=U4358815JK464150

8/4/2019 =U4358815JK464150

1/8

Journal of Computational Electronics 3: 512, 2004

c 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Modeling of Current Density Boundary Conditions for a HeterodimensionalContact of 3D-Metal to 2D-Semiconductor

RAJESH A. KATKAR AND GREGORY B. TAIT

Electrical Engineering Department, Virginia Commonwealth University, Richmond, VA 23284, USA

Abstract. We have investigated semi-classical carrier transport at the heterodimensional contact between a three-

dimensional (3D)-metal and two-dimensional (2D)-semiconductor system. Formulated for easy inclusion in nu-

merical device simulators, current density boundary conditions for the heterodimensional contact are theoretically

derived for both electrons and holes. Unlike conventional metal-semiconductor-metal (MSM) photodetectors, in

which planar surface electrodes are fabricated to establish contact with the semiconductor, a novel MSM device is

considered by utilizing recessed electrodes that directly contact the 2D-quantum well. The newly derived current

density boundary conditions have been incorporated in a commercial numerical device simulator to investigate the

steady-state and transient behavior of a novel quantum-well MSM photodetector. Results for devices with 1-m

electrode spacing at 2 V DC bias indicate a nominal 3-dB bandwidth of 25 GHz and low dark currents in the

quantum well of the order of a few femto-amps.

Keywords: GaAs/AlGaAs quantum well, current density boundary conditions, heterodimensional Schottky con-

tact, MSM photodetector

1. Introduction

A comprehensive research towards exploring the the-

oryand realization of low dimensional systems is aimed

at achieving superior performance of semiconductor

devices. Although optical and electronic properties of

these reduced dimensional systems are significantly

different from the bulk materials, the electronic con-

tact between these different dimensional systems is in-

evitable to transfer the information to and from the ex-

ternal circuits. Thus, an extensive study is required toanalyze the contact between a 3D-bulk metal and 2D as

well as 1D-semiconductor to keep pace with thecontin-

uous demand to improve speed, responsivity, transport

efficiency and overall device performance.

Various models have been developed in the past to

accurately describe the current density at the Schot-

tky boundary, a metal contact with the bulk semicon-

ductor. If the depletion region is sufficiently large,

the electrons undergo many collisions within this re-

gion. In such case, drift-diffusion models (put for-

ward by Schottky in 1938) more closely model the

current density in the bulk semiconductor as it takes

into account the scattering in the bulk. When the de-

pletion region is sufficiently small and the career free

path length is large with respect to the thickness of

the potential barrier, electron current from semicon-

ductor into the metal is governed by thermionic emis-

sion mechanism in which scattering is not very impor-

tant. Hence, thermionic emission model (put forward

by Bethe in 1942) more accurately models the cur-

rent density at the contact. Thus, the major models,

which most accurately describe the current density atthe Schottky contact, are based on the combined model

called the combined drift-diffusion/thermionic emis-

sion (DD/TE) model [1]. This combined DD/TE model

enables us to accurately describe the current density in

a self-consistent manner throughout the device. Vari-

ous revisions have been suggested to correct several

theoretical problems in the original DD/TE equations

due to various assumptions in its derivations. The uni-

directional normalization to calculate average electron

velocities was suggested as the first revision [1], while

the second revision [2] allowed the surface velocity of

8/4/2019 =U4358815JK464150

2/8

6 Katkar

the electron flux moving from semiconductor to metal

to vary under bias conditions. A third revision [3] wasmade to determine the separate flux surface velocities

and a fourth revision [4,5] introduced the fraction of

total carriers involved in each flux term. It also allowed

the fraction of total electrons that move from semicon-

ductor into the metal to vary with the applied voltage.

In this paper, we investigate the heterodimensional

Schottky contact where bulk metal is in contact with

the quantum well. In a quantum well, the energies are

discretized in the direction normal to the direction of

growth which affects the carrier density of states, their

concentration, carrier surface velocity and also the frac-

tion of carriers moving from semiconductor into the

metal andfrom themetal into thesemiconductor. In thiswork, we derive the two dimensional current density

expression and the boundary conditions for both elec-

trons and holes across this heterodimensional Schottky

contact.

