# Dependences of the electrical properties on the diameter and the doping concentration of the Si nanowire field effect transistors with a Schottky metal-semiconductor contact.

**ABSTRACT** A compact model of the current-voltage (I-V) characteristics for the Si nanowire field effect transistor (FET) taking into account dependence of the analytical electrical properties on the diameter and the concentration of the Si nanowire of the FETs with a Schottky metal-semiconductor contact has been proposed. I-V characteristics of the nanowire FETs were analytically calculated by using a quantum drift-diffusion current transport model taking into account an equivalent circuit together with the quantum effect of the Si nanowires and a Schottky model at Schottky barriers. The material parameters dependent on different diameters and concentrations of the Si nanowire were numerically estimated from the physical properties of the Si nanowire. The threshold voltage, the mobility, and the doping density of the Si nanowire and the Schottky barrier height at a metal-Si nanowire heterointerface in the nanowire FET were estimated by using the theoretical model.

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**rDelivered by Ingenta to: Korea Institute for Special Education IP : 163.152.133.22 Mon**. H Y Cha, H Wu, S D Chae, M G Spencer . 024307-3609.

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Copyright © 2010 American Scientific Publishers

All rights reserved

Printed in the United States of America

Journal of

Nanoscience and Nanotechnology

Vol. 10, 3609–3613, 2010

Dependences of the Electrical Properties on the

Diameter and the Doping Concentration of the Si

Nanowire Field Effect Transistors with a Schottky

Metal-Semiconductor Contact

Joo Hyung You1, Se Han Lee2, Chan Ho You1, Yun Seop Yu3, and Tae Whan Kim1?2?∗

1Nano Quantum Electronic Laboratory, Department of Electronics and Computer Engineering,

Hanyang University, Seoul 133-791, Korea

2Research Institute of Information Display, Department of Information Display Engineering,

Hanyang University, Seoul 133-791, Korea

3Department of Information and Control Engineering, Hankyong National University, Gyeonggi 456-749, Korea

A compact model of the current–voltage (I–V) characteristics for the Si nanowire field effect tran-

sistor (FET) taking into account dependence of the analytical electrical properties on the diameter

and the concentration of the Si nanowire of the FETs with a Schottky metal-semiconductor contact

has been proposed. I–V characteristics of the nanowire FETs were analytically calculated by using

a quantum drift-diffusion current transport model taking into account an equivalent circuit together

with the quantum effect of the Si nanowires and a Schottky model at Schottky barriers. The material

parameters dependent on different diameters and concentrations of the Si nanowire were numeri-

cally estimated from the physical properties of the Si nanowire. The threshold voltage, the mobility,

and the doping density of the Si nanowire and the Schottky barrier height at a metal-Si nanowire

heterointerface in the nanowire FET were estimated by using the theoretical model.

Keywords: Nanowire, Drift-Diffusion Transport, Schottky Diode, Analytical Circuit Model.

1. INTRODUCTION

Metal-oxide-semiconductor field-effect-transistors (MOS-

FETs) fabricated utilizing nanowires have attracted

considerable attention because of their potential applica-

tions in electronic and optoelectronic devices operating

at lower powers and higher temperatures.1–5The prospect

of potential applications of MOSFETs utilizing one-

dimensional nanowires has led to substantial researches

and development efforts to decrease their short channel

effect for the scale-down MOSFETs.6–8The semiconductor

nanowire field-effect-transistor (FET) has typically a metal-

semiconductor-metal structure. While the carrier transport

of the nanowire FET is dominant due to the themionic

field emission at the metal-semiconductor heterointerface

with a Schottky barrier height,8the quantum effect of the

nanowire should be considered for investigating the car-

rier transport of the FET with decreasing diameter of the

nanowire. Quantum mechanical simulations of nanowire

FETs have been performed by using the non-equilibrium

∗Author to whom correspondence should be addressed.

Green’s function formalism.9?10Even though several mod-

els for simulation of nanowire FETs have been introduced

to investigate their electrical and electronic properties,8–11

almost all of the models are not useful for applying the

circuit design simulator for the very large scale integration

based on nanowire FETs because of the inherent problems

encountered with the complicated computation procedure.

Very few studies on the analytical compact models by

using an analytical quantum transport model taking into

account an equivalent circuit together with a Schottky diode

model have been performed. Systematic studies concerning

the analytical compact models of nanowire MOSFETs are

very important in promising applications for applying cir-

cuit simulations and for understanding nanoscale electronic

devices.

