Arizona State University

Applied Project Report Presented in Partial Fulfillment of the Degree

PROFESSIONAL SCIENCE MASTERS IN NANOSCIENCE

——————————————————

PARTICLE-BASED DEVICE SIMULATIONS

OF GERMANIUM TRANSISTORS

——————————————————

Applied Project Report of:

Daniel R. Livingston

Committee Chair:

Dragica Vasileska, Ph.D.

Committee Members:

Jingyue Liu, Ph.D.

Stuart Lindsay, Ph.D.

John Venables, Ph.D.

November 2015

CONTENTS i

Contents

1 Introduction 1

2 Theoretical Background 2

2.1 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Quasi-Fermi Level Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Displaced Maxwellian Approximation . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Final Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Acoustic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Coulomb Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.3 Intervalley and Intravalley Scattering . . . . . . . . . . . . . . . . . . . . . . . 9

3 Monte Carlo Method as a Solution to the BTE 10

3.1 Monte Carlo Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Spatial Constrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Device Structure and Simulation Results 14

4.1 Simulation Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Bulk Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.2 Device Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Experimental Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 18

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A Properties of Bulk Germanium 21

B Code Repository 22

CONTENTS ii

Abstract

——————————————————

As conventional transistors quickly approached a fundamental limit where electrons were able to

tunnel across the source-drain potential barrier, it became imperative that new transistor designs

are created. One such design consideration was found in 15 and 22 nm technology node FinFET

devices. Though silicon has been the material of choice up to date, low-power of operation of state-of

the art devices allows germanium to be used as well. Bulk Germanium Monte Carlo simulations

are performed first and checked by comparing simulation to experimental values of velocity vs. field

characteristics. Device simulations are performed next and current conservation was verified.

1 INTRODUCTION 1

1 Introduction

Developments in nanotechnology have led to dramatic size decreases in electronic and mechanical

devices. This aggressive downsizing, popularized into the public lexicon through Moore’s Law, is

remarkably fit into a doubling of CPU transistor count per unit area every two years.1 In fact,

transistors with dimensions of just a few nanometers across are now openly considered.[1] The

continuation of progress in Moore’s Law has been so startling that Gordon Moore himself once

remarked that, “Moore’s law is a violation of Murphy’s law. Everything gets better and better.”

This is not to say, however, that as lithography techniques improve the same design and materials

can simply be used on a smaller scale, or that new manufacturing processes are the only hurdles to

overcome. Quantum and thermodynamic effects become more prescient the smaller the device scale,

and Lacroix et al. notes that devices on the order of a few nanometers will dissipate so much heat

through the joule effect that it will be comparable to the levels generated by a standard lightbulb.[1]

It is with these notions in mind that new device structures and materials must be considered. The

FINFET MOSFET architecture, for instance, was created to circumvent the current leakage that

similar sized planar transistors face by introducing a third dimension and effectively increasing the

potential barrier between the source and drain.

Device modeling comes in two broad categories: physical device models and equivalent circuit

models. Physical device models seek to simulate the physics underlying a structure – electron-phonon

scattering, heat dissipation, etc. – while equivalent circuit models use an electrical circuit framework

to model the electrical behavior.[4] The results of the latter are often limited in its applicability, so

physically-based device modelling is the consideration of the following report.

Since Kurosawa first introduced the Monte Carlo method to semiconductor device modelling in

1966, it has become the de facto process to determine characteristics of transistors.[4] The Monte

Carlo observes the motion, momentum, and energy of one or more electrons2 contained in a crystal

lattice, which are driven by an electric field and subjected to material-specific scattering processes.

The charge carriers are allowed to travel for some stochastic amount of time, termed the carrier

time-of-flight, before a scattering mechanism acts upon the carrier and scatters it to somewhere

else in the lattice.[7] The electrons are observed to see that they stay within device boundaries,

and carriers are added or removed to see the doping profile remain constant. The backbone of

Monte Carlo device simulations lie in the self-consistent solution of the Poisson Equation and the

Boltzmann Transport Equation; the former to calculate the electrostatic potential, and the latter to

determine carrier motion.

The final note to touch on is the study of germanium in this report. While silicon is famously the

material of choice for most semiconductors, the semiconductor revolution used almost exclusively

germanium until the early 1960’s. Researchers began to take interest in silicon for its higher band

gap and thermal conductivity, but it wasn’t until stability problems concerning manufacturing of

1Gordon E. Moore, Cramming More Components Onto Integrated Circuits (1965)2Where ‘electrons’ here and elsewhere in the text generally refers to the broader term charge carriers, which include

both electrons and holes.

