiltonian to compute both real and

complex band structures. Complex bands20–27 are known to

provide the k-space path in the bandgap for the carriers28

undergoing BTBT by connecting the valence band maximum

(VBM) to the conduction band minimum (CBM). This

approach provides attractive qualitative picture of the BTBT

problem, with explanations like why light hole band is

expected to contribute more than heavy hole band in carrier

tunneling, although they are degenerate. However, this is ap-

plicable only to very simple cases and in general does not

provide a quantitative picture of the problem. Even in simple

cases like relaxed semiconductor, complex bands may not

provide a useful solution away from the Brillouin zone cen-

ter.29 This directs us to use non-equilibrium Green’s function

(NEGF) formalism30 for the transport aspect of the tunneling

problem. This, although being computationally intensive,

provides generic solutions in the whole Brillouin zone for

arbitrarily complicated band structure.

The rest of the paper is organized as follows: In Sec.

II A, we first construct the tight binding Hamiltonian in

relaxed GaAs and find real and complex band structures. In

Sec. II B, we discuss the methodology to introduce strain in

the Hamiltonian. We then discuss the complex band struc-

tures in GaAs in a variety of strain and tunneling orientation

combinations in Sec. II C. The corresponding tunneling prob-

ability calculation using WKB method is discussed in Sec.

II D. We next move on to show how complex band picture

fails to account for off-zone center tunneling in Sec. III A.

Followed by this, in Sec. III B, we discuss the NEGF

approach, validate it with experimental results for relaxed

GaAs homojunction, and elucidate the contributions of off-

zone center k points to the total tunneling current density.

Finally, the paper is concluded in Sec. IV.

II. COMPLEX BAND STRUCTURE OF STRAINEDGaAs: ORIENTATION DEPENDENT TUNNELING

A. Real and complex band calculation from tightbinding Hamiltonian

GaAs has a zinc blende crystal structure having two

inter-penetrating face-centered cubic unit cell, with the Ga

basis atom at (0, 0, 0) and the As basis atom at ð14; 1

4; 1

4Þa0,

where a0 is the lattice constant. Each Ga (As) atom has four

As (Ga) nearest neighbor atoms. A 20 orbital sp3d5s* tighta)Electronic mail: kausik.majumdar@sematech.org

0021-8979/2014/115(17)/174503/8/$30.00 VC 2014 AIP Publishing LLC115, 174503-1

JOURNAL OF APPLIED PHYSICS 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

binding Hamiltonian including spin-orbit coupling is used

with these two basis atoms to calculate the bulk band struc-

ture. Any two-center overlap integral beyond the first nearest

neighbor is neglected in the Hamiltonian matrix. The on-site

energies and the Slater-Koster overlap integral parameters

are taken from existing literature.17 Throughout this paper,

we assume coordinate axes are aligned along h100i direc-

tions. To obtain the energy eigenvalues along h100i, the

Hamiltonian is diagonalized, keeping ky and kz as zero. The

calculated band structure of bulk GaAs along h100i is shown

in blue in Fig. 1(a), indicating a direct bandgap of 1.413 eV

at the zone center, in agreement with the expected bandgap

value at room temperature.

To understand the band-edge to band-edge tunneling

probability along h100i direction, we compute the complex

bands by keeping the real part of the k vector as zero. This

forces the component of the carrier momentum perpendicu-

lar to the direction of tunneling to be zero, allowing a true

band-edge to band-edge tunneling. To simplify the problem,

we also force Im(ky)¼ Im(kz)¼ 0 and Im(kx)> 0. Note that,

if we assume x is the tunneling direction and x� 0, solutions

for Im(kx)< 0 should be neglected as they correspond to

unbounded wave functions. Once the appropriate imaginary

parts of the k vector are defined to obtain the tunneling states

inside the bandgap of the semiconductor, we diagonalize the

Hamiltonian and filter out only the real energy eigenvalues,

ignoring the eigenvalues with nonzero imaginary parts. In

Fig. 1(a), we plot the corresponding complex bands in red

along h100i direction for bulk GaAs.

B. Strain effects

Application of strain changes the Hamiltonian in two

ways—(1) scaling in the off-diagonal elements due to a

change in the overlap integrals and (2) shift in the diagonal

elements due to a change in the on-site energies of the orbi-

tals. The calculation of the off-diagonal elements in the

strained Hamiltonian is relatively straight forward. This is

achieved by changing the Slater-Koster two-center integrals

based on the generalization of Harrison’s rule by Boykin

et al.,17 by writing U¼U0(d0/d)g, where g is a fitting parame-

ter. U and U0 are the strained and unperturbed two-center inte-

grals, respectively. d (d0) is the strained (relaxed) bond length.