This paper reports the dependence of the analytical elec-

trical properties on the diameter and the concentration

of the Si nanowire of the FETs with a Schottky metal-

semiconductor contact. Current–voltage (I–V) character-

istics of the nanowire FETs were analytically calculated

by using a quantum drift-diffusion current transport model

based on an equivalent circuit taking into account the

J. Nanosci. Nanotechnol. 2010, Vol. 10, No. 5

1533-4880/2010/10/3609/005 doi:10.1166/jnn.2010.2273

3609

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Dependences of the Electrical Properties on the Diameter and the Doping Concentration of the Si Nanowire

You et al.

quantum effect of the Si nanowires together with a Schot-

tky model in nanoscale Schottky barriers. The thresh-

old voltage, the mobility, and the doping density of the

nanowire and the Schottky barrier height in the metal-

semiconductor nanowire interface of the nanowire FET

were estimated by using the theoretical model.

2. DEVICE MODEL AND THEORETICAL

CONSIDERATIONS

Figure 1(a) shows an energy band diagram of a Si

nanowire FET with a metal-semiconductor Schottky con-

tact in a lateral direction. Si nanowire FET consists of one

Si nanowire connected between two metal electrodes of

the source and the drain. Studies concerning the Schottky

diode model taking into account thermionic emission (TE)

at a forward bias voltage and thermionic field emission

(TFE) at a reverse bias voltage for the nanowires with a

large diameter in the classical region has been described

elsewhere.12–15The current conduction of the nanowire

FET is dominant due to reverse-biased contacts at a low

voltage with increasing Schottky barrier height (SBH). The

current conduction of the nanowire FET becomes domi-

nant due to the electrical characteristics of the Si nanowire

with increasing bias voltage resulting from the decrease of

the SBH.

Figure 1(b) shows an energy band diagram of metal-

oxide-Si nanowire structure. When the applied gate volt-

age is sufficiently high, the Fermi level of the Si nanowire

is located above the conduction band. When the diam-

eter of the Si nanowire is enough small, because the

ground state of the nanowire is splited, the quantum

(a)

(b)

Fig. 1.

semiconductor Schottky contact with a lateral direction. (b) Energy band

diagram of the metal-oxide-Si nanowire with a vertical direction. The EC,

EF, Ei, EV, and Enrepresent the conduction band edge, the Fermi level,

the intrinsic Fermi level, the valence band edge, and the n-th eigenenergy

in the Si nanowire due to quantum effect, respectively. The Ldand Leff

represent the depletion length of the Si nanowire at the metal-Si heteroin-

terface and the effective channel length of the Si nanowire, respectively.

(a) Energy band diagram of the Si nanowire FET with a metal-

effect of the intrinsic characteristic for the Si nanowire

should be considered in the transport mechanism.8?16

The ballistic transport17–19and the drift-diffusion transport

mechanisms20?21for the conventional circuit simulator are

efficiently used to investigate the circuit-compatible ana-

lytical device model. Because the Si nanowire FET oper-

ates at lower drive currents, the ballistic transport model

does not apply to calculate the I–V characteristics of the

Si nanowire FET. Therefore, the drift-diffusion transport

model can be used to investigate the I–V characteristics

of the devices with a long channel length. The I–V ana-

lytical characteristics in terms of the applied bias voltage

by using the drift-diffusion transport model is given by20

Ids=

?

Leff

?Vds

0

QNW?Vgs?Vds? dV

(1)

where ? is the field dependent mobility, and Leffis the

effective channel length. QNWis the total charge of the

nanowire channel, which depends on the gate-source volt-

age (VGS) and the drain-source voltage (VDS). The QNWis

represented by the ?∗, ?∗

parameters taking into account the intrinsic characteristic

of the nanowire.20The ?∗, ?∗

mined by using the diameter, the doping concentration, and

the material parameter of the nanowire channel, regardless

of the applied bias voltage.

Figure 2 shows the numerically calculated geometry

parameters of ?∗, ?∗

and doping concentrations of the Si nanowires. The ?∗,

?∗

of the nanowire are analytically fitted to formulate the

nanowire FET compact model by using the circuit simula-

tion. The a1?a2?x0, and p for the ?∗as functions of diam-

eter (x) curves at various doping concentrations shown in

Figure 2(a) fitted by using the term of (a2+?a1−a2?/(1+

?x/x0?p?) are −1.19×10−13, 1.3×10−11, 104.79, and 2.6,

respectively. The value of ?∗linearly varies with changing

the ratio of the doping concentration of the Si nanowire,

and the complete term according to the doping concentra-

tion (N) and the diameter of the Si nanowire is expressed

by (a2+?a1−a2?/(1+?x/x0?p?)×?N/1016).

