electronic transport

7/25/2019 Electronic Transport

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ELECTRONIC TRANSPORT at the NANOSCALE

This chapter deals with transport through mesoscopic objects (few nm in size) where

electrons/holes propagate mostly ballistically.

I

V

Classical regime Quantum regime

Javier Rodrguez Viejo Fsica en la nanoescala Curso 2014/2015


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)EV(mk

02

2


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Field Effect Transistor



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L

If lL scattering events rareballistic regimetransport is well described by

QM mechanical fluxes and transmission

Nature of Transport



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Review. CLASSICAL approach: DRUDE model

Drude constructed his theory of electrical conductivity by considering the metal as a gas

of electrons and applying kinetic theory of gases (KTG).

In KTG molecules are considered spheres which move into straigth lines until they collide:

Basic assumptions:

1.- Electrons move freely between collisions, forming an ideal gas

2.- Collisions, with ions, are instantaneous and randomizing3.- Parameter is mean time between collisions

4.- Electrons in thermal equilibrium with rest of material

Total number of molecules encountered in

swept collision volume = npD2L

Number Density of Molecules = n

Average distance between

collisions, mc= L/(#of collisions)


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Main idea: scattering of electrons by positively charged nuclei

Ohms Law:

323 /10 cmnelec elec

elec

n

r 1

3

4 3

p mrelec10101

jEIRV

or /,/j, ALRAIELV

FREE e-in METALS: Drude Model (~ 1900)

Relate Ohms Law to atomic scale model: vnej

From Newtons equations of motion:m

Enej

m

Eevav

tt

2

so

Then, material conductivity given by:

m

net

21

Since ~106cm for typical metals, tau~10-14or 10-15s. Then mean free path l=v0t,

where v0is average speed, given by equilibrium thermodynamic distribution of electrons:

mlcm/svTkmv B107

0232

021

101or10so


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BZst

kd)k(f)k(vq

fvqj1

342

p

CONDUCTANCE IN 3D SOLIDS DIFFUSIVE REGIME

SEMICLASSICAL APPROACH --- Boltzmann Transport Equation

f

)k(ve

)k()(

kdf

)k(ve

)k(

j

n

x y zk k k

p

1

22

1

23

3

Which is the appropriate distribution function

to describe transport?

Boltzmann Transport Equation is an attempt to determine howfchanges with time



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CONDUCTANCE THROUGH SOLIDS BALLISTIC REGIME

We start with systems where phase coherence is preserved, but the Landauer concept

of conductance viewed as transmission can be generalized to regions with

considerable scattering.

1 2

Two-terminal conductance

1 2

Four-terminal conductance

A B

Reservoir 1 Lead A Lead B Reservoir 2

Sample

IT

R

Ideal conductors (no scattering)

Randomize the phase of the injected

or absorbed electrons through inelastic

processes. No phase relation


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L R

C

VD

L

RCHANNEL

DRL qV

VDimposes 2 different Fermi functions and every contact tries to drive the channel

to equilibrium

)E(fe

)E(f);E(fe

)E(f RkT/ERLkT/EL RL

00 1

1

1

1

From the point of view of the distribution functions



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CONDUCTANCE in 1D

We can write a conceptually similar equation but including the distribution functions(probability that each state is occupied) and the transmission coefficient (probability

that an incident electron passes through the sample and contributes to the current).

Current (1D) due to electrons impinging in the sample (barrier) from the left

Classically (3D) j=nqv

0 2

2p

dk

)k(T)k(v),k(feI LL

k

LL )k(T)k(v),k(feI 2

Number density of states

Converting to an integral over k (1D)

Integration is carried out over positive k values since only electrons impinging to the barrier

from the left are considered


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111

U)k(T)k(v),k(f

dE

dkdE

eI

p

Integrating over energy by

changing the variable of integration

dEv

dEdE

dkdk

1

The current due to electrons coming from the right is similar

222

U)k(T)k(v),k(f

dE

dkdE

eI

p

Therefore I =I1+I2

1

1

21

21

U

U

)E(TffdEe

)E(TvffdEdkdEeI

p

p

1

2



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Simplifying assumptions

Large bias: All incoming states from the rigth may be below U1and do not

contribute to the current.