The required changes in the bond angles are obtained through

the changes in the directional cosines of the strained lattice.

In addition to the off-diagonal terms, application of

strain also shifts the on-site energies of the orbitals, forcing a

change in the diagonal terms of the Hamiltonian. This is

because the strain is expected to lift the degeneracy among

the px,y,z orbitals and also among the dxy,yz,zx orbitals. This di-

agonal element shift problem has been discussed in detailed

in the literature.17,19,31 In this work, we leave the p orbitals

unchanged and adopt the simple d orbital energy shift as pre-

scribed by Jancu et al.19

Exy ¼ Ed½1þ 2adð�zz � �xxÞ� (1)

and

Ezx ¼ Eyz ¼ Ed½1� adð�zz � �xxÞ�; (2)

where ad is a single fitting parameter and �ii are the diagonal

elements of the strain tensor, the off-diagonal elements being

zero. However, instead of using a fixed ad,19 we found that

ad¼ 0.4 for tensile strain and ad¼�0.2 for compressive

strain reproduces the results obtained by Boykin et al.17 with

good agreement using a similar strain setting, as shown in

Fig. 1(b). Such a single parameter fitting with good accuracy

is attractive due to simplification of the multiple additional

tight binding parameter fitting problem.17 In the same figure,

we also plot the results obtained from k.p method for

comparison.17,18

C. Complex bands with different orientations relativeto strain

To understand the relative orientation effect between

applied h100i strain and tunneling direction, we consider

four general cases, namely, (1) biaxial strain in xy plane and

carrier tunneling along z, i.e., perpendicular to the plane of

biaxial strain (type I biaxial), (2) biaxial strain in xy plane

and carrier tunneling along x, i.e., in the plane of biaxial

strain (type II biaxial), (3) uniaxial strain along x and carrier

tunneling along the same direction (type I uniaxial), and (4)

uniaxial strain along in x and carrier tunneling along z, i.e.,

perpendicular to the direction of the strain (type II uniaxial).

For uniaxial strain �zz¼ �0, we take �xx¼ �yy¼�q�0, where qis the Poisson’s ratio (0.31 for GaAs). For biaxial strain of

magnitude �0 in xy plane, we obtain �xx¼ �yy¼ �0(1�q) and

�zz¼�2q�0.

The band structures of these four cases are summarized

in Figs. 2–5. In all the figures, the top row corresponds to

compressive strain (2% on the left and 4% on the right), while

the bottom row shows results for tensile strain of similar

FIG. 1. (a) Real (in blue) and complex

(in red) band structure of bulk GaAs. (b)

Bandgap versus strain in GaAs with the

present model in solid line, tight binding

model17 in circle and k.p model17,18 in

square, using D001¼ 0.934. Negative

and positive strains correspond to com-

pressive and tensile strains, respectively.

174503-2 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

magnitudes. We observe from Fig. 2 that biaxial compressive

strain pulls up the heavy hole (HH) band, suppressing light

hole (LH) band. On the other hand, biaxial tensile strain pulls

light hole band upward. The corresponding complex bands for

kx¼ ky¼Re(kz)¼ 0 and Im(kz)> 0 are shown in red. Clearly,

the branch from LH maximum connects to CBM in all the

cases. On the other hand, the branches from HH maximum

and split-off (SO) band maximum connect to higher bands at

the zone center. For compressive stress, this allows a cross

over between two branches of the complex bands [Figs. 2(a)

and 2(b)]. Such change in the band structure is expected to

result in an enhanced tunneling probability for the tensile

strain and suppress it for compressive strain.

We observe a qualitatively different band structure for

type II biaxial case where we plot E versus kx, i.e., in the

direction of one of the stress axes, as shown in Fig. 3. In par-

ticular, we note that spin degeneracy is lifted for nonzero k.

III–V semiconductors including GaAs lack the center of

inversion symmetry in the zinc blende lattice which strength-

ens spin splitting. Also the off-zone center k points exhibit

FIG. 2. Band structure along kz for

biaxial strain in {100} x-y plane (type I

biaxial) in GaAs with (a) 2% compres-

sive and (b) 4% compressive, (c) 2%

tensile and (d) 4% tensile strain.