2 THEORETICAL BACKGROUND 2

the silicon dioxide substrate were overcame that the material began to engulf the market.[10] The

relative cost of silicon was a strong factor in its market dominance.3 It is not only abundant –

the eighth most common element – but much simpler to process; a thin layer of silicon dioxide

substrate coats the surface very easily. Germanium, on the other hand, ranks much lower as the

fiftieth most common element, and processing is much harder. Irrespective of this, germanium

offers redeeming characteristics not found in silicon. The mobility of electrons and holes are much

higher for germanium (µe ≤ 3900 cm2/Vs, µp ≤ 1900 cm2/Vs) than for silicon (µe ≤ 1400 cm2/Vs,

µp ≤ 450 cm2/Vs), which allows Ge devices to function at a higher frequency.[15][16] The band gap is

also less than half that of silicon (0.66 eV against 1.11 eV, at T = 300 K)[12]. While a double-edged

sword – thermal pair production is lower in silicon – it also means that, for a sufficiently chosen

temperature range, germanium devices require less power to function than silicon.

2 Theoretical Background

2.1 Boltzmann Transport Equation

As device dimensions continue to shrink, quantum effects become increasingly more pertinent as the

channel lengths approach the characteristic wavelength of particles. While some quantum effects,

such as confinement in the inversion layers of silicon MOSFET devices, have been observed for some

time, they become second-order under standard room temperature and a strong driving field.

As this is the case, it is standard that charge transport is measured classically through the

Boltzmann Transport Equation (BTE). The BTE models electron behaviour in a crystal lattice,

through a seven-dimensional integral-differential:

∂f(r,k, t))

∂t+

1

h∇kE(k) · ∇rf(r,k, t) +

F

h· ∇kf(r,k, t) =

[∂f(r,k, t)

∂t

]coll

(2.1)

The equation is dependent on the particle distribution function, f(r,k, t), where r is the position

vector and k is the crystal momentum vector. The Boltzmann Transport Equation models the

evolution of the distribution function f , causing temporal and spatial adjustments in response to

force, collision, and advection.

The terms in the above equation are represented by:

Diffusion Particles will undergo diffusion if some concentration or temperature gradient is present.

(∂f/∂t)diff = 1h∇kE(k) · ∇rf(r,k, t))

Electromagnetic Force Particles will alter their momentum in response to forces from electric

and magnetic fields.

(∂f/∂t)forces = Fh · ∇kf(r,k, t), where F = q(E + v × B)

3A full historical perspective is beyond the work of this paper, but Walter Isaacson’s The Innovators (2014) offers

a detailed expose on this critical point in Silicon Valley history.

2 THEORETICAL BACKGROUND 3

Collisions Represents the forces acting between particles undergoing a collision, equal to the dif-

ference of in- and out- scattering processes.

(∂f/∂t)coll =∑k′S(k′,k)f(k′)[1− f(k)]− S(k,k′)f(k)[1− f(k′) = Cf

Due to the presence of both f(k) and f(k′), the collision term makes the Boltzmann Transport

Equation very difficult to solve. Fortunately, a simplification can be found in Fermi-Dirac statistics.

If we assume that the interacting phonons and electrons are in thermal equilibrium, then the following

approximation holds[9]:

S(k,k′)

S(k′,k)= exp

(Ek − Ek′

kBT

)(2.2)

where S(k,k′) is the transition rate between states k and k′.

Through the solution of the BTE, important averages and moments of interest can be found,

including particle, energy, and current density:

Particle Density n(r, t) =1

V

∑k

f(r,k, t)

Current Density J(r, t) = − e

V

∑k

v(k)f(r,k, t)

Energy Density W (r, t) =1

V

∑k

E(k)f(r,k, t)

The most difficult aspect of finding a solution is calculating what the distribution function is

that gives the probability of finding some electron with quasimomentum k at position r and time

t. It is commonly approximated by one of two approaches: the quasi-Fermi level concept and the

displaced Maxwellian approximation.

2.1.1 Quasi-Fermi Level Concept

For equilibrium conditions – where the electron and hole concentrations are equal to the intrinsic

carrier concentration squared, ne · np = n2i – the electron distribution function may be represented

through Fermi-Dirac statistics as

fn(E) =1

1 + exp((E − EF )/kBT )(2.3)

where EF is the Fermi-level energy. The hole distribution function is simply fp(E) = 1− fn(E).

Under non-equilibrium conditions, the state we are interested in for this discussion, the concen-

tration relationship is given by

ne · np = n2i exp

(EFn

− EFp

kBT

)(2.4)

2 THEORETICAL BACKGROUND 4

The variables EFn and EFp are known as quasi-Fermi level energies, for electrons and holes re-

spectively. Quasi-Fermi levels are useful in solid-state physics for describing displaced populations

of carriers in the valence and conduction bands, typically resulting from the use of an applied voltage.

The resulting non-equilibrium distribution functions come out to be

fn(E) =1

1 + exp((E − EFn)/kBT )(2.5)

fp(E) = 1− fn(E) =1

1 + exp((EFp

− E)/kBT) (2.6)

2.1.2 Displaced Maxwellian Approximation

This approach assumes that the distribution function maintains the same shape; however, it is

displaced according to the direction of the electric field.

f(r,k, t) = exp

(EFn

− EC0

kBT

)exp

(− h2

2m?kBT‖k − kd‖2

)(2.7)

where kd is the quasimomentum vector associated with group velocity vd. This approach is

typically a better approximation than the quasi-Fermi level concept, and useful for applications

where hydrodynamic equations are needed.