2. Current Density Boundary Conditions

for Electrons

A quantum well is formed by sandwiching a GaAs

layer between two AlGaAs layers. Note that, through-

out the calculations, z-direction is the direction of crys-

tal growth. The semi-classical electron current density

Je,x in x-direction in the QW can be given as

Je,x = q

vege feve,x dve (1)

where ve is the electron velocity and ve,x is the x-

component of electron velocity, ge is the 2D density

of states of electrons in velocity space and fe is the ve-

locity dependent distribution function for electrons. To

keepthe dark current low inMSMdevices, weuse a low

doping concentration in the semiconductor. Thus, an

approximate displaced Maxwellian distribution func-tion becomes

fe = expEc + n + 12 me [(ve,x vd,e)2 + v2e,y

EFnKB T

(2)

where Ec is conduction band edge, n is the elec-

tron sub-band energy in the quantized z-direction,

while vd,e is the effective electron drift-diffusion ve-

locity in x-direction which is identical to the displace-

ment of electron distribution along ve,x axis. vd,e is

calculated as

vd,e = Je,xqn

(3)

If we consider the negative x-directional and positive

x -directional velocity components of electrons sepa-

rately, (1) can be rewritten as [5]

Je,x = q

Iy

0ve,x=

ge fe,xve,x dve,x

q

Iy

ve,x=0

ge fe,xve,x dve,x

, where

Iy =

f

e,ydv

e,y. (4)

We introduce an average velocity term from (4) in the

following manner,

Je,x = q

Iy0 ge fe,xve,x dve,x

Iy0 ge fe,x dve,x

Iy

0

ge fe,x dve,x

q

Iy

0ge fe,xve,x dve,x

Iy 0 ge fe,x dve,x

Iy

0

ge fe,x dve,x

(5)

where

Iy0 ge fe,xve,x dve,x

Iy0 ge fe,x dve,x

= Je,xqn

ve,x and

Iy

0ge fe,xve,x dve,x

Iy

0ge fe,x dve,x

= Je,x+qn+

ve,x+(6)

Here ve,x and ve,x+ are the average velocities of elec-trons moving in negative x-direction and positive x-

direction respectively. We introduce the fraction of

electron terms (fraction of electrons moving in posi-

tive and negative x-direction) from (5) as,

Je,x =qve,x

Iy0 ge fe,x dve,xve

ge fedve

ve

ge fedve

qve,x+

Iy

0ge fe,x dve,x

vege fedve

ve

ge fedve

(7)

8/4/2019 =U4358815JK464150

3/8

Modeling of Current Density Boundary Conditions 7

where

Iy

0 ge fe,x dve,xve

ge fedve= n

n Fe,x and

Iy

0ge fe,x dve,x

vege fedve

= n+

n Fe,x+

(8)

Here Fe,x and Fe,x+ are the fractions of total elec-

trons moving in negative and positive x -directions re-

spectively. Thus electron current density expression be-

comes

Je,x = q(ve,x )(Fe,x )n q(ve,x+ )(Fe,x+ )n (9)

The average electron velocities in thex and +x direc-tion have been calculated for a displaced Maxwellian

(2) as,

ve,x = vd,e vt,eexp

2e1 erf(e)

ve,x+ = vd,e + vt,eexp

2e1 erf(e)

(10)

Note that sign for the error function in ve,x isused when vd,e 0, respectively. In case of the error

function in ve,x+ term, sign is used when vd,e >< 0,respectively.In (10), vt,e is the one-dimensional electron thermal

velocity given as

vt,e =

2KB T

me(11)

and, the term e is given as

e = |vd,e|

mn2KB T

= 1

vd,e

vt,e

. (12)

The fraction of electrons terms have been calculated

using (8) as,

Fe,x = 12

[1 erf(e)] Fe,x+ = 12

[1 erf(e)](13)

sign for the error function in Fe,x is used whenvd,e 0, respectively. In case of the error function in

Fe,x+ term,signisusedwhenvd,e >< 0, respectively.Consider the Schottky contact in which metal is on

the left and the semiconductor is on the right. Current

density due to electrons moving from metal into the

semiconductor is constant. Under equilibrium as to-tal current density is zero and Je,M>S is constant andis equal and opposite to Je,S>M . We can also calcu-late from (13) that under equilibrium, the fraction F

of electrons moving in either direction (x or+x di-rection) is exactly 1/2. ve,x+ , the average velocity of

electrons from metal to semiconductor under equilib-

rium, calculated by putting vd,e+ in (10), becomes vt,e.