At low Tthe electrons are highly degenerate and the distribution function is 0 or 1.

Only electrons with energies between 1and 2can pass the barrier.

If the bias isvery small (linear response regime) then the Fermi functions

can be expanded to lowest order in a Taylor series.

11

U)E(TfdEeI

p

1

2

p)E(TdE

eI

E

feV

feVff

21eV;eV2

1

2

121

Fermi level at equilibrium

1

22

U

dE)E(T

E

f

h

VeI


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At very low temperatures,fis much sharper than any features in T(E) and

)E(E

f

)(Th

e)(T

h

VeI 21

2 22 Eq. (1)

In a two-terminal conductance measurement, the voltage and current are measuredthrough the same set of leads. The voltage measured is eV=1-2which using eq. (1) yields

T

h

e

V

IG

22

Conductance is Transmission

1 2

Two-terminal conductanceSingle channel

Landauer formula



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h

eG

2

0

2

Quantum

of

Conductance

Transmission probability equal 1



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MULTICHANNEL

m

mTh

eG

22

x=0 a

V0

12

3

4

k

n=1

n=2

n=3

n=4

F

Summation over all modes or subbands that

contribute to conductance

Conductance only depends on fundamental constants and it is the same for all modes

In this case

22 2

h

eG

2DEG

current


i l d


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The actual voltage difference accross the scattering structure (sample) is eV=A-Bwhich isless than the voltage between the reservoirs, 1-2. The difference represents a contact

potential drop.

If we consider that the voltage drop is measured at the leads (four-terminal conductance) we

need to relate the potential difference A-Bin terms of the current flowing through the

structure (eq. (1)).

We can write the 1D density on the left lead as

)E(TffTdkdk)E(fn AA 21211

pp

fArepresents the near-to-equilibrium distribution function

in the left lead, characterized by a Fermi energy A

Average density in terms of the injectedcarriers from the left and right reservoirs

into the left lead.

Current conservation1+R=2-T

In the right lead

)E(TffTdkdk)E(fn BB 122

11

pp

1 2

Four-terminal conductance

A B

In four-terminal conductance


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In the low-T limit

TdE

dk

dE)T(dE

dk

dEdE

dk

dE

A

B

1

2

1

2 22

Neglecting the energy dependence of T and of the inverse velocity,

the above eq. can be integrated

)()T(BA 211 Eq. 2

R

T

h

eG);(

T

T

h

eI BA

22

1

2 Single channel

Landauer formula

fA1, f11

e-from right do nor contribute

Now, in addition to the potential drop across the scattering structure we have a contact

potential drop across the ideal leads due to charge build-up characterized by a contact

resistance

k.e

hR

c

9122 2


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QUANTUM POINT CONTACTS --- T=1

x direction


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Vh

ej

2

Number of channels

that contribute p

Wki Fmax

maxih

eG

22

More info: Read Van Wees et al. PRL 60, 848 (1988)

Quantized Conductance of Point Contacts

in a Two-Dimensional Electron Gas


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CONDUCTANCE THROUGH ROWS OF GOLD ATOMS

Read Ohnishi et al. Nature 395 780 (1988)

Quantized conductance through individual rows of suspended gold atoms


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TRANSPORT IN CARBON NANOTUBESRead Electron transport in very clean, as-grown suspended carbon nanotubes

Cao et al. Nature Materials 2005


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TRANSMISSION --- 0Vochanging k2by ik2. T=0; R=1


l b


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Square Potential barrier

)akexp(V

Eaksinh

)EV(E

VT 2

0

1

2

2

0

2

0 216

41

22

xxx eeexsinh

)akexp(T 22

)EV(mk

02

2

Electron tunnelling

STM li i STM


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STM tunneling current in an STM