Wave Vector ×109

-1 -0.5 0 0.5 1

Num

ber

of C

arr

iers

0

50

100

150

200

250

300

350

400

450

500k

x

ky

kz

Wave Vector ×109

-4 -2 0 2 4

Num

ber

of C

arr

iers

0

50

100

150

200

250

300k

x

ky

kz

Figure 1: Initial distribution (left) and final distribution (right) of

charge carriers, relative to momentum vector k for this work. Note the

spreading in the final distribution.

2.1.3 Final Notes

It is also important to note that the Boltzmann Transport Equation is semi-classical; classical in

the sense that a particle’s momentum and position are treated as exactly known, and quantum in

2 THEORETICAL BACKGROUND 5

the sense that scattering and particle dynamics are resultant from the Bloch nature of electrons and

the quantum-mechanical treatment of the electronic band structure. It cannot be represented as

purely quantum mechanical, as the distribution function presupposes that position and momentum

are exactly known, in violation of the Uncertainty Principle. The distribution function solves these

disparities by being sufficiently coarse-grained; where electron variables are found using probabilistic

averages.

For some class of particles that (i) follow Fermi-Dirac statistics and (ii) interact with one another

via scattering processes, the Boltzmann-Transport Equation provides an appropriate framework for

modelling.

2.2 Fermi’s Golden Rule

Scattering theory is based upon Fermi’s Golden Rule, which in turn, is derived from first-order

time-dependent perturbation theory. Fermi’s Golden Rule, abbreviated FGR, gives the transition

probability between two eigenstates and forms the basis for which scattering equations are built.

If we begin with the time evolution postulate, and consider the Hamiltonian as consisting of the

unperturbed Hamiltonian, H0, a perturbation potential H ′, and some perturbation parameter λ,

then the Shrodinger equation to be solved is

ih =∂

∂tψ |r, t〉 = (H0 + λH ′)ψ |r, t〉 (2.8)

It must be noted that the equation can only be solved for some weak influences, i.e. a weak

perturbing Hamiltonian part H ′. If this is not the case, Schrodinger’s equation becomes too difficult

to solve analytically.

Note that electrons in a periodic potential of a semiconductor lattice are Bloch wave functions,

mathematically represented as

ψn,k = un(k)eikr (2.9)

The cell periodic function, un(k), has the same periodicity as the lattice, and must satisfy

un(k+K) = un(k) (2.10)

The derivation of Fermi’s Golden Rule is very well known, and, in the interest of brevity, will not

be repeated here in full. A full derivation of FGR and the resultant scattering rates can be found

in Tomizawa [11]. The jth scattering rate, for a carrier in band n and state k to band m and state

k′, takes the form

Γj [n,k;m,k′] = limt→∞

∣∣⟨m,k′∣∣H ′ |n,k〉∣∣2h2

[sin(ζt)

ζt

]2

t (2.11)

which yields the profile found in figure 2:

2 THEORETICAL BACKGROUND 6

Figure 2: Plot of (sin(ζt)/ζt)2, for t = 0.1 ps and ζ = E/2h.

For the limit t → ∞ (weak scattering) this strongly resembles a Dirac δ-function, which gives

the final form for the scattering rate equation:

Γj [n,k] =2π

h

∑m,k′

∣∣⟨m,k′∣∣H ′ |n,k〉∣∣2 δ(Ek′ − Ek ∓ hω) (2.12)

The presence of the δ-function means that the scattering rate will be non-zero if and only if the

delta function itself is zero; that is, only if energy conservation is maintained. The presence of the

∓ symbol shows that there are two cases where this is true:

Emission If the negative sign is chosen, then the δ-function will be zero when Ek′ = Ek + hω, and

a phonon of energy hω was emitted

Absorption For the positive case, then the function is satisfied when Ek′ = Ek− hω, and a phonon

of energy hω was absorbed

This equation is used to calculate the decay rates for a broad range quantum systems, though

with the appropriate perturbation potential H ′ many different scattering rates can be found.

2.3 Scattering Mechanisms

In a crystal lattice, it is possible for the periodicity to be disturbed for some reason. When this

happens, lattice vibrations – or phonons – propagate throughout the material, which can interact

with a Bloch electron and cause scattering to occur. This scattering is quantum in nature, and

dubbed as electron-phonon interactions. At low temperatures, a semiconductor has only a few

phonons present at any given time. When an appropriately high electric field is applied, however,

phonon generation is spontaneous and responsible for a large amount of electron scattering in a

crystal.

For the periodic arrays of atoms found in a crystal, their vibrations can be characterized by a

superposition of individual normal mode oscillations – a pattern of motion in all waves of a system,

where their motion sinusoidally oscillates at a shared frequency and with a fixed phase relation.