FIG. 3. Band structure along kx for

biaxial strain in {100} x-y plane (type

II biaxial) in GaAs with (a) 2% com-

pressive and (b) 4% compressive, (c)

2% tensile and (d) 4% tensile strain.

174503-3 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

lower symmetry compared with the zone center. On top of

this, type-II biaxial strain forces additional asymmetry along

x direction. An interplay of all these effects helps to lift the

spin degeneracy away from the zone center, as shown in

Fig. 3. There is a branch of complex band that exists for

ky¼ kz¼Re(kx)¼ 0 and Im(kx)> 0 which still connects the

CBM and the VBM, allowing a direct calculation of band to

band carrier tunneling probability. Lifting of the degeneracy

between the two spins in the valence band away from the

zone center results in a spin dependent energy separation

between the conduction band and the valence band (or the

effective bandgap at a given k 6¼ 0). This difference in the

effective bandgap at nonzero k is in turn expected to result

in a spin dependent band to band tunneling probability.

Since electron spin is conserved during such band to band

tunneling, this non-trivial spin dependent tunneling could

have interesting implications in device applications.

In the case of uniaxial strain along x and carrier tunnel-

ing along the same direction (type I uniaxial), contrary to the

biaxial case, compressive strain pulls the LH bands up, while

FIG. 4. Band structure along kx for

uniaxial strain along x (h100i) direc-

tion (type I uniaxial) in GaAs with (a)

2% compressive and (b) 4% compres-

sive, (c) 2% tensile and (d) 4% tensile

strain.

FIG. 5. Band structure along kz for uni-

axial strain along x (h100i) direction

(type II uniaxial) in GaAs with (a) 2%

compressive and (b) 4% compressive,

(c) 2% tensile and (d) 4% tensile

strain.

174503-4 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

tensile strain pushes the LH bands down, as shown in Fig. 4.

The complex bands show similar behavior as in the biaxial

case, with a cross over between the HH and LH branches for

the tensile stress. In the case of tunneling direction perpen-

dicular to the direction of uniaxial stress (type II uniaxial),

due to symmetry breaking as in case 2, we observe similar

lifting of spin degeneracy away from the Brillouin zone cen-

ter, as shown in Fig. 5.

Fig. 6 shows the sensitivity of band edge positions at the

zone center with different types of strain. We consider the

conduction band and three bands from the valence band,

namely, HH, LH, and SO. Note that, the position of the con-

duction band edge is relatively less sensitive to strain, particu-

larly for uniaxial case. On the other hand, valence band has a

stronger sensitivity with strain with a clear lift in degeneracy

between HH and LH, as expected from the band structure dis-

cussion above. Although qualitatively the band shifts are simi-

lar for uniaxial and biaxial strains, we note that the calculated

bandgaps (by taking the difference between CBM and VBM)

show different trends, as illustrated in Fig. 7. Biaxial compres-

sive strain increases the bandgap of GaAs, while biaxial ten-

sile strain reduces the bandgap significantly. On the other

hand, uniaxial stress is found to reduce the bandgap for both

compressive and tensile strains. Such a difference is primarily

because of the relative sensitivity of CB and VB edges with

different types of strains.

D. Band edge tunneling probability from complexbands

In Fig. 8, we plot the band-edge to band-edge tunneling

probability (T) at the zone center for all the four cases, as a

function of strain and electric field (F). T is calculated using

WKB method as

FIG. 6. Energy position of the conduction band edge and three valence

bands (HH, LH, and SO) of GaAs at the zone center (k¼ 0) for both uniaxial

(in red circle) and biaxial (in blue square) strain conditions.

FIG. 7. Bandgap for uniaxial and biaxial strain in GaAs. Biaxial strain

monotonically increases the bandgap from tensile to compressive strain,

while uniaxial strain reduces the bandgap for both compressive and tensile

strain.

FIG. 8. (a) Qualitative description of

the band edge to band edge tunneling

probability calculation, depending on

the area of the shaded region defined

by the connecting complex band.

(b)–(e) 3D surface plot of band-edge to

band-edge tunneling probability calcu-

lated from WKB least path integral as

a function of strain and electric field,

for four different cases.

