Transport w 2-d oraz 3-d strukturach pprzewodnikowych wasnoci elektryczne

Transport w 2D & 1D

strukturach półprzewodnikowych

Jacek A. Majewski

Institut Fizyki Teoretycznej &

Interdyscyplinarne Centrum Modelowania Materiałów Universytet Warszawski ICM UW, Warszawa, 30 kwietnia, 2009 www.fuw.edu.pl

Rewolucja informatyczna

Wykładniczy wzrost

technologii informatycznych

Przetwarzana, zapamiętywana, i transmitowana gęstość informacji (liczba bitów na jednostkę pow.) rośniewykładniczo.

Ten szybki wzrost powoduje zmiany cywilizacyjne

wpływa silnie na światową gospodarkę

określa sposób w jaki pracujemy, bawimy się, etc.

Moore’s Law

Wykładniczy postęp w integracji

Liczba

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MOORE’a

Rozwój technologii obwodów scalonych

Pierwsze obwody scalone Texas Instruments

Procesor Pentium

Wielkość chipu prawie bez zmian

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Przeciętna cena tranzystora

Technologia CMOS osiąga skalę nano

Nanotechnologia krzemowa

Obecnie produkowany procesor Intel’a w technologii 90 nm

Influenza virus

Dłudość bramki ~ 2000 x mniejsza od średnicy ludzkiego włosa Grubość tlenku po bramką ~1.2 nm(5 atomowych warstw Si)

Nanotechnologia

ma już obecnie

duże znaczenie w dziedzinie

elektroniki, gdzie utrzymuje się trend

w kierunku

miniaturyzacji

1965 - 1970, Integrated Circuits based on

bipolar transistors

Since 1980, Integrated Circuits based on

CMOS technology dominate,

Field Effect Transistor (FET)

Development of Integrated Circuits Technology

CMOS Technology

CMOS = Complementary Metal Oxide Semiconductor

Metal Oxide Semiconductor p-Silicon Depletion

V

_G

= 0

x

E_c E_v

E

_F

p-Si

M O

S

Q

_M

x

SiO₂ (Semiconductor = Silicon)

CMOS Technology – Biased Gate

Metal Oxide Semiconductor p-Silicon Depletion

V

_G

> 0

x

E_c E_v

E

F

p-Si

Q

_M

_x

Semiconductor p-Silicon Depletion Metal Oxide Inversion

V

_G

>> 0

E_c E_v

E

F

p-Si

Q

_M 2DEG

Q

_n S Gate D

V

_G

≈

0

S Gate D

V

_G

>> 0

Current does not flow Current flows

State “0” State “1”

CMOS Technology

Logical circuits (processors) Memories

cutoff mode active mode

Operation of n-MOS FET in enhancement mode

(V

_G

>

V

_th

)

GS Gate Source Drain V_DS V I_D I_D V_G1 V_DS V_G2 V_G3 V_G4 g_m Current vs. voltage g_m

δ

ID

δ

V_{GS V} DS =

δ

V_DS

δ

I_D g_d V_GS = Small-signal transconductance Channel conductance

Transit time (frequency)

Basic characteristics of a

Field Effect Transistor (FET)

Current characteristic

CMOS Technology

n-MOS

&

p-MOS

electrons in the channel Complementary holes in the channel n+ n+ p+ p+ n well p-type body

Complementary MOS technology employs MOS transistors of both polarities CMOS devices are more difficult to fabricate than NMOS, but many more powerful circuits are possible with CMOS configuration

n-MOS transistor p-MOS

transistor

I

_D

V

_DS

Current characteristics of

p-MOS

&

n-MOS

Enhancement mode n-MOS transistors have a positive threshold voltage

V

_th

> 0

The operational behavior of a p-MOS enhancement transistor

represents the complement of the operation of the n-MOS. Compared to the n-MOS transistor, basically all voltage quantities are reversed.

CMOS circuits - Inverter

The complementary electrical character characteristics of n-MOS and p-MOS transistors make them well suited for the design of logical gates used in many digital designs.