These normal modes independently move in a harmonic oscillator fashion, which can be quantized.

The resultant quanta of vibrational mechanical energy is termed a phonon.

2 THEORETICAL BACKGROUND 7

Numerous scattering mechanisms exist within a semiconductor lattice, finding themselves within

the broader categories of defect scattering, carrier-carrier scattering, and lattice scattering.

Figure 3: A branching tree showing the many different types of semi-

conductor carrier scattering [9]

In defect scattering, scattering is caused by a break in the periodicity of the lattice – through

defects in crystal formation, some impurity in an otherwise homogeneous material, or the presence

of some substitutional or interstitial alloy.

When two charge carriers interact, such as an electron scattering off of a hole, carrier-carrier

scattering results; while lattice scattering is scattering between charge carriers and phonons. The

latter is always inelastic in nature. Different bulk materials will result in different scattering mech-

anisms – a device formed with gallium-arsenide, for example, will be subjected to polar scattering

due to the diatomic nature of the lattice. A homogeneous material, like undoped germanium, will

not. In all of these scattering processes, carriers may absorb or emit quanta of energy, in units of

hω.[9]

In a germanium Monte Carlo simulation, scattering can occur via acoustic phonons, optical

phonons, and due to ionized impurities. These processes can be either intravalley (where the elec-

tron’s initial and final states occur in the same valley) or intervalley (where the final state is in a

different valley than the initial).

2.3.1 Acoustic Scattering

One of the most important equations for deriving scattering rates is the lattice displacement vector,

u(r, t). The lattice displacement for position r and time t can be represented as

u(r, t) =∑q

(h

2ρΩωq

)1/2

eq

(aq + a+

−q)eiq·r (2.13)

2 THEORETICAL BACKGROUND 8

where q = k′ − k is the wave vector, ρ is the bulk material density, Ω is its volume, ωq is

the angular frequency of oscillation, eq is the unit polarization vector, and a+q and aq are the

quantum-mechanical creation and annihilation vectors. These two vectors represent the creation

and destruction of phonons.[11]

Recall the general scattering rate formula derived from Fermi’s Golden Rule,

Γj [n,k] =2π

h

∑m,k′

∣∣⟨m,k′∣∣H ′ |n,k〉∣∣2 δ(Ek′ − Ek ∓ hω)

Here, the critical missing piece for turning the general scattering rate formula into a specific

one is in the interaction potential, H ′. In the case of acoustic modes, the relative displacement of

neighboring ions – vibrations around some equilibrium position – cause an instantaneous change

in the energy band, leading to phonon scattering upon interaction.[7] This displacement causes

deformation of the unit cell, and can be represented as proportional to induced strain, or

H ′ = Ξd∇ · u(r, t) (2.14)

With these three equations, the transition rate due to acoustic phonons can be found. A complete

derivation can be found in Tomizawa [11], and is given as

S(k,k′) ≈ πΞ2dkBTLhcLΩ

k

qEkδ( q

2k± cos θ′

)(2.15)

for lattice temperature TL, material elastic constant cL, and θ′, the polar angle between vectors

k and q.

The scattering rate is the integrand of the transition rate, taken over all k′, and therefore

W (k) =Ω

(2π)3

∫S(k,k′)dk′ =

2πΞ2dkBTLhcL

N(Ek) (2.16)

in the elastic and equipartition approximation, where N(Ek) is the density of states. For a

parabolic band structure, it takes the form

N(Ek) =(2m?)3/2

√Ek

4π2h3 (2.17)

When a voltage of sufficient magnitude is applied, electrons may occupy much higher energy

levels. When this is the case, the parabolic assumption breaks down, and the non-parabolic approx-

imation must be used.[14] A small non-parabolicity factor α is introduced to the energy

Ek(1 + αEk) =h2k2

2m?= γ(k) (2.18)

In the case of germanium, α = 0.3 eV−1.[2] The non-parabolic density of states is therefore

N(Ek) =(2m?)3/2

√γ(Ek)

4π2h3

∂γ(Ek)

∂Ek=

(2m?)3/2√Ek(1 + αEk)

4π2h3 (1 + 2αEk) (2.19)

In this derivation of the acoustic scattering rate, the foundation for finding all other scattering

rates has been laid. This discussion will be concluded with a brief look at the two other significant

scattering mechanisms in germanium, Coulomb scattering and inter-/intravalley scattering.

2 THEORETICAL BACKGROUND 9

2.3.2 Coulomb Scattering

Coulomb scattering is a form of ionized impurity scattering, treated within the Brooks-Herring ap-

proach [9]. In the work presented here, it plays little significance into the overall results. As Coulomb

scattering is linearly related to the doping density of the bulk material, for a small amount of dopants

it can be considered negligible.[2] Further, it is dominant at low and cryogenic temperatures; in this

work, a standard temperature of T = 300 K was used.

