奈米尺寸金氧半場效電晶體通道背向散射:蒙地卡羅模擬與物理模型

國 立 交 通 大 學

電 子 工 程 學 系 電 子 研 究 所

碩 士 論 文

奈米尺寸金氧半場效電晶體通道背向散射:

蒙地卡羅模擬與物理模型

Nanoscale MOSFETs Channel Backscattering:

Monte Carlo Simulation and Physical Model

研 究 生：呂立方 Li-Fang Lu

指導教授：陳明哲 Prof. Ming-Jer Chen

奈米尺寸金氧半場效電晶體通道背向散射:

蒙地卡羅模擬與物理模型

Nanoscale MOSFETs Channel Backscattering:

Monte Carlo Simulation and Physical Model

研 究 生：呂立方 Student：Li-Fang Lu

指導教授：陳明哲 Advisor：Prof. Ming-Jer Chen

國 立 交 通 大 學

電子工程學系 電子研究所碩士班

碩 士 論 文

A Thesis

Submitted to Department of Electronics Engineering &

Institute of Electronics

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

In

Electronics Engineering

July 2008

奈米尺寸金氧半場效電晶體通道背向散射:

蒙地卡羅模擬與物理模型

研究生：呂立方 指導教授：陳明哲博士

國立交通大學

電子工程學系電子研究所

摘 要

藉由散射矩陣的方法，可以推導出背向散射係數，再利用蒙地卡

羅模擬証明。組成背向散射係數的兩個重要的參數─

λ0

和

A

─也在論

文中有進一步地討論；

A

的物理解析式模型建立在源極到通道上假設

的拋物線能障上，也藉由實驗以及蒙地卡羅模擬得到驗證。藉由蒙地

卡羅模擬得到在入射處的速度分佈推斷平均自由路徑的縮短源自於

載子熱效應，然而在拋物線能障的情況下因為沒有熱載子效應，使得

平均自由路徑維持定值。

Nanoscale MOSFETs Channel Backscattering:

Monte Carlo Simulation and Physical Model

Student：Li-Fang Lu Advisor：Prof. Ming-Jer Chen

Department of Electronics Engineering

Institute of Electronics

National Chiao Tung University

Abstract

Through the

scattering matrix approach, the backscattering

coefficient is derived and it is verified by Monte Carlo simulations. Two

important parameters,

λ0

and

A

, constituting the channel backscattering

have been taken into account. A parabolic barrier oriented compact model

has been physically derived for

A

. The validity of this compact model

has been corroborated experimentally and by Monte Carlo simulation

results. As for

λ0

, the carrier heating as the origin of reduced

mean-free-path is inferred on the basis of the simulated carrier velocity

distribution at the injection point. Strikingly, for the parabolic potential

case, the mean-free-paths remain consistent:

λ

’

=

λ

o

. This indicates the

absence or weakening of the carrier heating in the layer of interest, valid

only for a parabolic potential barrier

致 謝

感謝陳明哲教授，在這兩年的教導之下，充分地感受與學習到老師對做學問 的嚴謹態度，再加上跟老師不時的討論並指點我正確的方向，讓我得在做研究的 路途上一步一步地順利完成每一個階段的任務。在此我要由衷地向老師表達我的 感謝之意。 非常感謝博士班學長們給的許多指導和意見，謝振宇學長給予的專業方面的 指點，李建志學長在體育方面的競技，許智育、李韋漢學長在休閒、課業方面的 討論，以及實驗室碩二的同伴們的互相鼓勵。 這幾段話是沒辦法一一謝過所有關心我的人，故以此感言獻給每一位關心我 的人，感謝你們，謝謝！ 呂立方

Contents

Abstract (Chinese)

...

i

Abstract(English)

...

ii

Acknowledgement

...

iii

Contents

...

iv

List of Captions

...

vi

Chapter 1 Introduction

... 1

Chapter 2 Backscattering Coefficient and Monte Carlo

Simulation

... 4

2.1 Backscattering Coefficient r

c

... 4

2.2 Monte Carlo Simulation... 5

2.2.1 Monte Carlo Simulation... 6

2.2.2 Determination of r

c

... 6

2.2.3 Mean-Free-Path for Backscattering ... 7

2.3 Results and Verification ... 8

Chapter 3 A Parabolic Barrier Oriented Compact Model for

the k

B

T Layer’s Width

... 10

3.1 Discrepancy in Temperature Dependency... 10

3.2.1 Temperature Effect... 12

3.2.2 Gate Voltage Effect ... 13

3.2.3 Drain Voltage Effect... 13

3.2.4 Combination of Temperature and Bias Effects... 14

3.3 Model Verification... 14

3.3.1 Experimental Validation... 15

3.3.2 Simulation Validation... 15

3.4 Temperature Power Exponent Clarification ... 16

3.5 Results ... 17

Chapter 4 Re-examination of Mean-Free-Path for

Backscattering

... 18

4.1 Apparent Mean-Free-Path... 18

4.1.1 Mean-Free-Path Extraction in Linear Potential... 19

4.1.2 Mean-Free-Path Extraction in Parabolic Potential ... 19

4.2 Evidence for Carrier Heating ... 20

4.2.1 Velocity Distribution at the Injection Point ... 21

4.2.2 Flux Ratio at the End of k

B

T Layer ... 21

4.3 Results ... 22

Chapter 5 Conclusion

... 23

List of Captions

Table. 3.1 The corresponding key parameters extracted from open

literature simulation data, including channel length, V

tho

, DIBL,

V

G

, V

D

, and temperature………...…….………...…28

Fig. 1.1 Schematic diagram of channel backscattering theory. F is the

incident flux from the source,

l

is the critical length over which

a k

B

T/q drop is developed, r

C

is the channel backscattering

coefficient, and L is the conductor length………29

Fig. 2.1 Schematic diagram of a simulation picture. There are source side

contact on the left and drain side on the right. The conductor

length is L. Carriers are injected from the left contact with the

Hemi-Maxwellian Distribution………30

Fig. 2.2 Schematic diagram of two kinds of potential profile we adopted

in the simulations.

l

linear

and

l

parabolic

are the k

B

T/q layer

width for linear potential and parabolic potential, respectively. V

a

is the applied voltage…..………..…31

Fig. 2.3 Velocity distribution at the top of the barrier, x=0 for L=100 nm ,

T=300 K and V

a

= 0.6V in linear potential profile. The underlying

r

C

is equal to the area ratio of deep grey one to light grey

one.………...32

Fig. 2.4 Extracted mean-free-path for backscattering under near -

equilibrium long channel conditions. The bar represents the range

of simulated r

c

values……...………..…33

Fig. 2.5 Simulated r

C

(symbols) and calculation from (10) (dotted lines)

under linear potential profile………….………...………34

Fig. 2.6 Simulated r

C

(symbols) and calculation from (10) (dotted lines)

under parabolic potential profile………..35

Fig. 3.1 Schematic diagram of a parabolic potential profile near the

source.

l

represents the k

B

T/q layer width.