G

G

S

S

D

D

V

_DD

V

_IN

V

_OUT

n-MOS p-MOS

V_IN= 0 p-MOS = ON n- MOS = OFF V_OUT= V_DD

V_IN= V_DD p-MOS = OFF n- MOS = ON V_OUT= 0

IN ₁ OUT = IN

No current flow in steady state



low power dissipation in CMOS circuits

CMOS Technology –

“Computational state” and material system

Abakus (~3000BC)

_{Processor based on CMOS}

S Gate D

Current

The material system: CMOS transistors

The computational state: position of the beads

The material system: A set of stings in a frame

The computational state: electron current flow

S Gate D

CMOS Technology – Important Parameter

Important parameter – gate length

L

_G

determines stage of integration

switching speed

Nowadays it is necessary to remove ~1000 electrons from the channel

L

_G

2D-Electron Gas

Electron mobility Bulk GaAs at T=300 K – 9 000 cm2V-1s-1 2DEG at T=300 K in HEMT – 10 000-12 000 cm2V-1s-1

Electron mobility in 2DEG at low temperatures ( < 1 K) can reach 20 000 000 cm2V-1s-1 !!! E_F z M AlGaAs GaAs 2DEG _E c E_F z AlGaAs GaAs 2DEG E_c Gated AlGaAs/GaAs heterostructure AlGaAS/GaAs heterostructure

Classical in-plane transport

Ballistic, Coherent Quantum Transport

Sour ce Gate Drai n GaAs - substrate AlGaAs 2DEG

Mesoscopic devices – macroscopic with quantum effects present

Semiclassical Transport



The motion of an electron can be described

semi-classically by its position (r) and momentum (η

η

η

ηk) as

accurately as necessary, without violating the uncertainty

principle



provided

:



applied electric and/or magnetic fields vary slowly

over the dimension of the electron wave packet:

typical length >> ∆r =1/∆k >> a (lattice constant)



the phase information of the electron wavefunction

is not needed

Semiclassical Transport

1

( )

1

E

=

∇





=

_

+

×

_





d r

d t

d

q

d t

c

ε

k

k

k

v

H

ℏ

ℏ

The semiclassical electron motion, in the absence of collisions, is described by the following equations:

Band structure of semiconductors

Silicon GaAs

Brillouin Zone of zinc-blende semiconductors

Band structure of semiconductors

simplified models

Parabolic bands Spherical * 2 ) ( 2 2 m k k Ep η = Ellipsoidal * 2 2 * 2 2 * 2 2 2 2 2 ) ( z z y y x x p m k m k m k E k =η +η +η Warped bands

(

)

[

,θ,ϕ

]

2

[

1

(

θ,ϕ

)

]

g ak k Ek = µ Γ Γ Γ Γ L X1 X2

The distribution function

3 ( , 2 (2 ) , ) dN f t d d

π

= k r Ω Ωk r

represents the number of particles in the phase space volume dr dkaround point (r, k).

For a homogeneous solid in equilibrium, f(r,k,t)is equal to

1 0

( )

{exp[( ( )

) /

] 1}

−

=

−

_B

+

f k

ε

k

µ

k T

Fermi-Dirac distribution function. n

The central quantity in the semiclassical transport is the

distribution function f (r,k,t),

which gives the occupation probability of a “state” characterized by band index n, k-vector k, and space vector r.

The distribution function

( ) Ω Ω_r k d
d
t ( ) Ω Ω_r + k d
d
t dt A group of electrons moves rigidly through the phase space

under the influence of external fields.

Individual electrons are scattered intoor outof the group.

The total change in the distribution function f due to the electron scattering (“lattice collisions”)

To calculate the distribution function

in a given external fields, we examine its behavior

with time

( , , )

f r k t

The Boltzmann Equation

( , , ) ∂ ( , , ) = ∂ coll df r k t f r k t dt t



1 ( ) k r= ∇
εεεε k

ɺ ℏ 1 q k E v H c   = _ + × _  
_ɺ


ℏ

The evolution of the distribution function is given by the Boltzmann equation

In the steady state the local differential quotient is zero and we are left with the usual form of the Boltzmann eq.