Binnig

and

RorherAPL 1982

distance

sample

tip

A

Current

Changing the Voltage

)exp( 21 zdcUcI


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Conductance in a MIM


D bl P t ti l B i R t di d


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Double Potential Barrier Resonant diode

224

1 RL

RL

RL

RLpk

TT

TT

RR

TTTT

TL,TRsmall

20

12

2

2

21

2

11

)//(

T)EE(T)E(T

pkpk

pk

T(E)

E (eV)


Tunnelling in Heterostructures


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Tunnelling in Heterostructures

Double well

KRONNIG PENNEY MODEL opcional


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KRONNIG-PENNEY MODEL

)x(ue)x( ikx

periodic function

)ax(u)x(u Region I V(x)=0

Region II V(x)=Vo

baxfor)x(uk)x(udx

dik)x(u

dx

d

)x(ue)x(ue

dx

diH

III

I

ikx

I

ikx

002 22

2

2

2

2

mE2

2

p

axbafor)x(uk)x(udx

dik)x(u

dx

d

)x(ue)x(ueVdx

diH

IIIIII

II

ikx

II

ikx

02 22

2

2

02

2

)EV(m 022

p

x=0 a

a bV(x)

x

opcional

opcional


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The solutions are fo the form:

axbaforexsinhDxcoshC)x(u

baxforexsinBxcosA)x(uikx

II

ikx

I

0

Both the wavefunction and the derivative should be continuous at x=0 and x=a-b

CA)(u)(u III 00

ikb)ba(ik ebsinhDbcoshCe)ba(sinB)ba(cosA

The first derivatives of uIand uIIare

ikxikx

II

ikxikx

I

exsinhDxcoshCikexcoshDxsinhC)x(udx

d

exsinBxcosAikexcosBxsinA)x(udx

d

And imposing equality at x=0 and x=a-b

ikCDikAB)(udx

d)x(u

dx

d

x

II

x

I

00

0

ika

ebcoshDbsinhC)ba(cosB)ba(sinA

opcional

opcional


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000

0101

ikaika

ikaika

ebcoshebsin)bacos()ba(sin

ebsinhebcosh)ba(sin)ba(cos

)ba(cosbcosh)ba(sinbsinhkacos

2

22

For a delta function b0 and V0 with Vob ~cte.

As b0, sinhbb

2

0;2

cossincos

bamVP

m

aaaPka

Algebra

p



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Multiple quantum wells: Kronigg-Penney model

For a delta functionb0 and V0 with Vob ~cte.

As b0, sinhbb

2

02

bamVP;

m

acosaasinPkacos

Conductance in 2D systems


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Conductance in 2D systems

Letsassume a kind of a 2DEG where the states (wave functions) and the energy can

be written as

m

k

m

kU)k(

)k,k(k);z(ue)z,r(

zL

yxkz

rki

k,k z

22

2222

The current density from left to right

02

2

222 )k(T)k(v),k(f

dk

)(

kdeJ zzL

pp

Depends only on kz

m

kv zz

Depends on the

total energy

0

2222

2

2

2222

2 LLz

zz ,m

k

m

kUf

)(

kd)k(T

m

kdkeJ

z

pp

Density of states of 2DEG in the plane x,y

with band minima at UL+h2

kz2

/2m

22


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0

22

222 m

kUn)k(T

m

kdkeJ LLDz

zz p

m

kUE

dEv

dEdE

dkdk

L

z

2

1

22

using dE)E(TEne

JLU

LD

2

and the total

current density

dE)E(TEnEne

JL

U RDLD

22

Simplifying assumptions

Large bias: Right e-do not contribute

Low T: Fermi functionstep function

Small bias:

dE)E(TEm

h

eJ

L

LU L

p 2

p

pU

UdE)E(T

m

h

edE)E(T),E(f

m

h

eG

L2

2

2

2

p

22

mn D

electronic transport

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