L%

represents the

imaginary channel length corresponding to a certain position

where the parabolic potential drop from the top of the barrier is

equal to V

D

………36

Fig. 3.2 Comparison of calculation from (23) and experimental

l

versus V

G

. Data are created from 55-nm bulk n-MOSFETs by

means of a parameter extraction process. V

D

=0.5 V and 1.0 V…37

Fig. 3.3 Comparison of calculation from (22) and experimental

L%

versus

V

G

. Here

L%

is obtained directly from (23) with known T and V

D

.

These two figures are for V

D

=0.5 V and 1.0 V……...….…...38

quantity of the functional expression

1 1 1

0.25_{( ) (}2 _{/ ) (}2 _{/ )}2

D G th B B D

LV V −V − k T q k T qV

. The slope of the line furnishes

η

with a value of 4.1V

-0.25

…..……….………39

Fig. 4.1 The underlying

λ'

extracted through (25) with simulated r

C

as

input in linear potential profile…..………...…40

Fig. 4.2 Carrier velocity distribution at the injection point, x=0, for L=25

nm at V

a

=0.8 V and T=300 K in linear and parabolic potential

profile………..……….………41

Fig. 4.3 Carrier velocity distribution at the injection point, x=0, for L=50

nm at V

a

=1.0 V and T=150 K in linear and parabolic potential

profile………..……….………42

Fig. 4.4 Velocity distribution at the top of the barrier,

x

= A

for L=50

nm, T=200 K and V

a

= 0.2V in linear potential profile. The

underlying

b

( ) / ( )

A

a

A

is equal to the area ratio of deep grey one

to light grey one….………...43

Fig. 4.5 Ratio of the negatively-directed flux

b A

( )

to the positively -

directed flux

a A

( )

for the linear potential case. The dotted lines

are straightforwardly calculated by substituting (10) into (26)

Chapter 1

Introduction

The scaling down of silicon MOSFETs has promisingly moved from 180nm to 65nm node in the past decade. The ITRS also predicts that this scaling trend will reach 22nm technology node in the coming decade. This amazing evolution of silicon technology leads to challenges about how carriers transport in nanoscale devices, specially the losing efficiency of some classical-based carrier transport models such as drift-diffusion transport model. In the beginning of this promising decade, another aspect of carrier transport theory, named “Backscattering Theory”, was proposed by Mark Lundstrom [1]-[2]. This theory is different from drift-diffusion based model. Instead, current-voltage (I-V) characteristics are expressed in terms of the elementary parameters without depending on the carrier mobility. These parameters, such as carrier injection velocity and inversion carrier density at the virtual source, control the device performance. In other words, device performance can no longer be considered only by the channel itself.

In the Backscattering theory, we distinguish the carrier transport in the channel as a wave-like flux traveling from source to drain. According to the theory, the key region controls the flux ratio passing through the channel named

l

, which is the distance over which the channel potential drops by k TB

q (kB is the Boltzmann’s

constant and T is the temperature). Multiple backscattering events occur in this critical zone. Thus, a certain fraction

r

c of the injecting flux F is reflected and returns to the

scattering events and returning flux schematically. Backscattering theory relates the steady-state current to transmission and reflection coefficients. Consequently, the drain current per unit channel width, which is critically determined by

l

, can be expressed as (1 ) (1 ) c D inv inj c r I Q v r − = + (1)

where Qinv is the inversion-layer charge density per unit area at the top of the virtual

source, vinj is the thermal injection velocity at the top of the virtual source, and

r

c is

the channel backscattering coefficient through this critical layer. Backscattering theory also relates to both the near-equilibrium mean-free-path λ₀ (or called momentum relaxation length) and the critical length of the kBT layer:

0 c r λ = +l l (2)

Eq.(2) is valid only for drain voltage much larger than kBT/q. Here,

r

c can be

extracted experimentally by current-voltage (I-V) fitting [3] or by a temperature dependent method [4]. The temperature dependencies of the kBT layer’s width also

are clarified by methods of temperature oriented experiment and simulation [5]. However, these two important parameters in (2) must be discussed more clearly. First, expression of

l

used in the open lecture is in lack of gate voltage effect, and, we usually assume

l

as a function of lattice temperature, channel length and applied voltage (VD). Besides, the temperature dependent power order of

l

range from 0.5

to 1 [3]-[8]. Second, the usage of λ₀, called near-equilibrium mean-free-path for backscattering, must be clarified.

In this thesis, we present a new compact model for

l

on the basis of a parabolic potential barrier profile assumption around the source-channel junction. This compact model is physically derived and can elucidate the effects of channel length, gate

overdrive, drain voltage and the temperature on

l

. Additionally, 1-D Monte Carlo particle simulation [13] is performed to explore the backscattering coefficient

r

c,

which is related to the mean-free-path λ₀. Evidence about carrier heating can be found accordingly.

This thesis is organized as follows. In Chapter 2, we derive the backscattering coefficient through scattering matrix approach and Monte Carlo simulations are executed to verify the validity of the derivation. In Chapter 3, a parabolic barrier oriented compact model is physically derived and verified by experimental data and simulation results. In Chapter 4, another important parameter λ₀ is discussed. Finally, a conclusion of the work will be drawn in Chapter 5.

Chapter 2

Backscattering Coefficient

and Monte Carlo Simulation

In this chapter, the origin of the backscattering coefficient

r

c, under both low

applied voltage (Va <<kBT/q) and high applied voltage (Va >> kBT/q) conditions, is

derived using the scattering matrix approach [9]. Then, we will employ the existing particle-based Monte Carlo simulator to deal with the velocity distributions that are relevant to the desired backscattering coefficient

r

c.