The distribution function can be calculated for known external fields and known

collision processes

( , , )

( , , )

( , , )

r k coll

f r k t

r

f r k t

f r k t

t

∂

⋅∇

+ ∇

=

∂

_ɺ

The distribution function

Any quantity can be obtained from the knowledge of the distribution function

:

( )

( )

carrier density

current density:

mean energy :

:

( , )

( , , )

( , )

( ) ( , , )

( , )

( , )

( ) ( , , )

n t

f

t d

t

f

t d

t

n t

E

f

t d

r

r k

k

J r

v k

r k

k

r

r

k

r k

k

=

=

=

∫

∫

∫

2

2

2

2

1

3 3

π

π

E

The Boltzmann Equation - Scattering

Term

• Quantum mechanical probability (per unit time) of scattering between electronic states

k

and

k’

( , ') P k k

• Number of particles (electrons) that has been scattered during time

δt

from volume

d

3

_k

_into

_d

3

_k’

1

₃

[ ( , )

( , )]

3

(2 )

( , , )

′

Π

′

− Π

′

∫

∂

=

∂

coll

d k

k k

k k

f r k t

t

π

1

3

3

( )[1

( )] ( , )

3

(2 )

′

′

′

−

f k

f k

P k k dx kd k t

δ

π

( , ') :

k k

f k

( )[1

f k

( ')] ( , ')

P k k

Π

=

−

Scattering mechanisms

CARRIER-CARRIER Screened potential Plasmons DEFECTS Impurities Crystal defects Ionized or Neutral PHONONS Intravalley Intervalley DeformationElectrostatic Optical Acoustic Polar Piezoelectric Intervalley scattering Intravalley scattering

Fermi Golden rule

Scattering rate = transition probability for unit time

Microscopic Theory of Scattering

(

)

(

)

[

E

c

E

c

]

c

H

c

c

c

P

(k,

;

k'

'

)

=

2

π

k,

_I

k'

'

2

δ

k'

,

'

−

k,

η

Interaction matrix element

H_I= interaction Hamiltonian

k,k’ = carrier wave vector

c,c’ = crystal related coordinates (such as ion displacements)

Energy conservation

E(k,c) = energy of the system before scattering E(k’,c’) = energy of the system

after scattering

Intra-valley and Inter-valley scattering

Intra-valley Point Phonons for intra-valley scattering

, ,

X L

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

X

L

LA LA+TA LA+TA+LO+TO Inter-valley

Transition Phonons for inter-valley_scattering

L

X

X

Γ

Γ

Γ

Γ

L Γ Γ Γ Γ →→→→ L→→→→X ∆ ∆ ∆ ∆ ∆ ∆ ∆→→→→∆ X→→→→X LO + LA LO + LA (Opposite valley) (Non--opposite Valley) LO LA + TO LA LO (M_V >>>>M_III) (M_V <<<<M_III)

Phonon

Phonon

scattering:

scattering:

Deformation

Deformation

potential

potential

Optical phonons

Intravalley, spherical parabolic bands ( ) q op op op n E m E P ω ρ ω πη  µη     _{+ ±} ∆ = 2 1 2 1 2 3 2 3 ( ) v q op if op f op g n E E m E P  −∆      ± + ∆ = ω ρ ω πη 2 µη 1 2 1 2 3 2 3

Intervalley, spherical parabolic bands

gv= # equivalent valleys

Detailed knowledge of the band structure is very important !!

GaAs

GaAs

Linearization of the Boltzmann

Equation

• Splitting the distribution function into

• its equilibrium value f0 and a perturbation df (f1),

0 0 1 0 0 (1 0) 0 f f f f f φ f β f fφ ε ∂ = + = − = + − ∂ 0 0 0 (1 ) f f f f δ φ β φ ε ∂ ≡ − = − ∂ 3 0 3

[ ( )

( )]

( , )

(2 )

∂

′

′

′

= −

−

Π

∂

∫

f

d k

k

k

k k

t coll

β

φ

φ

π

• one can see that quadratic terms in df appear in the integrand of the collision term. These terms can be omitted for small perturbations, the collision term thereby being linearized.