If we assume the equilibrium electron density to be n0at the semiconductor interface, current density due to

electrons from metal to semiconductor becomes

Je,x+

=Je,M

>S

= qvt,e

n0

2

(14)

Thus, electron current density expression can be rewrit-

ten as

Je,x = qvs,eFn vt,e n0

2

(15)

where vs,e = |ve,x | is the electron surface velocityand F = Fe,x, the fraction of electrons moving fromsemiconductor into the metal.

When forward bias is applied to the Schottky con-

tact, the effective electron drift-diffusion velocity vd,ebecomes negative. At higher voltages, F increases to

1 while electron surface velocity vs,e is asymptotic to

|vd,e|. Hence under high forward bias, most of the elec-trons from the semiconductor side are thermally emit-

ted into the metal.

If the Schottky contact is subjected to reverse bias,

vd,e becomes slightly positive. For reverse bias volt-

ages, F = 1/2 and n reduces to zero. Hence, totalelectron current density is primarily composed of the

electrons emitted from the metal to the semiconductor

only.

Total number of electrons at the boundary is calcu-

lated self-consistently in the numerical simulator as

n =

vege f dve

=

1

L z

meKB Th2

n

exp

n

KB T

exp

EFn ECKB T

(16)

In (16), the term 1Lz

meKB Th

2

n exp( nKB T ) = N2DC

is the effective 2D density of states of the conduction

band.

8/4/2019 =U4358815JK464150

4/8

8 Katkar

Equation (15) allows us to implement the new cur-

rent density boundary condition for electrons at aheterodimensional Schottky contact in commercial nu-

merical device simulators.

3. Current Density Boundary Conditions

for Holes

While we calculate the hole current density, we must

take into account thefactthat it iscomposedof transport

of heavy holes andlightholes. Hole current density Jp,xin x-direction can be given as

Jp,x = q i=l,h

vPi

gpi (1 fpi )vpi,x dvpi (17)where vpi is the hole velocity (i represents heavy holeor light hole), gpi is the 2D-hole density of states in v-

space while fpi is the velocity dependent distribution

function for holes.

The hole distribution function can be approximated

by a displaced Maxwellian distribution given as

1 fpi

= expEV n,pi +12

mpi [(vpi,xvd,pi )

2

+v2pi,y ] E f pKB T

(18)

where vd,pi is the effective drift-diffusion velocity of

theholes and mpi is the massof the hole (light or heavy)in the nth valence sub-band.

As we calculated for electrons, considering posi-

tive and negative x-directional velocity components of

holes, the hole current density expression canbe rewrit-

ten as

Jp,x = q i=l,h

Iiy0

vpi,x=gpi (1 fpi,x )vpi,x dvpi,x

+

vpi,x=0gpi (1 fpi,x )vpi,x dvpi,x

(19)

If we follow the same steps as we did for electron cur-

rent density in (4)(15), we get

Jp,x = q

i=l,h[(vpi,x )(Fpi,x )p q(vpi,x+ )(Fpi,x+ )p]

(20)

The final expression for hole current density for the

Schottky contact in which metal is on the left whilesemiconductor is on the right, can be written analo-

gously to (15) as

Jp,x = qvt,p

p0

2 vs,pF p

(21)

where vs,p = |vp,x | is the hole surface velocity,F = Fp,x is the fraction of holes moving from semi-conductor into the metal and vt,p is the combined (i.e.

light and heavy) hole thermal velocity which can be

expressed, taking into account only the first sub-band,

as

vt,p =

i=l,h

mpi exp 1,pi

KB T

vt,pi

i=l,h

mpi exp 1,pi

KB T

,

vt,pi =

2KB T

mpi(22)

At equilibrium, the hole surface velocity approaches

the combined hole thermal velocity value, while frac-

tion F reaches 1/2 and hole current density subse-

quently becomes zero.