Section 2.1 Backscattering Coefficient

The underlying carrier transport from the top of the source-channel barrier, or called the virtual source, down to the drain contact can be treated using the scattering matrix approach [9]. However, there are some assumptions that must be introduced. First, we assume that the flux is only emitted from the virtual source; in other words, there is no flux emitted from the drain contact. Second, we assume that the flux traveling from the top of the virtual source down the potential drop is not affected by the electric field. Third, we assume steady-state conditions, which means right-directed flux a(x) and left-directed flux b(x) are position dependent, while being independent of time. The recombination-generation processes are also neglected. Under these assumptions, the change of the positive-directed flux a and negative-directed flux b due to the backscattering event over the dx path interval at the position x can be described as

0 0 1 ( ) B db a q b dx = −λ + λ +k T εl (8)

where ε_l is the electric field and λ₀ is the mean-free-path for backscattering. We know that the net flux a – b, as a result of the flux conservation through all sections, has to be kept constant. To solve (8) in the zone of interest where: k TB

q ε_l ≈ l [1], we have 0 0 0 1 exp( ) ₁ (0) (0) ( ) 1 exp( ) (1 ) exp( ) x b a b x x x λ λ λ − − = + + − − + − l l l l l l (9)

From the second assumption, we can know that the left-directed flux b at the drain contact is equal to zero, b(L)=0. Thus, the complete expression of the

r

c can be

written as 0 1 exp( / ) 1 exp( / ) c L r L λ − − = + − − l l l (10)

Apparently, for applied voltage Va >> kBT/q, also L>>l , (10) reduces to (2). For Va

<<kBT/q or L<<l , opposite to the former case, (10) reduces to the near-equilibrium

case prevailing over the entire channel [11]

0 c L r L λ = + (11)

In the citation [1], reproduction of the simulation results in the transition region between the low and high field limits can be conducted via the product of (11) and (2), and is some what empirical. Here, (10) can be readily used for not only high Va and

low bias but also the transition region. In the citation [10], the same expression is solved from the Boltzmann transport equation in the framework of a “relaxation length” approximation.

Section 2.2 Monte Carlo Simulation

In order to verify (10), we adopt the Monte Carlo simulation. This particle-based Monte Carlo technique is recognized as a powerful method for accurately describing the carrier transport in semiconductor materials and devices within the semi-classical approximation. Here, we use Monte Carlo simulation program to solve the complicated scattering events in a silicon bulk conductor. This program named Demons [13] can provide the rich information regarding the carrier velocity distributions at any positions in the conductor. Furthermore, the positively-directed flux and the negatively-directed flux can be separately determined, which means the flux ratio at any position can be easily obtained.

Section 2.2.1 Simulation Methodology

Before performing Monte Carlo simulation, some assumptions and boundary conditions must be clarified: (i) carriers are only injected one by one at the origin x=0 in the steady state from the left contact and are of hemi-Maxwellian distribution; (ii) termination condition: at the origin x=0, left contact absorbs backscattered carriers without further reflections, and at the end of the conductor, x=L, the positively-directed carriers are absorbed; (iii) the potential profile in the conductor is not self-consistent but frozen, just like other group adopted [10],[12]; and (iv) the conductor is divided into 100 grids, as shown in Fig.2-1, to analyze the positive-directed flux and negatively-directed flux in each grid. Figure 2-1 schematically depicts this simulation picture.

In this work, we perform Monte Carlo simulation at three different lattice temperatures, (150, 200 and 300 K), four different conductor lengths, (15, 25, 50 and 100 nm). Moreover, these simulations are executed in two potential profile cases: linear and parabolic potential, which are displayed in Fig.2-2. In our group previous work [5], linear potential profile case has been performed on 80-nm silicon conductor

of 1012 cm-3 doping to examine the temperature dependencies of the width spanned by A . However, parabolic potential case is the significant augmentation. It has been experimentally shown to be a good approximation to build near –source channel conduction-band profile of MOSFET in saturation [7]. Besides, by using the basis of parabolic potential profile around the source-channel junction barrier of nanoscale MOSFETs, a compact new model of kBT-layer width is physically derived [14]. We

will discuss this compact model in the subsequent chapter.

Section 2.2.2 Determination of

r

c

From the definition of backscattering coefficient, we have

C ( ) ( ) flux n v r flux n v − − + + − = = + (12) where 0 0 0 0 ( ) ( ) ( ) ( ) f v vdv f v dv v v _{f v vdv} f v dv −∞ − −∞ ∞ + ∞ =

∫

∫

∫

∫

, and 0 0 ( ) ( ) f v dv n n _{f v dv} − −∞ ∞ + =

∫

∫

.

n± is the electron density for positively-directed / negatively-directed flux on the top of the barrier, and v± is the average velocity for positively-directed and negatively-directed flux on the top of the barrier. Figure 2-3 shows the simulated velocity distribution of conductor length 100 nm with 1012 cm-3 doping, 300K temperature and 0.6 V applied voltage in linear potential case at the x=0. Here, the backscattering coefficient

r

c is just equal to the area ratio of deep gray one to light

gray one. Likewise, we can see that the injected carriers still retains hemi-Maxwellian distribution but is split into two distinct components: one of the longitudinal effective

mass and one of the transverse effective mass.

Section 2.2.3 Mean-Free-Path for Backscattering

If we want to use (10) to evaluate the simulation

r

c data, we need information

for two important parameters: mean-free-path for backscattering λ₀ and corresponding kBT-layer’s width A . However, according to the backscattering

framework [1-2], A can be explicitly expressed as a function of the thermal energy kBT, the conductor length L, and the applied voltage Va. For this reason, we have

B a k T L qV =

A for the linear potential profile and B a

k T L

qV

=

A for the parabolic one.

Since the kBT-layer’s width is clearly defined, in order for calculations from (10)

to have good agreement with simulated

r

c, mean-free-path for backscattering λ0 should be obtained accurately. We can extract λ₀ via (11) on the longest conductor (L = 100 nm) in the low electric filed, Va <<kBT/q, that is to say, L>>l . Figure 2-4

shows the extracted results for three different temperatures of 150, 200 and 300 K. In the figure, the bar represents the range of simulated

r

c values. The average

r

c of a

linear potential profile is close to that of the parabolic one, which indicates λ₀ is essentially independent of the potential profile we adopted in the low field regime. The average

r

c over these two potential profiles leads to the average λ0 of 56, 105, and 155 nm for 300, 200 and 150 K, respectively. The extracted average value of λ₀ at 300 K is identical to that of our group previous work [5].

Since the mean-free-path for backscattering is extracted via (11) in the low electric field or near-equilibrium condition. The corresponding A is B

a k T L qV = A for

the linear potential profile and B a

k T L

qV

=

A for the parabolic one. Figure 2-5 and 2-6

show the simulated

r

c in symbols and (10) in dotted lines. Comparison between linear

potential profile and parabolic one, it is easy to find out that there is discrepancy in the linear potential profile between calculations and simulation values. This deviation increases with decreasing conductor length and decreasing lattice temperature. However, Figure 2-6 shows good agreement between the simulated

r

c and that of (10).