The linearized collision term takes a particularly simple form

The Relaxation Time Approximation:

Drude’s Model of Carrier Transport

0 1 ( ) ( ) ( ) f k
= f k
+ f k
1 ( , , ) coll f f r k t t ττττ ∂ = − ∂

0 1 1 ( ) k f qE f f ττττ − ⋅∇
+ = −

The distribution function:

Furthermore, assuming a constant electric field E and a spatially uniform charge electron distribution, the Boltzmann transport equation becomes

=

J

q nE

µ

The relaxation time approximation:

* q m τ µ= Mobility:

Ohm’s law from the Boltzmann transport equation

Another way of looking at this is to consider f as a Taylor series for f0

Displaced distribution 3 3 3 3 2 1 0 ( ) ( ) ( ) ( ) * * * ( ) ( ) k k d k f k d k f k q n m m J qn qn E m d kf k d kf k =

∫

=

∫

=

∫

∫

1( ) k 0( ) f k =q Eττττ ⋅∇
f k


0 0 ( ) ( ) _k ( ) ( ) f k = f k +q Eττττ ⋅∇
f k + = f k+q Eττττ





…

Methods to solve the Boltzmann Equation

a) The Variational Method

c)

The Monte Carlo Method

The variational principle for the linearized form of the Boltzmann equation

b) Relaxation time approximation

Drift-diffusion equations

Monte Carlo procedure

The Boltzmann equation is solved by a stochastic

evolution (r,k,n) (r’,k’,n) (r’,k’’,n’) Scattering time Initial

state Free flight _{State after}

the scattering

∆t = 1 fs

F

Generation of random numbers with given

distribution: The direct technique

To sample x in (a,b) distributed according to the normalized probability distribution f(x) we do the following:

1) we generate a number

r

uniformly distributed in (0,1)

2) then, x_ris chosen according the relation

( )

=

∫

( )

=

r x r

f

x

dx

x

F

r

0 F (x ) 0 1 r xr dF dx a b but …

1) we need to solve the integral 2) we need to invert the relation

r=F(x)

f(

x

)

a b x

Generation of random numbers with given

distribution: The rejection technique

To sample x in (a,b) distributed according to the normalized probability distribution f(x) we do the following:

1) we choose a constant Γ≥f(x)

2) we generate two uniformly distributed random numbers in (0,1), r, r’

3) we define x₁=a+(b-a)r,y₁=Γr’

4) if y₁≤f(x₁) then x₁is retained as a choice of x, otherwise x₁ is rejected and

steps 2)-4) are repeated

f( x ) a b x (x1,y1) (x2,y2) accepted reject

When f(x) is strongly peaked, many numbers of pair (x,y) might be generated before a successful trial, resulting in a large

expense of CPU time

Free-Flight (τ)

During the free flight carriers do not suffer any scattering.

r and k coordinates change according semiclassical theory

k

r

F

k

∂

∂

=

−

=

E

dt

d

e

dt

d

η

1

;

Determination of the flight duration

The probability that the electron will suffer its next scattering during the time dt around t is given by

( )

[

]

⋅

∫

[ ( )]

=

− t dt t P

e

dt

t

P

dt

t

0 ' '

)

(

k

k

P

Probability of flight from 0 to t without scattering Probability to suffer a scattering in (t,t+dt)

For simulation purposes, we need to generate t according to the P(t) distribution …too difficult !!

Free flight: Self-scattering

Γ=1/τ0

real scattering

self-scattering

In order to simplify the generation of the free flight, we define a new fictitious scattering (N+1 th) such that

( ) ( ) 1 const. 0 1 1 = = Γ = =

∑

+ = τ N i i P Pk k

For a constant scattering rate, the probability that the electron will suffer its next scattering during the time dt around t is given by

0 ) ( 0 τ

τ

t e dt dt t − ⋅ = P

Now we can easily generate the free-flightτ distributed according P(t) via the direct technique

( )

r

ln

0

τ

τ

=

−

with r uniformly distributed in (0,1) WhereΓ is the P(k) max for the energy interval considered in the simulation

