L4 ECE-ENGR 4243/6243 09222015 FJain

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Derivation of current-voltage relation in 1-D wires/nanotubes (pp. 90-102A)

Ballistic, quasi-ballistic transport—elastic and inelastic length and phase coherence (pp. 89-90)

Universal Conductivity Fluctuation (UCF) in in 1-D wires/nanotubes in the presence of quasi-ballistic transport (pp. 102B-104).

Overview

Derivation of current due to 1D subband or channel i. page 91-92

2

V2π

egI

2

sy

Show conductance quantized as e2/h, g is 2 due to spin

3

V2π

egI

2

sy

v)ne(I y

v

(1)

n = half of the number of carriers per unit length (carrier density)e = electron charge

= increase in velocity due to constriction.

fE

0

2

1*

s dE2E

m

2π

gf(E)N(E)dEn

2

1

i

*s

2

1

2

*s

21

2

1

2

*

1D )E-2(E

m

2π

g

2E

m

2π

g

πE

12m(E)ρ

yL

N(E) = 1D(E) =

Here, Ef

EdE0

E

0

2/11/2-f

]2[E and f(Ef)=1 at 0°K.

2

1

f

*s E

2

m

π

g

n (1b)

Fermi energy is expressed in terms of carrier velocity Vf

2f

*f Vm

2

1E

Substituting Eq. 1c in Eq. 1b,

(1c)

2

1

2f

**

s Vm2

1

2

m

π

gn

2π

Vmg f*

s 2(a)

Now we need an expression for the increase in velocity due to constriction or applied voltage.v

e*external voltage V = eV= 2f

*2f

* Vm2

1vVm

2

1 (2b)

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Derivation of current-voltage relation in 1-D wires/nanotubes continued

e * external voltage V = eV = v)(Vmvm2

1f

*2* (3)

v v)(Vm f* If << Vf, , eV = as we neglect the first term in Eq. 3.

vf

*Vm

V*eOr = (4)

Substituting equations (2a) and (4) into equation 1:

f*

f*

sy Vm

V*ee

2π

VmgI

2π

Veg 2s (5a)

22s

2s

yy 2e

h

eg

h

eg

2πR

I

V

(non magnetic field case where gs = 2) (5b)

Multiple sub-bands or multiple channelsThe total resistance R is expressed as:

G= ih

e 2

R

1

R

1 2i

1i x

(6)

(6)

V2π

egI

2

sy

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Ballistic, quasi-ballistic, and diffusive Transport

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L and W are the length and the width of the wire or constriction; le is the elastic mean free path between impurity scattering processes, and l is the inelastic or phase-breaking mean free path between phonon scattering events.

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Ballistic: l >> le >> L, W. Electrons can only experience the boundaries of the wire, and quantum states extend from one end to the other. Any occupied states carry current from one end to the other. When there is no applied voltage, the left and the right currents cancel each other. If a small voltage V is applied the only states present on the left and the right ends are those who have chemical potentials L and R on the left and on the right respectively. This imbalance gives a net current that is proportional to the chemical potential difference, L R = eV. For the electrons in the quantum wire, their paths are determined by scattering on the potential walls of the wire in a perfectly ballistic way. If W is small compared with the Fermi wavelength, then only one of a few channels can be occupied.

Quasi-ballistic and Universal Conductance Fluctuation (UCF) regimes: Refer to figure 1(b). In this regime, there are a few impurities in the wire, and transport is via channel but scattering introduced by the impurities mixes the modes, and increase the reflection probability of electrons entering the wire. It is also possible for electrons via multiple scattering on the walls and on a few impurities to be trapped in states that are localized on the scale of le. These states have no contact to her reservoirs, and they do not contribute to the transport. The conductance in this regime depends on the precise positions of the impurities and the potentials that define the wire, and conductance changes in the order of e2/h when the potentials or the sample is changed. Quasi-ballistic regime: l >> L >> le; UCF: l >> L >> le >> W.

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Weakly Localized Regime: l >> L >> W >> le

In this situation, multiple scattering on the impurities dominates, so wire modes no longer have meaning. Electrons are localized both longitudinally and transversely on the length scale le. The electrons no longer see the one-dimensionality of the wire. NO states exist that extend from one end of the wire to the other end. This 2D weakly localized regime has no conductivity at low temperatures.

Diffusive regime: L, W >> lElectrons diffuse through the wire, and the transport of electrons through the wire is by scattering between the localized states, and this requires inelastic scattering. Mobility is determined by the average density of impurities, but at high temperatures when inelastic length is smaller than elastic length, phonon scattering determine mobility. Refer to Figure 1(a).

L5 Universal conductance fluctuation (UCF)

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L R

Wire

+ _

VA

LY

LX

L

Z

E Eqb Ef

Ef '

Ef No bias

IA

qVA

pp. 120BFig. 1. Biased quantum wire between two reservoirs (top). The location of Fermi levels (bottom).

Q.Wire

ρ2D

Ef New

Efo

E

BarrierqV

ρ1D

Step-like density of states

E to-1/2 density of states

Fig. 2. Density of states in wire and point contact (represented by quantum well like reservoirs).When energy states extend from left hand reservoir to the right hand reservoir all across the wire,conductance is determined by quantum mechanical transmission probability of state between µR +µL wire represent a barrier between two reservoir

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Scattering of carriers:Impurity/defect/grain boundary scattering, Ionized impurity –scattering is elastic (no energy exchange and maintain phase relationship)Phonon scattering (acoustical phonons and optical phonons—quantum of sound waves)-electrons scattering with phonon is inelastic. Energy can increase or decrease. Electron wave no longer the same. It is known as phase breaking scattering. No longer ballistic.

Few Impurities

Reflection of carrier depends on the impurities

Quasi Ballistic Transport

le

10lc

W = 20nmL = 200nmLe = 40nm

L =5le

11

12

13

Ballistic or quasi-ballistic transport is wave like. Like microwaves in a waveguide.

Non-locality or global nature of transport. (page 103)

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Em

q-=velocity drift average = v

n

cn

(16)

E-=v nn

The carrier mobility μn = m

q

n

c, it depends on the scattering processes .

Chapter 2 ECE 4211

DIlatticen

1+

1+

1=

1

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(12)Without f(E) we get density of states

Quantization due to carrier confinement along the x and z-axes. Looking at the integration

)(*/2

2Ef

Ly

dk

Vnzkz

nxkx

y

Carrier density n or p

nzkz

nxkx

y

Ly

dk

V /2

2

2

22

2 y

zxy

y dk

LLL

dk

V13

dEE

m

LLL

dk

V zxy

y

2

1

2

14

/2

220

This simplifies

19

nzkz

nxkx

ydk

V 2

2

E

m

LL kxnxkznzzx2

21

Go back to Eq. (12), the density of states in a nano wire is

where E=E-Enx-Enz

Density of state (quantum wires) review L3 p85 Density of state (quantum

wells) review L3 p85

n

enz

z

e EEULh

m.

2

Density of states

N(E)

E1/2

EE2E1

N(E)

EE2E1

Density of states in 0-D (Quantum dots)The k values are discrete in all three

directions

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yk zk xk

nznynx EEEEV

)(2

Energy levels in nanowire (discrete due to nx and nz)Y-axis gives energy width

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1. Discrete value of nx=1 and then add nz= 1, 2, 3

2. discrete value of nx=2 and then add nz= 1, 2, 3

3. Add to each discrete value of nx, nz an energy width as shown in density of state plot (for one level).