In order to discuss this discrepancy, we have to investigate two important parameters, A and λ0, in (10). First, according to the open literature [1-2], when Va is much larger than kBT/q, the definitions of the kBT-layer’s width in both potential profiles

have been defined. Second, the other parameter λ₀ is called near-equilibrium mean-free-path for backscattering. We extracted λ₀ via (11) in the low electric field or near-equilibrium condition, and it is the only mean-free-path used for all operating conditions. In [10], the similar expression of (10) is also derived via Boltzmann transport equation in the framework of a “relaxation length” approximation, and within that approximation, mean-free-path λ₀ must be independent of carrier energy. In other words, λ₀ has to be constant no matter how large the applied voltage is. However, Monte Carlo simulation results at room temperature in linear potential profile show that λ₀ should change. Certain relationship,λ'=λ γ₀/ with γ =1.5 to 2.0, is discovered [15]. We will discuss this issue later.

Chapter 3

A Parabolic Barrier Oriented Compact Model

for the k

B

T Layer’s Width

The channel backscattering theory [1-2] establishes a connection between the kBT-layer’s width, which locates in the beginning of the conductive channel near the

source, and the drive performance of the device. However, the ability to quantitatively determine the width of this critical zone is necessary. We have discussed the backscattering coefficient through scattering matrix approach and performed Monte Carlo simulation in Chapter 2. Although _{l can be explicitly expressed as a function} of the conductor length L, the thermal energy kBT, and the applied voltage Va, this

expression can not elucidate the effect of gate voltage on _{l . Moreover, the influence} of temperature on _{l should be discussed more clearly. In this chapter, a parabolic} potential barrier oriented compact model for kBT-layer’s width in nano-MOSFETs is

derived, and will be verified by experiment and simulation.

Section 3.1 Discrepancy in Temperature Dependency

Mostly adopted model of _{l can be quoted in the literature [6],}

(

B

)

a

k T

L

qV

α

=

l

(13)

where _{l is the k}BT-layer’s width and L is the conductor channel length. In the

previous chapter, we can easily define _{l in both linear and parabolic potential} profile with α= and 1 α=0.5, respectively. However, some division among the magnitude of the temperature exponent α should be clarified. In the first place, the I-V characteristics of a simulation double-gate MOSFET at room temperature can be

best fitted by α=0.57 [6]. In addition, comparable α=0.5 has been produced on experimental bulk n-MOSFETs in a temperature range of 233 to 298 K [5], [7]. Besides, α=0.75, which covers the same range as the former case, has been experimentally determined [3]. Still, α= has already been adopted in a 1 temperature dependent backscattering coefficient extraction method [4]. Another open literature [8] also shows that _{l is approximately proportional to the temperature} form 100 to 500 K, which means α= . Apparently, the issue of widely-ranged 1 α

values must be addressed. Furthermore, it is difficult for (13) to elucidate the effect of gate voltage. We will make a simple approach by assuming a parabolic potential near the source-channel junction. Then a compact model with the channel, gate overdrive, drain voltage, and temperature as input parameters will be physically derived.

Section 3.2 Parabolic Barrier Profile

A parabolic potential profile close to the source is schematically plotted in Figure 3-1. This potential profile that extends to the remaining channel can be described as

2

( )

D

( / )

V x

=

V

x L

%

(14)

The origin x=0 is the peak of the barrier.

L%

indicates the imaginary channel length corresponding to a certain position where the parabolic potential drop from the top of the barrier is equal to V . At the other side, we neglect the barrier height with respect _D

to the source side because of the large drain voltage. Then, by substituting

x

= l

and V x( )=k T q_B / into (14), which means a local potential drop being equal to

thermal energy, the

l

expressed in terms of imaginary

L%

, applied voltage V , and _D

1 2

(

B

)

D

k T

L

qV

= %

l

(15)

Obviously, once

L%

can be expressed as a function of channel length, gate voltage, drain voltage and temperature, the expression of

l

in terms of these parameters can be obtained.

There exists a set of specific drain voltage V_D0 and gate voltage V_G0, at a given temperature T₀, which can ensure L% = L:

0 0 0

(

,

_{D D}

,

_{G G}

)

L T

%

=

T V

=

V

V

=

V

=

L

(16) That is to say, this imaginary

L%

just locates on the real channel length. Then, we will discuss each component in later sections.

Section 3.2.1 Temperature Effect

First of all, the thermal energy layer width, corresponding to L% →L, T → , T₀

and V_D →V_D0, is denoted as

l

₀.

l

₀ can be expressed as

1 0 2 0 ( ) B Do k T L qV = l (17)

Then only the temperature changes individually from T₀ to T . By dividing (15) by (17), a power-law relation can be found:

1 2 0 ( / )

L% =L T T (18) Here, we have assumed that the potential profile does not change with temperature. In other words, the local electric field from the peak of the barrier to the end of kBT-layer

theory [1], the local electric fields across this section are equal to k T_{B 0}/q_{l and 0}

/

B

k T q_{l at} T₀ and T, respectively.

Section 3.2.2 Gate Voltage Effect

Once the potential profile is assumed, we can easily differentiate (14) twice with respect to position x, which leads to

2

2 2

( ) 2 _D

d V x V

dx = L% (19)

Under the same V , and according to Poisson’s equation, (19) relates to the charge _D

density. From [6], we can know that (19) also can be linearly related to the underlying inversion-layer density, which is proportional to the gate overdrive. It turns out that

1 2 0 ( ) ( ) G th G th V V L L V V ⎛ − ⎞ = ⎜ ₋ ⎟ ⎝ ⎠ % (20)

Here, the term, (V_G0−V_th) represents the specific gate overdrive that ensures (16).

Section 3.2.3 Drain Voltage Effect

The local electric field can be obtained by differentiating (14) once. Thus, if the drain voltage increases from V_D0 to V , local electric field, _D 2V xD₂

L% , must be larger

that 2V xD₂0

L . If the local electric fields were equal, we could know that 0.5

0 ( _D/ _D )

L%= L V V . Because V_D >V_D0, the electric field is increased. Thus the power exponent must be no more than 0.5. As a result, one obtains

0.25 0 ( _D/ _D )

The power exponent ν, which is equal to 0.25, has been experimentally determined in the previous work of our group [7]. For V_D <V_D0, the same expression as (21) can be obtained.