Choice of the scattering mechanism

impurity acustic def. polar optical intervalley self-scattering 0 Γ

After the free flight the carrier has a given energy E(k)

We look-up at the scattering table for this energy and by extracting a uniformly distributed numberrin (0,1) we chose the scattering mechanism:

Real scattering

we update the carrier state (E,k,band,valley) according to the scattering mechanism

Self-scattering

the carrier state remains unchanged (fictitious scattering)

rΓ

r

polar optical scattering

Synchronous Ensemble Monte Carlo

As soon as electron-electron interactions (also Poisson) or

explicitly time dependent phenomena occur,

synchronization of ensemble should be accounted

time 1 2 to+∆t to+2∆t to+3∆t to+4∆t to τ …

We define a simulation time step ∆∆∆∆t, after which

distribution function, potential etc. are updated

In the Ensemble Monte Carlo simulation many carriers

are simulated simultaneously

MC simulation of bulk GaAs

Γ L

X

At high electric field, electrons are scattered by phonons into satellite valleys

Hot Carriers

At high electric field, the kinetic energy of carriers is much larger than the thermal value

MC Simulation of Bulk GaAs

:

Velocity-Field Characteristics

For electric fields > 3-4 kV/cm, inter-valley transitions induce negative differential mobility.

High-field drift velocity: Theory

Electric field [kV/cm] 1 10 100 1000 6 10 7 10 D ri ft v e lo c it y [ c m /s ]

_InN

Saturation velocity is determined by dominant energy loss mechanism -> polar optical phonon energy

⇒ ⇒ ⇒

⇒ High transit time frequencies

MC simulation: PM-HEMT (I

)

In_0.25Ga_0.75As/Al_0.23Ga_0.77As:GaAs HEMT

2D simulation Electrons (ΓΓ,X,L) ΓΓ Holes (HH,LH,SO) Multigrid for Poisson Eq.

Electrons get hot at the Gate-Drain region

MC simulation: PM-HEMT (II)

In_0.25Ga_0.75As/Al_0.23Ga_0.77As:GaAs HEMT Potential profile - gate region

VD=2V High electric field at the

MC simulation: PM-HEMT (III)

Channel shrinks towards the end of the gate region Transfer of electrons to high energy valley

Transfer of electrons into surrounding barrier material

How far can we push Si CMOS?

Is it possible to keep exponential

growth?

S. E. Thompson & S. Parthasarathy, materialstoday, June 2006

Si technology industry time line:

possible time frame for new device types

What lies What lies What lies What lies beyond? beyond? beyond? beyond? Ge, III-V channel materials Carbon nanotubes Quantum Electronic Device Single Electron Transistor QED SET spintronics

Nanotechnology Eras

Ultimate electronics

Gate leakage current

S Gate D

Challenges – Gate Leakage

Gate Oxide

Thinner gate oxides produce faster transistors The limit of Gate Oxide (SiO₂) has been reached

30 nm transistor has 0.8 nm gate oxide layer

Thinner oxides leak more

Gate oxide can get so thin that it no longer acts as a good insulator

High-K Gate insulator

n+ n+

High-K Gate Dielectric

New material replaces SiO₂

Thicker physical film but the same capacitance 10 000 lower gate leakage for the same capacitance Alternative gate dielectrics to reduce gate leakage TiO₂, HfO₂, ZrO₂, Ta₂O₅

New Structures and Materials for

Nanoscale MOSFETs

Non-planar MOSFETs

Enhanced Hole mobility for Uniaxial Strained-Si

Strain in Si is introduced by Si_1-xGe_xin the source/drain.

Higher hole mobility

Why Germanium MOS Transistors?

Higher (low field)

Higher (low field)

Higher (low field)

Higher (low field) mobilities

mobilities

mobilities

mobilities

Heteroepitaxial Growth of Ge on Si

With H With H With H

With H₂₂₂₂anneal, dislocations are confined to the anneal, dislocations are confined to the anneal, dislocations are confined to the anneal, dislocations are confined to the Si/

Si/ Si/

Si/GeGeGe interface leaving defect free top Geinterface leaving defect free top interface leaving defect free top interface leaving defect free top GeGeGeGe layers.layers.layers.layers.