For completeness, however, it is included. As noted by Aubrey-Fortuna [2], for low temperatures

and a small field, ionized impurity scattering will have a significant impact on charge carrier velocities

– even for a very low concentration of impurities. The screened Coulomb potential, then, is

Vi(r) = − Ze2

4πεre−λr (2.20)

where λ = 1/LD is the screened Debye scattering length, equivalent to the inverse of the Debye

length LD =√εVT /(q(ne + np)). Introducing this potential to Fermi’s Golden Rule, under the

non-parabolic approximation, results in the scattering rate[9]

Γk =

√2niZ

2e4

ε2sc√m?dEβ

√E(1 + αE)(1 + 2αE)

1 + 4(E(1 + αE)/Eβ)(2.21)

for Eβ = h2q2D/2m

?d and qD = 1/LD = λ.[9]

2.3.3 Intervalley and Intravalley Scattering

In this section, scattering by optical intervalley and intravalley phonons is considered. To refresh,

when an electron moving through a lattice is scattered optically, its final state may either be in the

same valley as which it began, or it may be scattered to a different valley altogether. In the case

where the electron’s initial and final states are in the same valley, it is known as intravalley ; for the

converse case, the nomenclature is intervalley.

Germanium has three valleys: Γ, X, and L.4 Intravalley scattering for this band structure can only

happen in the < 111 > L valley – selection rules elsewhere forbid optical intravalley transistions. [2]

This scattering rate is given as

Γop(E) =(m?

d)3/2Ξ2

D√2πh2ρ(hωop)

[Nq

Nq + 1

]√E ± hωop

×√

1 + α(E ± hωop) [1 + 2α(E ± hωop)]

(2.22)

Intervalley scattering is much broader than intravalley. For germanium, there are two intervalley

scattering processes between the L valleys, one between L and Γ, one between L and X, two between

X valleys, and one between X and Γ. The general intervalley scattering rate for all of these valley

4The X valley is sometimes denoted in other literature as ∆.

3 MONTE CARLO METHOD AS A SOLUTION TO THE BTE 10

processes is

Γiv(E) =Ziv(m

?d)

3/2Ξ2D√

2πh2ρ(hωiv)

[Nq

Nq + 1

]√E ± hωop + ∆Eiv

×√

1 + α(E ± hωop + ∆Eiv) [1 + 2α(E ± hωop + ∆Eiv)]

(2.23)

for the number of final possible valleys, Ziv, and the energy transition between valleys, ∆Eiv.

Energy (Normalized)0 0.5 1

Γ (

1/s

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy (Normalized)0 0.5 1

Γ (

1/s

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy (Normalized)0 0.5 1

Γ (

1/s

)0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: Cumulative scattering rates for Ge valleys Γ, X, and L re-

spectively. For comparison’s sake, the non-normalized maxima of the

scattering rates are: Γmax = 2.69 · 1013 (Γ-valley); Γmax = 9.73 · 1012

(X-valley); Γmax = 3.85 · 1012 (L-valley)

3 Monte Carlo Method as a Solution to the BTE

For over 40 years, various forms of ensemble Monte Carlo techniques have been used to numerically

solve non-equilibrium transport in semiconductor devices and bulk materials, in addition to other

particle-based fields such as electron and radiation transport.[13]

A bulk semiconductor is essentially a large many-body problem, with upwards of 1017 cm−3 free-

carrier electrons. A proper simulation, considering each carrier as interacting, requires calculations

on the order of N2/2 with an O-factor of O(N2). It is obvious, then, that the enormous complexity

of this problem requires approximations if a solution is to be found in any reasonable time-scale. For

a homogenous bulk semiconductor, one can approximate a solution to such a many-body system by

considering only a single particle undergoing many scattering events. The basis of the Monte Carlo

algorithm lies in generating a Markovian random walk in momentum space, which is interrupted by

scattering processes. Upon termination, a scattering type is chosen and final energy and momentum

3 MONTE CARLO METHOD AS A SOLUTION TO THE BTE 11

is changed to reflect the physics. This process is repeated until the simulation elapses. The figure

below illustrates the flow of the program.

Parameter Initialization

Scattering Table Construction

Carrier Initialization

Histogram Calculation

Free-flight Scatter

Histogram Calculation

Write Data

t < tmax

Data Initialization

Compute Charge

Solve Poisson Eq.

Carrier Dynamics with Monte Carlo kernel

t < tmax

Collect Data

Figure 5: (a) Flow of Three-Valley Bulk Monte Carlo Simulation (b)

Flow of Single-Particle Device Simulation. References Monte Carlo car-

rier dynamics are the body of the leftmost figure.[9]

The core of the Monte Carlo lies in the free flight scatter() subroutine. For every time step, a

random flight time is assigned to a particle, and upon termination, a scattering process is chosen

according to the probability density profile. The joint probability density of a particle scattering in

the time interval dt after a free-flight time t is given by

P (t)dt = Γ [k(t)] exp

[−∫ t

0

Γ [k(t′)] dt′]dt (3.1)

where Γ [k(t)] is the scattering rate, defined as the sum of scattering rates for each individual

mechanism.