Section 3.2.4 Combination of Temperature and Bias Effects

Through the combination of (18), (20), and (21), these power-law relationships establish a unique expression for L% :

0.5 0.25 0.5 ( ) D B G th V k T L L V V q η ⎛ ⎞ = _{⎜ ⎟} − _{⎝ ⎠} % (22)

Combining (15) and (22) can further lead to 0.5 0.5 0.25 0.5 ( ) D B B G th D V k T k T L V V q qV η ⎛ ⎞ ⎛ ⎞ = _{⎜ ⎟ ⎜ ⎟} − _{⎝ ⎠ ⎝ ⎠} l (23) where 0.5 0.5 0 0 0.25 0 ( _{G th}) _B D V V k T V q η − ⎛ ⎞ − = _{⎜ ⎟}

⎝ ⎠ . Furthermore, η is fixed and also the only fitting parameter in this expression. Indeed, it is supposed that η is a constant value over the channel length, gate and drain voltage, and temperature. Thus, the capability of this parabolic potential barrier oriented compact model will be more general.

Section 3.3 Model Verification

On the one hand, the experimental l in this section was obtained from 55-nm bulk n-MOSFETs by means of a parameter extraction process proposed by our group, and the details can be found in [3], [5], [7]. On the other hand, we quoted the rich literature [8],[19], in which Monte Carlo simulations including quantum corrections to the potential and calibrated scattering models are used to study carrier transport in bulk and double-gate silicon-on-insulator MOSFETs. Also cited are those of the open

literature [2], [6], [16], [17], [18].

Section 3.3.1 Experimental Validation

Figure 3-2 shows the experimental l , which is obtained from 55-nm bulk n-MOSFETs with tox=1.65nm, Npoly=1x1020cm-3, and NA=7x1017 cm-3, versus gate

voltage for two drain voltages of 0.5 and 1.0 V and three different temperatures of 233, 263, and 298 K. With these experimental l values, temperature T, and applied drain voltage VD, we can obtain the corresponding L

~

using (15). These results are shown in Figure 3-3 for two drain voltages versus gate voltage. The corresponding near-equilibrium threshold voltage, denoted V_tho for 233, 263, and 298K are 0.360, 0.345, and 0.328 V, respectively. The drain-induced barrier lowering (DIBL) magnitude for 233, 263, and 298K are 120, 123, and 130 mV/V, respectively. Throughout this work, the threshold voltage Vth at higher drain voltages is equal to

tho D

V −DIBL V× .

Section 3.3.2 Simulation Validation

First, in Ref. [8], the extracted l at VD = VG = 1 V is available in a wide range

of the channel length from 14 to 37 nm and also a wide range of the temperature from 100 to 500 K. The underlying threshold voltage Vtho and DIBL are reasonably 0.3 V

and 110 mV/V, respectively. Second, the citation [19] can further provide the relevant information at room temperature: l from 2.0 to 7.0 nm, L from 14 to 65 nm, VD (=

VG) from 1.0 to 1.2 V, and DIBL from 11 to 230mV/V. The above information is listed

in Table 3-1. Moreover, we have also extracted l from the published literature [6], [8], [16-18], on double-gate device simulation and l can be extracted directly from the channel potential profiles. The corresponding key parameters are: (i) L = 10 nm,

nm, Vtho ≈ 0.33 V, DIBL ≈ 25 mV/V, VD = 0.2 V, VG = 0.55 V, and T = 300 K [6]; (iii)

L = 25 nm, Vtho ≈ 0.3 V, DIBL ≈100 mV/V, VD = 0.8 V, VG = 0.5, 0.8, and 1.0 V, and T

= 300 K [16]; (iv) L = 15 nm, Vtho ≈ 0.2 V, DIBL ≈120 mV/V, VD = 0.7 V, VG = 0.7 V,

and T = 300 K [17]; and (v) L = 15 nm, Vtho ≈ 0.3 V, DIBL ≈ 77 mV/V, VD = 0.7 V, VG

= 0.7 V, and T = 300 K [18]. The above key parameters are also listed in Table 3-1. With all these key parameters, a scatter plot can be created as shown in Figure 3-4 in terms of the experimental and simulated l versus the quantity of the functional expression LV_D0.25(V -V ) (k T/q) (k T/qV ) . Apparently, all data seem to _{G th} -0.5 _B 0.5 _{B D} 0.5 fall on or around a straight line. The slope of this line furnishes η with a value of

0.25

4.1V− . Still, η remains constant, regardless of the channel length, gate and drain voltage, and temperature.

Section 3.4 Temperature Power Exponent Clarification

Some remarks can now be made to clarify the confusing α values in the open literature [3]-[8]. First of all, it is noticed that in case of bulk n-MOSFET two different values of α were produced: one of 0.5 [5], [7] and one of 0.75 [3]. This difference can be attributed to the different subband treatments during the parameter extraction process: Schrödinger-Poisson equations are numerically solved in [5], [7] whereas in [3] this was done by a triangular potential approximation [20]. Therefore, different subband levels can lead to different average thermal injection velocities, which in turn give rise to different l values. Secondly, according to (22) a temperature range of 233 to 298 K in case of 55-nm bulk device [5], [7] is not large enough to affect the calculated L~. In other words, L~ is considerably insensitive to such a narrow temperature range. Consequently, the resulting apparent temperature power exponent was limited to 0.5 as reported in the previous work [4], [5]. Indeed, with the known η as input, fairly good reproduction can be achieved as depicted in

the α ≈ 0.57 case [6]: since the room temperature of operation was involved alone, the temperature effect of L~ can no longer be examined. Only in a wide temperature range as done in the comprehensive study of [8], [19] can the linear relationship of

T

∝

l as shown in Figure 3-4 actually occur. Thus, from the aspect of temperature dependencies, excellent coincidence with the data as shown in Figure 3-4 stresses that the existing backscattering coefficient extraction method [4] is valid.

Section 3.5 Results

This new compact kBT-layer’s width model, which links the width of thermal

energy kBT layer to the geometrical and bias parameters of the devices, is physically

derived on the basis of a parabolic potential profile around the source-channel junction barrier of nanoscale-MOSFETs. Moreover, this proposed model is supported not only by experimental data, but also by various simulation works presented in the open literature. Only one fitting parameter remains constant in a wide range of channel length (10 to 65 nm), gate voltage (0.4 to 1.2 V), drain voltage (0.2 to 1.2 V), and temperature (100 to 500 K), which means that the capability of this compact model is universal.