174503-5 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

T ¼ e�2Ðjkjdx ¼ e

�ð2=qFÞÐ Ec

EvjkjdE

; (3)

which assumes constant F in the depletion region. Note that

the integral in the exponent can be calculated from the

derived complex bands above and is schematically shown by

the shaded area in Fig. 8(a). Clearly, the magnitude of the

shaded area is governed by the bandgap and also by the cur-

vature of the complex band, which can parametrically be

denoted by kp, the peak imaginary k through which the com-

plex band traverses. Noting that the curvature of the complex

band follows that of real bands at the band edges, both the

magnitude of kp and its position in the energy scale are

strong functions of the effective masses of carriers at the

CBM and the VBM. We also note that we only consider tun-

neling through the path of least action,28 which means that

any cross over in complex bands (particularly when the HH

band comes above LH band) would mean a carrier shifts

from one complex branch to the other during tunneling.

From Fig. 8, we find that for biaxial strain, T monotonically

increases from compressive to tensile strain for both type I

and type II. On the other hand, for uniaxial strain, T is a non-

monotonic function of strain, in agreement with the bandgap

trends discussed previously. Biaxial type I tensile strain is

found to provide the maximum tunneling probability among

all the cases studied.

III. OFF-ZONE CENTER TUNNELING CONTRIBUTIONS

A. Complex band picture away from the zone center

Till now we have only considered tunneling from VBM

to CBM at the Brillouin zone center and the picture can be

well understood under the light of complex band structure,

as shown above. However, off-zone center tunneling plays

an important role in determining the total BTBT current den-

sity. Unfortunately, complex band picture does not provide

an unambiguous picture of the off-zone center tunneling sce-

nario. To illustrate the point, in Fig. 9, we show the off-zone

center complex bands for relaxed GaAs, with Re(kx)

¼Re(ky) 6¼ 0, keeping Re(kz)¼ 0 (assuming tunneling along

z direction). Clearly, away from the zone center, there is no

complex band branch that directly connects the VBM to

CBM. This leaves it difficult to calculate the tunneling prob-

ability beyond the simple case of zone center. This off-zone

center tunneling problem has been tackled in the Kane’s

model1 by assuming a “perpendicular energy component”(E?) that adds a multiplicative factor to the band-edge tun-

neling probability and taking an integration over E?.

However, in a full-band simulation, NEGF allows a more el-

egant way to capture such off-zone center tunneling.

B. NEGF approach to tunneling

In order to find the relevant Green’s function (G), we

first construct a basis set consisting of n monolayers32 to

write down the tight-binding Hamiltonian (H) as a function

of kx and ky, assuming tunneling direction along z. The num-

ber n depends on the thickness of the bulk n-type and p-type

regions and the width of the depletion region sandwiched in

between. For simplicity, we assume a constant field in the

depletion region. We obtain the Green’s function of the system

using GðE; kx; kyÞ ¼ ½ðEþ jfÞI � Hðkx; kyÞ � RlðE; kx; kyÞ�RrðE; kx; kyÞ��1; where the self-energies for the left (Rl) and

right sides (Rr) of the two terminal device are calculated using

an iterative approach.33 f is an infinitesimal positive quantity.

The (kx, ky) resolved current density is then found as

Jðkx; kyÞ ¼q

h

ðE

TðE; kx; kyÞ½flðEÞ � frðEÞ�dE; (4)

where TðE; kx; kyÞ ¼ TraceðClGCrG†Þ. fl and fr are the left

and right contact Fermi-Dirac probabilities given by

FIG. 9. Complex band structure for relaxed GaAs with varying kx¼ ky, with Re(kz)¼ 0. The lack of connection between VBM and CBM makes it difficult to

calculate the WKB integral.

FIG. 10. Calibration of simulated Zener current density using NEGF (line)

and corresponding experimental data (symbol) for relaxed GaAs homo-

junction diode at room temperature. Inset: The MBE grown P-i-N layer

stack of the diode.

174503-6 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

flðrÞ ¼ 1

1þeðE�llðrÞÞ=kBT . ll(r) is the left (right) contact Fermi level.

Note that the two spins are already taken into account in the

tight-binding Hamiltonian. Cl and Cr are obtained from the

self-energies as Cl ¼ j½Rl � R†

l � and Cr ¼ j½Rr � R†

r �.To validate the calculation, we calibrated the NEGF pre-

dicted total tunneling current density in the Zener region

(reverse bias) with measured GaAs homo-junction tunnel

diode data at room temperature in the external bias range of

0–0.5 V, as shown in Fig. 10. The GaAs diode is fabricated

using molecular beam epitaxy (MBE) grown layers on a

GaAs substrate, as shown in the inset of the figure. The target

in-situ doping values are around 4� 1019 per cc. We

describe the device in the simulation as discussed above, by

constructing the Hamiltonian which adequately covers the

whole depletion region and extending both the edges into the

bulk doped regions. We extracted a series resistance of

4� 10�8X-m2 from the high bias regime of the measured

I–V curves. No other fitting parameter was used in the trans-

port simulation. The simulated current densities at different

bias points are found to be in good agreement with the meas-

ured data.