When forward bias is applied to theSchottkycontact,

the hole drift-diffusion velocityvd,pi becomes negative.At higher voltages, F increases to 1 while hole surface

velocity vs,p is asymptotic to the effective hole drift-

diffusion velocity |vd,pi |. Hence under high forwardbias, a majority of holes from the semiconductor side

is emitted into the metal.

Total number of holes at the boundary is calculated

self-consistently in the numerical simulator as,

p =i=l,h

vpi

gpi (1 fpi )dvpi

=1

L z

KB T

h2

i=l,h

mpi

nexp

n,piKB T exp

EV EF p

KB T

(23)

In (23), the term 1Lz

KB T h2

i=l,h (m

pi

n exp( n,piKB T ))= N2DV is the effective 2D density of states of the va-

lence band.

Equation (21) allows us to implement the new cur-

rent density boundary condition for holes at a heterodi-

mensional Schottky contact in commercial numerical

device simulators.

8/4/2019 =U4358815JK464150

5/8


4. Results and Discussion

ConventionalMSM devicesuse planer metal electrodes

which are deposited at the top of the deviceand theyare

not in actual contact with the 2DEG [68]. This pro-

duces non-uniform electric field across the device and

creates weak electric field intensity regions deep inside

the device. Carriers move slowly deep inside the device,

effectively increasing their transit time and reducing

device speed. In this work, we have investigated MSM

devices with recessed electrodes to establish a direct

contact between 3D metal and 2-D quantum channel

[5,912]. This makes the electric field in the semicon-

ductor layers of the device more uniformly strong and

more uni-directional, as shown in Fig. 1. Thus, the car-

rier motion is mostly constrained in the direction of

the field, which ultimately increases the speed of the

device.

Various numerical device simulations with the new

boundary conditions incorporated, were performed

with the aid of Silvacos device simulation software

[13]. For illustrative purposes, we have considered sim-

ple single-quantum-well structures formed by sand-

wiching a GaAs layer between two AlGaAs layers. For

the devices studied, the thickness of both AlGaAs lay-

ers have been kept constant at 500 A while the GaAs

Figure1 . Electric fieldprofile insideGaAs/AlGaAsquantum-well MSM photodetector with recessed anode(left vertical boundary) and cathode

(right vertical boundary) contacts. The 150-A GaAs quantum well is located between the AlGaAs confining layers.

layer thickness has been varied at 50, 100 and 150 A.

The DC-bias performance of the devices is shownin Fig. 2. The electron barrier height for the metal-

quantum well (GaAs channel) contact is 0.8 eV while

for metal-AlGaAs (with 40% Al concentration) con-

tact, it is 1.12 eV. Bias voltage of 2 V is applied across

the device. Figure 2 shows the dark current generated

in all three devices with different quantum well thick-

ness. The sub-band energies of the quantum well are

inversely proportional to the well thickness. The cur-

rent density decreases exponentially with the barrier

height at the metal-semiconductor contact. Thus, cur-

rent flowing in the device with smaller QW thickness is

considerablyless than that in the device with larger QW

thickness, primarily because of their different sub-band

energies (both conduction band and valence band) that

effectively increase the barrier height. Smaller cross-

sectionalarea of GaAs QW in devices with smaller QW

thickness is also another factor for the current being

lower. The current flowing in AlGaAs layers is very

small compared to the GaAs channel current, due to

the higher barrier height, despite the AlGaAs cross-

sectional area being an order of magnitude larger than

GaAs channel area. As AlGaAscurrent is much smaller

than the GaAs current, total current is primarily due

to the GaAs current contribution, although some real

8/4/2019 =U4358815JK464150

6/8

10 Katkar

Figure 2. Dark current characteristics of quantum-well MSM photodetectors with 1-m electrode spacing.

Figure 3. Transient quantum-well MSM photocurrent response to 820-nm light impulse.

8/4/2019 =U4358815JK464150

7/8


Figure 4. Frequency response of 50-A quantum-well MSM photodetector at 2-V DC bias and 1-m electrode spacing.

space transfer of electrons is observed. The dark cur-

rent we simulated in these single-quantum-well MSM

structures is in the range 1 10161 1014 A/m(Fig. 2).