Carrier time-of-flight values are also generated according to this profile. A pseudo-random num-

3 MONTE CARLO METHOD AS A SOLUTION TO THE BTE 12

ber generator, properly seeded to approach a uniform distribution between zero and one, is related

to P (t) and time-of-flight according to the equation

r =

∫ tr

0

P (t)dt (3.2)

where tr is the time of flight value to be solved for. By defining an artificial scattering rate

Γ (self-scattering), which makes the total scattering rate energy independent and observing that

r ≡ 1− r, where r is a random number uniformly distributed between 0 and 1, it can be shown that

[9]

tr = − 1

Γln r (3.3)

where Γ is an artificial scattering rate, constant in time, composed of the sums of the real

scattering rate Γ[k(t′)] and self-scattering rate Γself [k(t′)]. Self-scattering is an arbitrary scattering

rate chosen such that the state is constant throughout the scattering process.

3.1 Monte Carlo Device Simulation

When considering a device simulation, as opposed to one in a bulk semiconductor, several condition-

als must be applied for an appropriate ensemble Monte Carlo. First, the motion of the particles must

be spatially restricted in a way that conforms to the device characteristics of the simulation. This

is contrasted to a bulk Monte Carlo, where it is assumed the spatial boundaries approach infinity.

Second, the Poisson equation, considered on a mesh and applied to the boundary conditions, must

be solved at the end of every time step.

3.2 Spatial Constrictions

In a bulk Monte Carlo, semiconductors regions are assumed as boundless. To more accurately model

the device in question, it is obvious that an infinite Cartesian plane is not sufficient. By design, there

are inherent boundaries within a device, so allowing carriers to move in a laissez-faire fashion will

not produce results characteristic of the device in question. A roughly constant number of electrons

(correspondent to the doping density) must be maintained within the device boundaries, so three

conditionals to ensure this are implemented:

Reflection If a carrier is scattered onto one of the device faces, a functional equation is applied

to the spatial and momentum vectors to maintain the particle within device limits. (This

condition is excepted in the cases of penetration into the source, drain, or gate contacts.)

Exiting If a carrier passes through the drain, it is deleted.

Entering Particles may be injected into the source and drain (as well as deleted) to maintain

consistency between the number of carriers in the simulation and the defined doping density.

3 MONTE CARLO METHOD AS A SOLUTION TO THE BTE 13

n+ n+n

x

z

SOURCE DRAINGATE x

Figure 6: Illustration of particle reflection (bottom-left) and deletion

(top-right) for a standard MOSFET

3.3 Poisson Equation

In order to accurately measure the potential voltage profile, the Poisson equation must be decoupled

from the Boltzmann Transport Equation and solved at every time step ∆t, according to the configu-

ration of charge carriers determined in the Monte Carlo. The time step must be much smaller than

the inverse of the plasma frequency in order to satisfy a stability criterion5. While the ensemble

Monte Carlo takes place along a spatial continuum, when Poisson’s equation is solved it is done so

over a mesh of grid size ∆r = ∆x,∆z. The discrete nature of solving the Poisson equation is due

to simplification of computational workload.

The Poisson equation for a homogeneous material is

∇2V (r) = − ρεs

(3.4)

where V (r) is the potential, ρ is the charge density, and εs is the material dielectric constant.

For a two-dimensional simulation, Poisson’s Equation can also be represented as

∂2V (r)

∂x2+∂2V (r)

∂z2= − ρ

εs(3.5)

Discretizing this on equally spaced meshes, for the general case ∆x 6= ∆z, the 2D case expands

to

Vi−1,j − 2Vi,j + Vi+1,j

∆x2+Vi,j−1 − 2Vi,j + Vi,j+1

∆z2= −ρi,j

εs(3.6)

where i and j represent coordinates on the two-dimensional mesh. Poisson’s Equation is now in

a suitable position to be solved. Written in matrix form,

A~V = −∆x2

εs~ρ (3.7)

5Having a stability criterion is a requirement of stability theory, which is a eponymous framework used to describe

how stable a given system is.

4 DEVICE STRUCTURE AND SIMULATION RESULTS 14

Many different numerical methods can be used to solve this linear equation. The simplest and

most common are the successive over-relaxation (SOR) and alternating direction implicit (ADI)

methods. [8] For a two-dimensional, rectangular, equally-spaced mesh, it is recommended by

Tomizawa to use the Fourier analysis and cyclic reduction (FACR) method.[11] For all other cases,

solutions include the Gaussian-Seidel (GS) method, the multi-grid method, and the incomplete

Choleski conjugate gradient (ICCG) method. [11]

Figure 7: Potential values on a 2D, equally spaced grid [11]

4 Device Structure and Simulation Results

The simulation was performed at T = 300 K for an ultra-pure Germanium MOSFET.[6] The device

had a depth of 102 nm, a width of 300 nm, with a doping profile of np = 1024 m−3 for the source

and drain, and ne = 1021 m−3 for the bulk. Bulk properties were considered over a time frame of

10−10 s, and device properties were performed over 10−12 s. Properties intrinsic to germanium can

be found in Appendix A.