In Fig. 11, we resolve the total J as a function of kx and

ky around the Brillouin zone center, for two different bias

conditions. The results have some key features: (i) Only a

very small fraction of the whole Brillouin zone is found to

contribute to the total J. This is attributed to the fact that as

one moves away from the one center, the separation

between the valence band edge and the conduction band

edge increases, which in turn exponentially reduces the tun-

neling probability. (ii) The relative contributing area in the

k space increases with the increase in the reverse bias. This

is because as the reverse bias increases, the effective elec-

tric field in the depletion region also increases, which in

turn allows tunneling to higher energy states. Thus, states

away from zone center with relatively higher effective band

gap start to contribute at higher electric field. Qualitatively,

this is in agreement with the parallel momentum picture in

Kane model. (iii) However, one striking difference from the

Kane type model is that the peak of J does not necessarily

occur at the Brillouin zone center, as seen for the higher

reverse bias in Fig. 11(b). This may be attributed to the

complex nature of the valence band, with varying tunneling

contributions from LH and HH bands due to the varying

position of the band edges in the energy scale and also

varying curvature of bands at different k (or different effec-

tive mass at different k) values away from the zone center,

as explained previously in Fig. 9. We will not discuss the

off-zone center effects of tunneling here any further for the

different strain cases, however, noting that the effective

band gap increases away from the zone center for all the

different strain situations discussed earlier (Figs. 2–5) and

also higher reverse bias tends to increase the depletion field

irrespective of the strain type, the above observations (i)

and (ii) are qualitatively expected in different strain condi-

tions as well. However, a quantitative picture of the off-

zone center tunneling at different strain conditions is

beyond the scope of the present work and is a topic of sepa-

rate study.

IV. CONCLUSION

In conclusion, we used an sp3d5s* tight binding

Hamiltonian to investigate the band structure effects on the

band to band tunneling in relaxed and strained III–V semi-

conductors, with GaAs as an example. The major findings

can be summarized as follows:

(1) Only dxy,yz,zx orbital on-site energy shifts with a strain de-

pendent single fitting parameter (coupled with usual

off-diagonal overlap integral scaling) was able to repro-

duce band edge shifts with strain correctly. This simple

approach reduces the effort in multiple additional tight

binding parameter fitting.

(2) Valence band shows rich structure with different strain

conditions. Four different strains and relative orientation

effects were studied, which can be implemented in a de-

vice. Biaxial tensile strain with a tunneling direction

FIG. 11. Color plot of the distribution of tunneling current density in the region close to zone center for relaxed GaAs homo-junction at room temperature at a

reverse bias of (a) 0.1 V and (b) 0.5 V. The contributing range of k space is small compared with the size of the Brillouin zone and increases with the applied

reverse bias.

174503-7 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37

perpendicular to the plane of strain is found to produce

the highest tunneling probability.

(3) It was found that one can obtain spin dependent

tunneling away from the zone center by choosing the

right orientation with respect to strain—this effect can

be attributed to the symmetry breaking in the

Hamiltonian. This finding could have interesting device

applications.

(4) It was explicitly shown that away from the zone center,

complex bands do not connect the band edges and thus it

becomes ambiguous to calculate tunneling probability.

However, full quantum approaches like NEGF can

adequately take the whole Brillouin zone into account in

the expense of increased computational requirement.

(5) Tunneling current density calculated from NEGF formal-

ism was resolved in k-space to provide fundamental

insights on the role of parallel k states in the tunneling

phenomenon. It was found that only those states, which

are close to the zone center, contribute sufficiently to the

total tunneling current, which qualitatively agrees with a

Kane-type picture. With an increase in field, the contribut-

ing area increases. Interestingly, the peak current contrib-

uting state did not necessarily occur at the zone center.

Finally, all the results presented in this paper are on a

single III–V material, namely, GaAs. However, noting that

all the binary III–V semiconductors have zinc blende crystal

structure with lack of center of inversion symmetry and ex-

hibit qualitatively similar band structure, the implications

discussed in the paper are expected to hold for all of them,

with a possible quantitative difference.