The fitting values of a1, a2, x0, and p for the ?∗

functions of x curves shown in Figure 2(b) are −0.2381,

2.59×10−10, 104.79, and 2.6, respectively. The fitting val-

ues of a1, a2, x0, and p for the ?∗

shown in Figure 2(b) are −0.2328, 4.42×10−9, 96.28, and

2.6, respectively. The fitting values of a1, a2, x0, and p for

the ?∗

3.75×10−9, 7.21×10−8, 91.3, and 2.6, respectively. The

?∗

tion of the Si nanowire. Because the term of (a2+?a1−

a2?/(1+?x/x0?p?) contains the doping conentration and the

diameter of the Si nanowire, the compact model of the Si

nanowire FETs is more compatible than the conventional

model.

0, ?∗

1, and ?∗

2, which are geometry

0, ?∗

1, and ?∗

2can be deter-

0, ?∗

1, and ?∗

2for various diameters

0, ?∗

1, and ?∗

2taking into account the physical model

0as

1as functions of x curves

2as functions of x curves shown in Figure 2(b) are

0, ?∗

1, and ?∗

2are independent of the doping concentra-

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J. Nanosci. Nanotechnol. 10, 3609–3613, 2010

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You et al.

Dependences of the Electrical Properties on the Diameter and the Doping Concentration of the Si Nanowire

0 50 100 150 200

0

3

6

9

12

(a)

α* (nC/m)

Diameter (nm)

0

4

20

40

60

0

1

2

3

0 50100150 200

0.00

0.04

0.08

0.12

λ2* (nC/m)

λ1* (nC/m)

(b)

Diameter (nm)

λ0* (nC/m)

Fig. 2.

diameters and doping concentrations of the Si nanowire. The rectangles,

upward triangles, downward triangles, and diamonds represent numeri-

cally calculated results, and the dashed, dotted, dash dotted, solid lines

indicate the fitting data at doping concentration of 1×1017, 5×1017,

1×1018, and 5×1018cm−3, respectively. Numerically calculated geom-

etry parameters of (b)?∗

doping concentrations of the Si nanowire. The circles represent numeri-

cally calculated results, and the solid lines indicate the fitting data.

(a) Numerically calculated geometry parameter of ?∗for various

0, ?∗

1, and ?∗

2for the various diameters and the

Fig. 3.

FET with a Schottky metal-semiconductor contact.

Schematic diagram of the equivalent circuit for the Si nanowire

Figure 3 shows a schematic diagram of the euivalent cir-

cuit consisting of two back-to-back Schottky diodes con-

nected by the Si nanowire FET. This circuit applies the

Kirchoff’s law. The formular for the simulation designed

from the routines is integrated into the SMARTSPICE for

the TFE and the drift-diffusion calculation.

3. RESULTS AND DISCUSSION

Figure 4(a) shows the electrical characteristic curves of the

difference between the conduction band and the ground

state energies (?E) of the nanowire dependent on the

diameter of the Si nanowire. When the diameter of the Si

nanowire is small enough to get quantum effect, because

the quantum states of the Si nanowire become discre-

ate, the quantum effect of the nanowire FET should be

consider to calculate accurately electrical characteristics.

However, when the diameter of the nanowire is enough

large, the quantum states of the nanowires becomes con-

tinuous, resulting that the electronic states of the nanowire

FET have classical phenomena. The carrier transport of

the nanowire FET with a large diameter at a reverse-bias

voltage is dominant due to TFE. The arrow shown in

Figure 4(a) repesents the quantum-transition diameter of

24 nm for the Si nanowire.

0 50 100150 200

0.0

0.2

0.4

0.6

(a)

∆E (eV)

Diameter (nm)

24 nm

0 50100150 200

0.0

0.5

1.0

1.5

2.0

5 x 1016 cm–3

1 x 1017 cm–3

5 x 1017 cm–3

1 x 1018 cm–3

5 x 1018 cm–3

ψT (eV)

Diameter (nm)

(b)

Fig. 4.

conduction band and ground state energy dependent on the diameter of

the Si nanowire. (b) Characteristics curve of the threshold potential in

the surface point (?T) dependent on the diameter and the doping density

of the Si nanowire.