New Materials for Si- Technology

Moore’s Law increasingly relies on material innovations

Epitaxial Growth of Vertical Nanowires

Controlled Growth and Structures of

_{Molecular-Scale Silicon Nanowires}

(a) TEM images of 3.8-nm SiNWs

grown along the <110> direction (c) cross-sectional image (b) & (d) models based on Wulff construction

High Performance Silicon Nanowire

Field Effect Transistors

Yi Cui, et al. NANO LETTERS 3, 149 (2003)

Comparison of SiNW FET transport parameters with those for

state-of-the-art planar MOSFETs show that

“SiNWs have the potential to exceed substantially conventional devices, and thus could be ideal building blocks for future nanoelectronics.”

Nanocable device concepts

Gargini-type nanotransistor

high k gate dielectric, e.g. ZrO2

nanorod,

e.g Si/Ge or an alloy

metal gate electrode possible MOS transistor structure

P. Gargini, (Director - Technology Strategy Intel Fellow) ‘Enlightenment beyond Classical CMOS’ ISS US 2002.

Si NW FETs: Experimental Status: ‘Top Down’

Ge Nanowires Synthesized

by Low Temperature CVD

Ge Nanowire FET with High K Gate Dielectric

Key Challenge: Controlled growth Key Challenge: Controlled growthKey Challenge: Controlled growth Key Challenge: Controlled growth

Technolo_{gia półp} rzewodn_ików Supramol ekularna c hemia CZAS SKALA

10 nm?

2015 ?

Elektronika oparta o nano-struktury

półprzewodnikowe i duże molekuły

“Top down”

“Bottom up”

?

?

Czy uda się utrzymać wykładniczy

charakter wzrostu?

Nanorurki węglowe podstawą przyszłych

technologii informacyjnych ?

Delft, Tans, et al.,

Nature_{393 (1998)}

Harvard, Javel et al.,

Nano Letters_{393 (1998)}

Parametry tranzystorów lepsze od 2D FETs stosowanych w obecnych procesorach opartych o technologię CMOS

Quantum Coherent

Transport Theory

Source Drain Gate

V

V

_G

µ

₁

µ

₂ Channel

Quantum transport – the study of current flow

on an atomistic scale

Physics of open systems

Conductance from Transmission

( , ) ( , ) F L F F f E µ =f E E +V fF( ,E µR)=fF( ,E EF) Device Lead Lead Contact _Contact

µ

µ

µ

µ

L= EF+ V

L

R

µ

µ

µ

µ

R= EF

µL- electrochemical potential = chemical potential + voltage V – external bias

E_F– Fermi energy (chemical potential in the absence of bias)

The Landauer approach – very useful in describing mesoscopic transport The current through a conductor (device) is expressed in terms of probability that an electron can transmit through it

Leads are reservoirs of electrons in which energy- and momentum relaxation processes are so effective that the electron system remains in equilibrium even under a given applied voltage bias

1 ( , ) exp[ ( )] 1 F f E E µ β µ ≡ − +

The electron concentration in the leads is so high that the electrostatic potential in each lead is taken to be constant (as for the case of metal)

Electric current through the device

Conductance from Transmission

||, , k n m || || || || , 0 2 ( ) ( ( , , ) ) L L n m k c e I v T E f E k n m L > µ − =

∑∑

− || || || || , 0 2 ( ) ( ( , , ) ) R R n m k c e I v T E f E k n m L < µ − =

∑∑

− || || || || || , 0 2 ( )[ ( ( , , ) ) ( ( , , ) )] L R R L n m k c e I I I v T E f E k n m f E k n m L > µ µ = + =

∑ ∑

− − −

The number of electrons in state

||

2 ( ( , , )f E k n m −µL) /Lc

per unit length The total contribution to the electric current from the electrons entering from the left

Similarly, for the electrons entering from the right

The total electric current through the device

L

E V

F

µ

= +

R

E

F

µ

=

Quantized Resistance of a ballistic

waveguide - Experiment

Measured conductance vs gate voltage