A light impulse of wavelength 820 nm and intensity

10,000 W/cm2 at 2V bias is shined on the device. At

820 nm, all the electron-hole pairs are generated in the

GaAs channel. Figure 3 shows the total current tran-

sients in all three devices. We can see the increase in

the total current in the devices according to the increase

in well thickness. Figure 4 shows the 3 dB electrical

bandwidth response of the 50 A QW MSM device de-

rived by Fourier Transform of the full transient cur-

rent responses in Fig. 3. We observe comparable 3 dB

bandwidths for the three QW device thicknesses in this

study. Thenominal 3 dB-bandwidthis 25 GHzfor these

devices with 1-micron electrode finger spacing at 2V

DC bias.

5. Conclusion

Current density boundary conditions for heterodimen-

sional Schottky contacts (a contact between 3D-metal

and 2D-semiconductor) are derived in this work. The

boundary conditions have been successfully incorpo-

rated in a numerical device simulator to investigate the

dark current characteristics and the transient response

of MSM quantum-well photodetectors. The simulation

results show low dark currents and encouraging 3 dB-

bandwidths at low DC bias voltages. The extension of

this work towards modeling of similar boundary con-ditions between heterodimensional Schottky contact of

3D-metal to 1D-semiconductor can be accomplished

by the straightforward incorporation of equivalent 1-D

density of states and energy terms for the conduction

and valence sub-bands.

References

1. C.R. Crowell and S.M. Sze, Current transport in metal semi-

conductor barriers, Solid State Electronics, 9, 1035 (1966).

2. J. Adams and T.W. Tang, A revised boundary conditions for the

numerical analysis of Schottky barrier diodes, IEEE Electron

Device Lett., EDL-7(9) 525 (1986).

3. R.B.Darling, High field, non linearelectron transportin lightly

dopes Schottky barrier diodes, Solid State Electronics, 31(6),

1031 (1988).

4. D.B. Ameen, Global simulation of a GaAs metal-

semiconductor-metal photodetector for the conversion of opti-

cal signals into microwaves, Ph.D. Thesis, Virginia Common-

wealth University, 2000.

5. D.B. Ameen and G.B. Tait, Convolution based global sim-

ulation technique for millimeter-wave photodetector and pho-

tomixer circuits,IEEE Trans. Microwave Theory Tech., 50(10),

2253 (2002).

6. J.Lu, R. Surridge,G. Paluski,H. Van Driel,and J.M. Xsu, Stud-

ies of high speed metal-semiconductor-metal photodetector

8/4/2019 =U4358815JK464150

8/8

12 Katkar

with a GaAs/AlGaAs/GaAs heterostructure, IEEE Trans. Elec-

tron Dev., 40(6), 1087 (1993).

7. B. Nabet, A. Cola, F. Quaranta, M. Cesareo, R. Rossi, F.

Fucci, and A. Anwar, Electron cloud effect on current injec-

tion across a Schottky contact, Appl. Phys. Lett., 77(24), 4007

(2000).

8. X. Chen, B. Nabet, F. Quaranta, A. Cola, and M.

Curie, Resonant-cavity-enhanced heterostructure metal-

semiconductor-metal photodetector, Appl. Phys. Lett., 80(17),

3222 (2002).

9. F. Castro, A. Anwar, and B. Nabet, Schottky contact between

metaland 2D electron gas:Deviceapplications to low-noiseopti-

cal detectors,Proceedings of SBMO/IEEE MTT-S International

Microwave and Optoelectronic Conference (Natal, Brazil) Aug.

1997, p. 323.

10. T. Yttterdal, M.S. Shur, M. Hurt, and W.C.B. Peatman, En-

hancement of Schottky barrier height in heterodimensional

metal-semiconductor con tacts, Appl. Phys. Lett., 70(4), 441

(1997).

11. B. Nabet, A heterojunction metal-semiconductor-metal pho-

todetector, IEEE Photonics Tech. Lett., 9(2), 223 (1997).

12. A. Anwar, B. Nabet, J. Culp, and F. Castro, Effects of electron

confinement on thermionic emission current in a modulation

doped heterostructure,J. Appl.Phys., 85(5), 26632666 (1999).

13. Atlas Device Simulation Software, Silvaco International, Santa

Clara, CA, 2003.

Documents

=U4358815JK464150