Chapter 4

Re-examination of Mean-Free-path for Backscattering

When applying (1) to nanoscale MOSFETs to predict relevant current-voltage characteristic, we need to quantify the key backscattering coefficient

r

c. However,

there are two important parameters needed in

r

c calculation. We have discussed the l

on the basis of a parabolic potential profile around the source-channel junction barrier and obtained a compact model of _{l with the channel length, gate overdrive, drain} voltage and temperature as input parameters. On the other hand, another parameter

0

λ has not been discussed yet. In Section 2.2.3, we extracted this information under near-equilibrium low field condition. When comparing (10) to compare with Monte Carlo results, it seems to have some differences between calculated and simulated

r

c

in the linear potential profile. In parabolic case, Figure 2-6 shows good agreement between calculation and simulation results. In this chapter, we will discuss the near-equilibrium mean-free-path for backscattering.

Section 4.1 Apparent Mean-Free-Path

At the end of Chapter 2, we pointed out that in the citation [12], the Monte Carlo simulations at room temperature in case of non-degenerate statistics on a linear channel potential profile have exhibited certain relationship: λ'=λ γ0/ with γ =1.5 to 2.0. Here λ is the apparent mean-free-path that constitutes the '

following expression in the high field case:

'

c

r

λ

=

+

l

l

(24)

Thus, some clarifications are demanded.

In order to achieve this goal, we perform Monte Carlo particle-based simulation on a silicon bulk conductor like what we have done in Chapter 2, but pay more attention on the velocity distribution on the top of the barrier and flux ration at the end of the kBT layer.

Section 4.1.1 Mean-Free-Path Extraction in Linear Potential

First of all, we extract the mean free path from the simulated

r

c using (25),

instead of (10): 1 exp( / ) ' 1 exp( / ) c L r L λ − − = + − − l l l (25)

We denote the apparent mean free path as λ . The validity of (25) has been '

proven in Chapter 2. By substituting the simulated

r

c into (25), the underlying λ '

can be obtained as depicted in Figure 4-1 versus applied voltage. From Figure 4-1, we can see that: (i) λ falls below ' λ₀, and decreases with increasing applied voltage, leading to λ'=λ γ₀/ with γ = 1.5 to 2.5; and (ii) on average, λ decreases with '

decreasing conductor length. However, it is supposed that the upper limit of λ0 can be

recovered only with increasing conductor length or decreasing applied voltage. On the other hand, the ratio of λ to ' λ₀ appears to be a weak function of the lattice temperature. The same argument has been produced by the recent Monte Carlo simulations [12] devoted to a linear potential profile at lattice temperature of 300 K as we mentioned at the end of Chapter 2.

For a parabolic potential profile, the corresponding simulated

r

c is

displayed in Figure. 2-6. Here, the lines are calculated from (25) with λ of 56, '

105, and 155 nm for 300, 200, and 150 K, respectively. Good reproduction of the data for different conductor lengths, different temperatures, and, especially, different applied voltages can be gotten. Moreover, the reproduction can all be achieved with simply λ'=λ₀, without adjusting any parameters. This means no matter how the quasi-ballistic transport prevails in the kBT layer (λ'> A), the quasi-equilibrium

conditions still govern the backscattered carriers.

Section 4.2 Evidence for Carrier Heating

In Section 4.1, we see that the discrepancy in the mean-free-path in linear potential profile. This inconsistency can be contributed to the presence of the carrier heating. In citation [10], all the derivations are carried on the basis of the “relaxation length” approximation, which means that the mean-free-path is a constant, and is independent of the carrier energy or carrier temperature. However, as carriers transport under applied electric field, their energy should be changed, especially in the linear potential profile. Because of the lack of weak field regime near the injection point, the carriers experience a larger electric field in the linear one, and thus the deviation between the lattice temperature and carrier temperature would be possible. On the other hand, for a parabolic potential profile, as showing Figure 2-2, there is significant fraction of the kBT layer, which can be determined as the weak field

regime. Take this into consideration, although the deviations from the lattice temperature would be possible as entering into the remainder, the overall carrier heating in the kBT layer should be weakened. Consequently, when the carriers are

injected into the channel at the beginning of the kBT layer, they immediately undergo

mean-free-path is therefore no longer independent of the carrier energy.

Section 4.2.1 Velocity Distribution at the Injection point

Concerning the difference between the lattice temperature and carrier temperature, the carrier velocity at the injection point is helpful as demonstrated in Figure 4-2 and 4-3. Figure 4-2 shows the velocity distribution of L=25nm at Va=0.8 V

and 300 K for two potential profiles. Figure 4-2 clearly shows that two significant differences between the potential profiles. First, the injected single hemi-Maxwellian velocity distribution is retained in the positively-directed carriers with the parabolic one, but it is split into two distinct components in the linear one: One of the longitudinal effective mass and one of the transverse effective mass. Second, the distribution of the negatively-directed or backscattered carriers appears to be wider in the linear potential profile than the parabolic one. The same result can be seen in Figure 4-3 for L=50nm at Va=1.0 V and 150 K. This is evidence of carrier heating.

Section 4.2.2 Flux Ratio at the End of kBT Layer

Carrier heating evidence also can be found at the end of kBT layer (x= A). First

of all, we replace x with A in (9), leading to 1 1 1 0 0 0 1 1 (0) (0) ( ) 1 (1 ) e b a b e e λ λ λ − − − = + + − + l − l l l (26)

where a(0) and (0)b represent the incident and reflected flux at the origin, respectively, and ( )b _{l is the negatively-directed flux at the end of the k}BT layer. (26)

also can be cited in the literature [10], which was derived on the basis of a Maxwillien shape distribution in both the forward and backward directions with in the context of the relaxation length approximation. In order to get the ratio of negatively-directed flux to positively-directed flux, we substitute (10) into (26) for r_c=a(0) / (0)b . Then,

accounting for the conservation of the current, the ratio ( ) / ( )b _l a _{l can be}

straightforwardly calculated. Monte Carlo simulation program can provide the velocity distribution at the end of the kBT layer. For example, Figure 4-4 shows

velocity distribution of L=50 nm, T=200 K, and Va=0.2 V at x= A. The velocity

distribution also is split into two components, and velocity distribution of backscattered carrier is wider. The ( ) / ( )b _l a _{l is the ratio of deep grey area to light}

grey area, like what we did to determine

r

c at the injection point in Section 2.2.2. The

calculated lines from (26) are shown in Figure 4-4 for linear potential profile. Also plotted in the figure are those of the Monte Carlo simulation. However, the calculated values are seen to fall below the simulation ones. This reveals the fact that the reflected flux at the end of kBT layer can be enhanced in the presence of carrier

heating. Only with decreasing applied voltage can the deviation between the simulation and model calculation be shortened.

Section 4.3 Results

We executed Monte Carlo simulations on a silicon bulk conductor to re-examine the channel backscattering theory in bulk nano-MOSFETs. Through these simulations, some important points can be addressed again:

(i) The near-equilibrium mean-free-path for backscattering λ₀, is independent of the potential profile.