4.1 Simulation Verifications

4.1.1 Bulk Characteristics

Figure 8 shows the bulk profile for charge carrier energy and velocity per valley over the course of the

simulation. The transient state of the simulation occurs over the approximate period [0, 0.3] · 10−10

seconds. Note that as the simulation progresses, there is a convergence on the steady state solution

for both plots. As the electric field for this case is applied only in the y-direction, E = (0, 7, 0) · 105

V/m, carriers accelerate accordingly. The steady state solution for this electric field is vy = −1.454 ·105 m/s.

The bottom half of figure 8 belays a note of interest for germanium. The profile for the L−valley

contains the least noise – as this valley is energetically favored, it has a higher concentration of

charge carriers than the other two.

4 DEVICE STRUCTURE AND SIMULATION RESULTS 15

Time (t) ×10-10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Energ

y (

eV

)

0

0.2

0.4

0.6

0.8

1

EΓ

EL

EX

×10-10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1V

elo

city (

m/s

)

×105

-6

-4

-2

0

2

Vx

Vy

Vz

Figure 8: Average carrier velocity (top) and energy (bottom) in different

valleys as a function of time

4.1.2 Device Characteristics

It is critical to ensure that a constant number of charge carriers are in the device over some amount

of time ∆t. Figure 9 shows the absolute value of the sum of charges in the device over the simulation

run. The parallel nature of the lines shows that a roughly equal number of charges are being deleted

as they are added, an important stability check for maintaining the simulation.

Figure 10 shows the overall profile of electrons in the source (right) and drain (left). That the

concentrations are roughly equal adds further support to the claim that charge carriers are behaving

appropriately. The x− and z− position space refers to the mesh grid. This can easily be converted

to meters – the length of an x grid space is 2 nm, and a z grid space length is 0.5 nm.

4 DEVICE STRUCTURE AND SIMULATION RESULTS 16

Simulation Time (s) ×10-12

0 0.5 1 1.5 2 2.5 3 3.5 4

Cu

mu

lative

Ch

arg

e

-5000

0

5000

10000

15000

20000

Figure 9: Total count of charge carriers added to and deleted from

source (dark blue) and drain (light blue).

010203040

X Node

506070200

150

100

Z Node

50

0

0.5

1

2.5

2

1.5

0

×1025

Ele

ctr

on

De

nsity (

m-3

)

Figure 10: Electron distribution across the device. ∆x = 2 nm, ∆z =

0.5 nm.

4 DEVICE STRUCTURE AND SIMULATION RESULTS 17

4.2 Experimental Verifications

Electric Field (V/m) ×105

0 1 2 3 4 5 6 7

Drift V

elo

city (

m/s

)

×104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Livingston (300 K)Aubrey-Fortuna (240 K)

Electric Field (V/m) ×105

0 1 2 3 4 5 6 7

Energ

y (

eV

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Livingston (300 K)Aubrey-Fortuna (190 K)

Figure 11: (a) Charge carrier drift as a function of electric field at

T = 300 K. Compared with the work of Aubrey-Fortuna et al. (T = 240

K) [2]. (b) Average L-valley carrier energy as a function of electric field,

against the work of Aubrey-Fortuna

The plots in figure 11 are compared against the work of Aubry-Fortuna et al. The discrepancy in

drift velocity (fig. 11(a)) is presumed to be caused by a lack of Coulomb scattering, which becomes

more present at lower temperatures.[2]

Figure 11(b) compares the average energy of carriers in the L-valley in this work and Aubry-

Fortuna’s. There is nearly a ten-fold difference between the two. It may be the case that some of this

discrepancy is due to difference in lattice temperature, but it is doubtful that 110 K temperature

difference can account for an order of magnitude in carrier energy. Further work is needed to precisely

pinpoint the problem.

5 CONCLUSION 18

200

150

Z Node

100

50

07060

50X Node

4030

2010

0-1

-0.5

0

0.5

1

Co

nd

uctio

n B

an

d P

rofile

(e

V)

Figure 12: Potential gradient over the device. Note the potential

plateau at the source (far left) curving into the drain (right).

Figure 12 shows how the potential varys across the MOSFET. For electrons in the source, an

applied potential – such as gate voltage – provides the energetically favorable conditions to transverse

the potential crest in the depletion region and into the drain.

5 Conclusion

The characteristics of germanium, as a bulk material and as a MOSFET device, were investigated in

this report. Using the Monte Carlo method to investigate bulk and device-specific characteristics is

an active field, with researchers investigating charge carrier behaviour at cryogenic temperatures[2],

extremely pure germanium (99.99% 70Ge)[3], and device dimensions on the order of 2 - 20 nm.[1]

The verification of the work presented here lays a framework for future investigation of phenomena

similar to these.