ACKNOWLEDGMENTS

The author wishes to acknowledge the experimental

data support from S. L. Rommel, RIT.

1E. O. Kane, J. Appl. Phys. 32, 83 (1961).2D. Pawlik et al., Tech. Dig. - Int. Electron Devices Meet. 2012, 27.1.1.3B. Romanczyk, P. Thomas, D. Pawlik, S. L. Rommel, W.-Y. Loh, M. H.

Wong, K. Majumdar, W.-E. Wang, and P. D. Kirsch, Appl. Phys. Lett.

102, 213504 (2013).

4A. C. Seabaugh and Q. Zhang, Proc. IEEE 98, 2095 (2010).5A. M. Ionescu and H. Riel, Nature 479, 329 (2011).6G. Dewey et al., Tech. Dig. - Int. Electron Devices Meet. 2011, 785.7D. Mohata, B. Rajamohanan, T. Mayer, M. Hudait, J. Fastenau, D.

Lubyshev, A. W. K. Liu, and S. Datta, IEEE Electron Dev. Lett. 33, 1568

(2012).8R. Li, Y. Lu, G. Zhou, Q. Liu, S. D. Chae, T. Vasen, W. S. Hwang, Q.

Zhang, P. Fay, T. Kosel, M. Wistey, H. Xing, and A. Seabaugh, IEEE

Electron Device Lett. 33, 363 (2012).9K. Boucart and A. M. Ionescu, IEEE Trans. Electron Device 54, 1725

(2007).10Y. Sun, S. E. Thompson, and T. Nishida, J. Appl. Phys. 101, 104503

(2007).11C. P. Kuo, S. K. Vong, R. M. Cohen, and G. B. Stringfellow, J. Appl.

Phys. 57, 5428 (1985).12E. Yablonovitch and E. O. Kane, J. Lightwave Technol. 6, 1292 (1988).13M. P. C. M. Krijn, Semicond. Sci. Technol. 6, 27 (1991).14M. Grundmann, O. Stier, and D. Bimberg, Phys. Rev. B 52, 11969

(1995).15I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89,

5815 (2001).16C. M. N. Mateo, A. T. Garcia, F. R. M. Ramos, K. I. Manibog, and A. A.

Salvador, J. Appl. Phys. 101, 073519 (2007).17T. B. Boykin, G. Klimeck, R. C. Bowen, and F. Oyafuso, Phys. Rev. B 66,

125207 (2002).18C. G. V. de Walle, Phys. Rev. B 39, 1871 (1989).19J.-M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Phys. Rev. B 57, 6493

(1998).20Y. C. Chang, Phys. Rev. B 25, 605 (1982).21T. B. Boykin, Phys. Rev. B 54, 8107 (1996).22P. Mavropoulos, N. Papanikolaou, and P. H. Dederichs, Phys. Rev. Lett.

85, 1088 (2000).23J. K. Tomfohr and O. F. Sankey, Phys. Rev. B 65, 245105 (2002).24P. H. Dederichs, P. Mavropoulos, O. Wunnicke, N. Papanikolaou, V.

Bellini, R. Zeller, V. Drchal, and J. Kudrnovsky, J. Magn. Magn. Mater.

240, 108 (2002).25T.-S. Xia, L. F. Register, and S. K. Banerjee, Phys. Rev. B 70, 045322

(2004).26N. F. Hinsche, M. Fechner, P. Bose, S. Ostanin, J. Henk, I. Mertig, and P.

Zahn, Phys. Rev. B 82, 214110 (2010).27R. K. Pandey, K. V. R. M. Murali, S. S. Furkay, P. J. Oldiges, and E. J.

Nowak, IEEE Trans. Electron Device 57, 2098 (2010).28S. Laux, 13th International Workshop on Computational Electronics

(2009), p. 1.29G. Klimeck, R. C. Bowen, and T. B. Boykin, Phys. Rev. B 63, 195310

(2001).30M. Luisier and G. Klimeck, J. Appl. Phys. 107, 084507 (2010).31C. Tserbak, H. M. Polatoglou, and G. Theodorou, Phys. Rev. B 47, 7104

(1993).32K. Majumdar and N. Bhat, J. Appl. Phys. 103, 114503 (2008).33J. Guo, S. Datta, M. Lundstrom, and M. P. Anantram, Int. J. Multiscale

Comput. Eng. 2, 257 (2004).

174503-8 Kausik Majumdar J. Appl. Phys. 115, 174503 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.47.65.106 On: Sat, 18 Oct 2014 11:54:37