(ii) λ in a linear potential profile is lower than ' λ0 of parabolic one due to the presence of the carrier heating. Evidence is highlighted by both the velocity distribution at the injection point and the flux ratio at the end of the kBT layer.

Chapter 5

Conclusion

The backscattering coefficient is derived through the scattering matrix approach and verified by Monte Carlo simulations. Two important parameters, λ0 and A , constituting the channel backscattering coefficient have been taken into account. A parabolic barrier oriented compact model has been physically derived for A . The validity of this compact model has been corroborated experimentally and by Monte Carlo simulation results. As forλ₀, the carrier heating as the origin of reduced mean-free-path is inferred on the basis of the simulated carrier velocity distribution at the injection point. Strikingly, for the parabolic potential case, the mean-free-paths remain consistent: λ’= λo. This indicates the absence or weakening of the carrier

References

[1] M. S. Lundstrom, “Elementary scattering theory of the Si MOSFET,” IEEE Electron Device Letters, vol. 18, pp. 361-363, July 1997.

[2] M. S. Lundstrom and Z. Ren, “Essential physics of carrier transport in nanoscale

MOSFETs,” IEEE Trans. Electron Devices, vol. 49, pp. 133-141, Jan. 2002.

[3] M.-J. Chen, H.-T. Huang, Y.-C. Chou, R.-T. Chen, Y.-T. Tseng, P.-N. Chen, and C.

H. Diaz, “Separation of channel backscattering coefficients in nanoscale MOSFETs,” IEEE Trans. Electron Devices, vol. 51, pp. 1409-1415, Sept. 2004.

[4] M.-J. Chen, H.-T. Huang, K.-C. Huang, P.-N. Chen, C.-S. Chang, and C. H. Diaz,

“Temperature dependent channel backscattering coefficients in nanoscale MOSFETs,” in IEDM Tech. Dig., pp. 39-42, Dec. 2002.

[5] M. J. Chen, S. G. Yan, R. T. Chen, C. Y. Hsieh, P. W. Huang, and H. P. Chen,

“Temperature-oriented experiment and simulation as corroborating evidence of MOSFET backscattering theory,” IEEE Electron Device Lett., vol. 28, pp.177-179, Feb. 2007.

[6] A. Rahman and M. S. Lundstrom, “A compact scattering model for the nanoscale

double-gate MOSFET,” IEEE Trans. Electron Devices, vol. 49, pp. 481-489, March 2002.

[7] M. J. Chen, R. T. Chen, and Y. S. Lin, “Decoupling channel backscattering

coefficients in nanoscale MOSFETs to establish near-source channel conduction-band profiles,” in Silicon Nanoelectronics Workshop, pp. 50-51, June 2005.

“ Understanding quasi-ballistic transport in nano-MOSFETs : Part I – Scattering in the channel and in the drain,” IEEE Trans. Electron Devices, vol. 52, pp. 2727-2735, Dec. 2005.

[9] Mark Lundstrom, Fundamentals of Carrier Transport, second edition, School of

Electrical and Computer Engineering Purdue University, West Lafayette, Indiana, USA: Cambridge University Press, 2000.

[10] R. Clerc, P. Palestri, and L. Selmi, “On the physical understanding of the

kT-layer concept in quasi-ballistic regime of transport in nanoscale devices,”

IEEE Trans. Electron Devices, vol. 53, pp. 1634-1640, July 2006.

[11] S. Datta, Electronic Transport in Mesoscopic System, Cambridge, U.K.:

Cambridge Univ. Press, 1995.

[12] P. Palestri R. Clerc, and D. Esseni, L. Lucci, and L. Selmi,

“Multi-subband-Monte-Carlo investigation of the mean free path and of the kT layer in degenerated quasi ballistic nanoMOSFETs,” in IEDM Tech. Dig., , pp.945-948, Dec. 2006.

[13] http://www.nanohub.org

[14] M. J. Chen and L. F. Lu, “A parabolic potential barrier oriented compact model

for the kBT-layer’s width in nano-MOSFETs,” IEEE Trans. Electron Devices, vol. 53, pp. 1265-1268, May 2008.

[15] R. Clerc, P. Palestri, and L. Selmi, “On the physical understanding of the

kT-layer concept in quasi-ballistic regime of transport in nanoscale devices,”

IEEE Trans. Electron Devices, vol. 53, pp. 1634-1640, July 2006.

[16] E. Fuchs, P. Dollfus, G. L. Carval, S. Barraud, D. Villanueva, F. Salvetti, H.

Jaouen, and T. Skotnicki, “ A new backscattering model giving a description of the quasi-ballistic transport in nano-MOSFET,” IEEE Trans. Electron Devices, vol. 52, pp. 2280-2289, Oct. 2005.

[17] J. Saint-Martin, A. Bournel, and P. Dollfus, “On the ballistic transport in

nanometer-scaled DG MOSFETs,” IEEE Trans. Electron Devices, vol. 51, pp. 1148-1155, July 2004.

[18]D. Querlioz, J. Saint-Martin, K. Huet, A. Bournel, V. Aubry-Fortuna, C. Chassat, S. Galdin-Retailleau, and P. Dollfus, “On the ability of the particle Monte Carlo technique to include quantum effects in nano-MOSFET simulation,” IEEE Trans.

Electron Devices, vol. 54, pp. 2232-2242, Sept. 2007.

[19] S. Eminente, D. Esseni, P. Palestri, C. Fiegna, L. Selmi and E. Sangiorgi,,

“ Understanding quasi-ballistic transport in nano-MOSFETs : Part II – Technology scaling along the IRTS,” IEEE Trans. Electron Devices, vol. 52, pp. 2736-2743, Dec. 2005.

[20] K. N. Yang, H. T. Huang, M. C. Chang, C. M. Chu, Y. S. Chen, M. J. Chen, Y.

M. Lin, M. C. Yu, S. M. Jang, C. H. Yu, and M. S. Liang, “A physical model for hole direct tunneling current in p+ poly-gate pMOSFETs with ultrathin gate oxides,” IEEE Trans. Electron Devices, vol. 47, pp. 2161–2166, Nov. 2000.