(a) Characteristic curve of the difference energy, ?E, between

J. Nanosci. Nanotechnol. 10, 3609–3613, 2010

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2obtaned by using the

Dependences of the Electrical Properties on the Diameter and the Doping Concentration of the Si Nanowire

You et al.

Figure 4(b) shows the electrical characteristics curves

of the surface potential at the threshold point (?T) depen-

dent on the diameter and the doping density of the Si

nanowire. The value of ?Tvaries by changing the diameter

and the doping density of the Si nanowire. When the diam-

eter of the Si nanowire is enough small (below 5∼10 nm),

the splitting value of the ground state of the Si nanowire

increase, resulting in an increse of the ?T. The threshold

potential at the surface point of the Si nanowire rapidly

decreases with increaing diameter. When the diameter of

the Si nanowire is above 24 nm, because the difference

between the conduction band and the ground state energies

is approximately zero, ?Tis dominant due to the value of

the fermi level dependent on the doping density of the Si

nanowire.

Figure 5(a) shows drain-source current-gate-source volt-

age (IDS−VGS) characteristics of the Si nanowire FET with

the source and drain electrodes by using a Ti metal at the

VDSof 0.5 V.20The empty downward triangles shown in

the Figure 5(a) represent the simulated data of Ref. [20],

and the solid lines in the Figure 5(a) indicate the sim-

ulation data calculated by using our formulated model.

The values of ?∗, ?∗

numerical method are 5.23556×10−21, 1.195570×10−11,

4.341334×10−10, and 7.39715×10−9C/m, respectively.

The analytically simulated results of the Si nanowire FETs

obtained from our model are in reasonable agreement with

0, ?∗

1, and ?∗

0.0 0.2

0.4

1E-8

1E-6

1E-4

0.01

1

100

(a)

IDS (µA)

VGS (V)

Simulation data from Ref. [20]

Simulation data of our model

0.00.5 1.01.5

0

2

4

6

(b)

IDS (µA)

VDS (V)

VGS = 3 V

VGS = 4 V

VGS = 5 V

Fig. 5.

teristics curve of Si nanowire FET at a drain-source voltage of 0.5 V.

(b) Drain-source current-drain-source voltage (IDS–VDS) characteristics

curve of the n-type Si nanowire FET at various gate voltages.

(a) Drain-source current-gate-source voltage (IDS–VGS) charac-

the numerically simulated data of Ref. [20]. Figure 5(b)

shows the IDS–VDScharacteristics of the n-type Si nanowire

FET22at a gate voltage of 3, 4, or 5 V. The empty rect-

angles, circles, and triangles shown in the Figure 5(b) rep-

resent the experimental data of Ref. [22], and the solid,

dashed, and dotted lines in the Figure 5(b) indicate the sim-

ulation data. The Si nanowire FET used in this simulation

consists of the n-Si nanowire and the heavely doped Si

contacts. When the diameter and the doping concentration

of the Si nanowire are 20 nm and 1×1017cm−3, respec-

tively, the values of ?∗, ?∗

the analytical method are 7.4381×10−14, 8.1549×10−12,

2.8974×10−10, and 4.7059×10−9C/m, respectively. The

mobility of the FET device estimated from the obtained

model is 4.5×10−3cm2/V·sec. The simulated results of

the Si nanowire FETs are resonable agreement with those

of experimental data within a 5% error.22

0, ?∗

1, and ?∗

2obtained by using

4. CONCLUSION

The I–V characteristics of the nanowire FETs were

analytically calculated by using a quantum drift-diffusion

current transport model taking into accout an equiva-

lent circuit together with the quantum effect of the Si

nanowires and a Schottky model at Schottky barriers. The

threshold voltage, the mobility, and the doping density of

the Si nanowire and the Schottky barrier height at metal-Si

nanowire heterointerface in the nanowire FET were esti-

mated by using the proposed theoretical model. The carrier

transport of the nanowire FET at the reverse-bias volt-

age wth a large diameter was dominant due to TFE. The

threshold potential at the surface point of the Si nanowire

rapidly decreased with increasing diameter. The simulated

results of the Si nanowire FETs were resonable agreement

with those of experimental data. These results provide

useful informations for optimum design in the fabrica-

tions of large scale nanowire-complementary metal-oxide-

semiconductor and nanowire-memory circuits.

Acknowledgment: This work was supported by the

Korea Science and Engineering Foundation (KOSEF)

grant funded by the Korea government (MEST) (No. R0A-

2007-000-20044-0).

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You et al.

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Received: 1 January 2009. Accepted: 1 July 2009.

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