5.1 Future Work

As device dimensions continue to shrink and power efficiency becomes more important, germanium

in conjunction with other device structures such as FINFET offer a lucrative field for investigation.

The next phase of this work is to expand from two-dimensions into three, and investigate the

characteristics of different purities of germanium in conjunction with different sub-25 nm node device

designs. Multigate devices in particular, such as the FINFET, Tri-Gate, and GAAFET architectures,

are particularly interesting and offer strong promise for the future of the semiconductor industry.[5]

REFERENCES 19

References

[1] Lacroix D., Joulian K, Lemonnier D. Monte Carlo Transient Phonon Transport in Silicon and

Germanium at Nanoscales. Phys. Rev B, 72(6), 2005.

[2] Aubrey-Fortuna V., Dollfus P. Electron Transport Properties in High-Purity Ge Down to

Cryogenic Temperatures. Journal of Applied Physics, 108(12), 2010.

[3] Kuleev I.G., I. I. Kuleev, A. N. Taldenkov, A. V. Inyushkin, V. I. Ozhogin, K. M. Itoh, and

E. E. Haller. Normal Processes of Phonon–Phonon Scattering and the Drag Thermopower in

Germanium Crystals with Isotopic Disorder. Journal of Experimental and Theoretical Physics,

96(6), 2003.

[4] Snowden C. M. Semiconductor Device Modelling. Rep. Prog. Phys., 48:223 – 275, 1985.

[5] Liu T.J.K. FinFET History, Fundamentals and Future, June 11 2012. 2012 Symposium on

VLSI Technology Short Course. Hilton Hawaiian Village, Honolulu, HI.

[6] Ghosh B., Wang X., Fan X.F., Register L.F., Banerjee S. Monte Carlo Study of Germanium n-

and pMOSFETS. IEEE Transactions on Electron Devices, 52(12):547 – 552, 2005.

[7] Jacobini C., Reggiani L. The Monte Carlo Method for the Solution of Charge Transport in

Semiconductors with Applications to Covalent Materials. Rev. Mod. Phys., 55(3), 1983.

[8] Vasileska D., Goodnick S. Materials Science and Engineering, Reports: A Review Journal,

R38(5):181 – 236, 2002.

[9] Vasileska D., Goodnick S., Klimeck G. Computational Electronics. Morgan & Claypool Pub-

lishers, San Rafael, CA, 2006.

[10] Seidenberg P. From Germanium to Silicon, A History of Change in the Technology of the

Semiconductors. ed. Andrew Goldstein & William Aspray (New Brunswick: IEEE Center for

the History of Electrical Engineering), 1997.

[11] Tomizawa K. Numerical Simulation of Submicron Semiconductor Devices. Artech House, Nor-

wood, MA, 1993.

[12] Kittel C. Introduction to Solid State Physics, 6th Ed. John Wiley & Sons, New York, NY, 1986.

[13] Mittal A. Monte-Carlo Study of Phonon Heat Conduction in Silicon Thin Films. Ohio State

University, 2009.

[14] Smirnov S. Physical Modeling of Electron Transport in Strained Silicon and Silicon-Germanium.

Vienna University of Technology, 2003.

[15] Ioffe Institute. Physical Properties of Germanium (Ge). http://www.ioffe.rssi.ru/SVA/

NSM/Semicond/Ge/index.html.

REFERENCES 20

[16] Ioffe Institute. Physical Properties of Silicon (Si). http://www.ioffe.rssi.ru/SVA/NSM/

Semicond/Si/index.html.

A PROPERTIES OF BULK GERMANIUM 21

Appendix

A Properties of Bulk Germanium

Lattice constant (A) a0 5.657

Longitudinal Mass (m0) aml 1.59

Transverse Mass (m0) amt 0.0815

Non-parabolicity α 0.3

Density (kg/m3) ρ 5327

Sound Velocity (m/s) vs 5.4 · 103

Dielectric Constant ε 16.2

Band Gap (eV) 0.66

Acoustic Intravalley Valley Ξac (eV)

Γ 5.0

L 11.0

X 9.0

Intervalley Valleys Ξi→j (eV) Phonon Energy (eV)

Γ→ L 2.8 · 1010 0.0276

Γ→ X 1.0 · 1011 0.0276

L→ L 3.0 · 1010 0.0275

X → X 9.46 · 1010 0.0370

Table 1: Material properties of a bulk germanium crystal [2][14][15]

Figure 13: Band structure of germanium [15]

B CODE REPOSITORY 22

B Code Repository

The code that was used to generate these results is too large to be contained in any meaningful

fashion within this work. Both the bulk germanium Monte Carlo and the device simulator have

been uploaded to distinct BitBucket repositories. They can be accessed at the following links:

Bulk Germanium Monte Carlo (C++): https://bitbucket.org/appliedproject/bulk-mc

Particle-Based Device Simulator (FORTRAN 95): https://bitbucket.org/daniellivingston/

particle-based-device-simulator