L (nm) Vtho (V) DIBL (mV/V) VD (V) VG (V) T (K) M. S. Lundstrom, et al. [2] 10 0.33 140 0.6 0.6 300 A. Rahman, et al. [6] 20 0.33 25 0.2 0.55 300 P. Palestri, et al. [8] 14~37 0.3 110 1 1 100~500 E. Fuchs, et al. [16] 25 0.3 100 0.8 0.5~1.0 300 J. Saint-Martin, et al. [17] 15 0.2 120 0.7 0.7 300 D. Querlioz, et al. [18] 15 0.3 77 0.7 0.7 300 S. Eminente, et al. [19] 14~65 0.3 11~230 1.0~1.2 1.0~1.2 300

Table. 3-1

( )

Ec x

l

F

c

r F

(1− r_c)F / B k T q

L

0

Fig.1-1

linear

l

parabolic

l

-5x107-4x107-3x107-2x107-1x107 0 1x107 2x107 3x107 4x107 5x107

0.0

0.2

0.4

0.6

0.8

1.0

Doping =10

12

#/cm

3

0

x

=

L=100nm,T=300K V

_a

=0.6V

Dis

tribut

ion

Velocity(cm/s)

Linear Potential

Fig.2-3

40

60

80

100

120

140

160

180

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0

c

L

r

L

λ

=

+

T=150K

T=200K

r

c

Linear Potential

T=300K

L=100nm

Parabolic Potential

λ

₀

(nm)

Fig.2-4

0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 V a(V) L=15nm L=25nm L=100nm L=15nm L=25nm L=50nm L=100nm Lines: (10), λ 0=56nm r c Linear Potential T=300K

}

Monte Carlo Simulation L=50nm 0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 Lines: (10), λ 0=105nm L=15nm L=25nm L=50nm L=100nm L=15nm L=25nm L=50nm L=100nm

}

r c V a(V) Linear Potential T=200K Monte Carlo Simulation 0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 V_a(V) Lines: (10), λ 0=155nm L=15nm L=25nm L=50nm L=100nm

}

L=15nm L=25nm L=50nm L=100nm r c Linear Potential

T=150K Monte Carlo_Simulation

(a)

(b)

(c)

0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 λ₀=λ' Lines: (10), λ 0=56nm V_a(V) L=15nm L=25nm L=50nm L=100nm L=15nm L=25nm L=50nm L=100nm

}

r c Parabolic Potential T=300K Monte Carlo Simulation 0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 λ₀=λ' Lines: λ 0=105nm V_a(V) L=15nm L=25nm L=50nm L=100nm L=15nm L=25nm L=50nm L=100nm

}

r c Parabolic Potential T=200K Monte Carlo Simulation 0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 λ₀=λ' Lines: λ 0=155nm V_a(V) L=15nm L=25nm L=50nm L=100nm L=15nm L=25nm L=50nm L=100nm

}

r c Parabolic Potential T=150K Monte Carlo Simulation

(a)

(b)

(c)

Fig.2-6

(

)

G

th

V

>

V

D

V

≈

/

B

k T q

( )

V x

l

L%

0

D

V

Fig. 3-1

0.4 0.6 0.8 1.0 1.2 0 4 8 12 16 20

}

(23)

VG(V)

298K

263K

233K

V

_D

=0.5V

Expriment 0.4 0.6 0.8 1.0 1.2 0 4 8 12 16 20

}

V

_D

=1.0V

VG(V)

(23)

298K

263K

233K

Experiment

Fig.3-2

0.4 0.6 0.8 1.0 1.2 1.4 20 30 40 50 60 70 80 90 100

}

(nm

)

V

_D

=1.0V

VG(V)

(22)

298K

263K

233K

Experiment 0.4 0.6 0.8 1.0 1.2 1.4 20 30 40 50 60 70 80 90 100

}

V

_D

=0.5V

VG(V)

(nm)

(22)

298K

263K

233K

Expriment

Fig.3-3

0 1 2 3 4 5 0 4 8 12 16 Experiment _{Ref. [2]} Ref. [6] Ref. [8] Ref. [16] Ref. [17] Ref. [18] Ref. [19] 1 1 0 .2 5 _{2 2} D B B 0 .5 G th D

V

k T

k T

L

( V -V

)

q

q V

⎛

⎞

⎛

⎞

⎜

⎟

⎜

⎟

⎝

⎠ ⎝

⎠

(nm-V

0.25

)

η

= 4.1

(

V

-0.25

)

}

Simulation

Fig.3-4

0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 λ₀=56nm V_a(V) L=15nm L=25nm L=50nm L=100nm λ ' (nm) Linear Potential T=300K 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 λ₀=105nm V_a(V) L=15nm L=25nm L=50nm L=100nm λ ' (nm) Linear Potential T=200K 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 120 λ₀=155nm V_a(V) L=15nm L=25nm L=50nm L=100nm λ ' (nm) Linear Potential T=150K

(a)

(b)

(c)

Fig.4-1

-3x107 -2x107 -1x107 0 1x107 2x107 3x107 4x107 5x107 0.0 0.2 0.4 0.6 0.8 1.0 Linear Potential L=25nm,T=300K V a=0.8V D istribution Velocity(cm/s)

x=0

-3x107 -2x107 -1x107 0 1x107 2x107 3x107 4x107 5x107 0.0 0.2 0.4 0.6 0.8 1.0

x=0

L=25nm,T=300K Va=0.8V D ist ribut ion Velocity(cm/s) Parabolic Potential

(a)

(b)

Fig.4-2

-3x107 -2x107 -1x107 0 1x107 2x107 3x107 4x107 5x107 0.0 0.2 0.4 0.6 0.8 1.0

x=0

L=50nm,T=150K V a=1.0V D is tribut ion Velocity(cm /s) Linear Potential -3x107 -2x107 -1x107 0 1x107 2x107 3x107 4x107 5x107 0.0 0.2 0.4 0.6 0.8 1.0

x=0

L=50nm ,T=150K V a=1.0V Di st ri bu ti o n Velocity(cm /s) Parabolic Potential

(a)

(b)

Fig.4-3

Fig.4-4

-5x107-4x107-3x107-2x107-1x107 0 1x107 2x107 3x107 4x107 5x107

0.0

0.2

0.4

0.6

0.8

1.0

x

= l

L=50nm,T=200K V

_a

=0.2V

Dis

tribut

ion

Velocity(cm/s)

Linear Potential

a( )

l

b( )

l

a(0)

c

r a(0)

a( )

L

L

− l

l

0.0

0.2

0.4

0.6

0.8

1.0

0.01

0.1

Monte Carlo

Simulation

}

L=

10

0n

_m

L=

15

nm

L=

25n

m

L=

50

nm

L=15nm

L=25nm

L=50nm

L=100nm

T=200K

λ

_0=105nm

Linear Potential

b

(

l

)

/a

(

l

)

V

_a

(V)

Fig.4-5