⎟⎠⎞

⎜⎝⎛+=

n

Zz

πβαε

2cos2)(

⎟⎠⎞

⎜⎝⎛+=

na

Zaz

πβαε

2cos2)(

( )kaz cos2)( βαε +=

⎟⎠⎞

⎜⎝⎛=

na

Zk

π2⎟⎠⎞

⎜⎝⎛=

an

Zk

π2

Vetor da rederecíproca

Propriedades de Transporte

• SEMICONDUTORES

• METAIS

• NANOESTRUTURAS

FIGURE 19.1 Schematic representation of the apparatus used to measure electrical resistivity. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

ELECTRICAL CONDUCTION

OHM’S LAW

RIV =

Resistivity

l

RA=

lI

AV=

ou

Where R is the resistance of the material thought which the current is passing, l is the distance between the two points at which the voltage is measured, and A is the cross-section area perpendicular to the direction of the current.

Condutividade elétrica

Condutividade elétrica

VA

IL==

1

εrr

=J

l

V=ε

Densidade de corrente J

Intensidade de campo elétrico

Il

VA=

Resistividade elétrica Condutância G

RG

1=

FIGURE 19.2 Schematic plot of electron energy versus interatomic separation for an aggregate of 12 atoms (N=12). Upon close approach, each of the 1s and 2s atomic states splits to form an electron energy band consisting of 12 states. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

ENERGY BAND STRUCTURE IN SOLIDS

FIGURE 19.3 (a) The conventional representation of the electron energy band structure for a solid material at the equilibrium interatomic separation. (b) Electron energy versus interatomic separation for an aggregate of atoms, illustrating how the energy band structure at the equilibrium separation in (a) is generated. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

FIGURE 19.4 The various possible electron band structure in solids at 0 K. (a) The electron band structure found in metals such as copper, in which there are available electron states above and adjacent to filled states, in the same band. (b) The electron band structure of metals such as magnesium, wherein there is an overlap of the filled valence band with an empty conduction band. (c) The electron band structure characteristic of insulators; the filled valence band is separated from the empty conduction band by a relatively large band gap (>2 eV). (d) The electron band structure found in the semiconductors, which is the same as for insulators except that the band gap is relatively narrow (<2 eV). (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

Influence of temperature

aT0t +=

( )iii c1cA −=ρ

ββαα += VVi

Influence of impurities

Rule-of-mixtures expression

Where 0 and a are constants for each particular metal.

Where A is a composition-independent constants that is a function of both the impurity and host metals, and ci is the impurity concentration.

Where the V’s and ’s represent volume fractions and individual resistivities for the respective phases.

FIGURE 19.5 For a metal, occupancy of electron states (a) before and (b) after an electron excitation. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

CONDUCTION IN TERMS OF BANDS AND ATOMIC BONDING MODELS

FIGURE 19.6 For an insulator or semiconductor, occupancy of electron states (a) before and (b) after an electron excitation from the valence band into the conduction band, in which both a free electron and a hole are generated. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

Singlewall Nanotube

Bethune et al. Nature 367, 605 (1993)

Band Structure of Metallic CNTs

Armchair (6,6) CNT

J

J’

Band Structure of the Si(001) Surface

Dangling bonds of thereconstructed Si(001)

J

Célula com 8 moléculas de H2O

DO-O = 2.67 ÅALAT = 4.418921 ÅB/A = 0.980298 ÅC/A = 1.618664 Å

DO-H (H2O)= 1.00 ÅDO-H = 1.67 Å

Densidade de carga - ice Ih

BandasGap direto = - 6.69 eV

ELECTRON MOBILITY

Mobility

εμ= edv

een μ=

dittotal ++=

Electrical conductivity

Electric resistivity of metals

Where μe is called the electron mobility.

Where n is the number of free or conducting electrons per unit volume, and |e| is the absolute magnitude of the electrical charge on an electron.

(Matthiessen’s rule)(Matthiessen’s rule)

In which t, i, d represent the individual thermal, impurity, and deformation resistivity contributions, respectively.

ε J

=

μv

J=

vA

I μ =

l

t

A

I μ =

FIGURE 19.5 The electrical resistivity versus temperature for copper and three copper-nickel alloys, one of which has been deformed. Thermal, impurity, and deformation contributions to the resistivity are indicated at –100 0C. (William D. Callister, JR. Materials Science and Engineering an Introduction, John Wiley & Sons, Inc.)

IRV =

A

IjjE =→=

vv

A

LR

A

LILEV

=⇒

==

vr

Lei de Ohm

Se n (elétrons/volume) movem-se com velocidade :

Em um tempo dt os elétrons vão caminhar uma distância: vdt na direção de v.

n(vdt)A irão atravessar uma área A à direção do fluxo.

Como cada elétron carrega uma carga –e a carga que atravessa a área A em um tempo dt será dtAven−

j=-n e v

1. Na ausência do campo externo cada elétron move-se uniformemente em linha reta, seguindo as leis do movimento de Newton. Desprezar a interação elétron-elétron é conhecida como aproximação de elétrons independentes e desprezar a interação elétron-núcleo é conhecida como aproximação de elétrons livres.

Modelo de Drude

2. Colisões no modelo de Drude são eventos instantâneos que altera a velocidade do elétron instantaneamente.

3. O elétron sofre uma colisão com uma probabilidade por unidade de tempo 1/. ( é o tempo de relaxação)

colisão há t0t0 →=

colisão. a após nteimediatame e velocidadv0 →

m

tEev −=

−=−=m

Eet

m

Eev média Na

⎟⎠

⎞⎜⎝

⎛ −−=m

Eeenj

Ej comovv

=

m

en 2 =

3/1

s 4

3r ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ndefinindo

π

22 en

mou

en

m1

=

=

=

Estudo Fundamental de um gás de elétrons

( ) ( )rrzyxm

vvhψεψ =⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

+∂∂

− 2

2

2

2

2

22

2

( ) ( )LxLx ψ=+ψ Caixa cúbica de lado L

Condições de

Born −von−Karman

ψ x, y, z+ L( ) =ψ x, y, z( )

ψ x, y+ L, z( ) =ψ x, y, z( )

ψ x+ L, y, z( ) =ψ x, y, z( )

⎧

⎨⎪⎪

⎩⎪⎪

1( )

ψK

vr( ) =

1

Vei

vK • vr

como energia ε

vk( ) =

h2 K 2

2m

Note que ψK

vr( ) ′e autoestado de vp

h

i

∂∂v

rψ v

r( ) =hvK ψ v

r( )

vp =h

vK , vv=

hvK

m

A condição de contorno (1)

ei Kx L =e

i K y L =ei K z L =1

ez =1 se z=2π in, n→ inteiro.

K

x

2πnx

L,K y =

2πny

L,K z =

2πnz

L nx,ny,nz{ } inteiros

N0 de valores primitivos de vK=

Ω

2πL( )

3

onde Ω volume do espaço K, 2π L( )

3volume de cada K

Ω

2π L( )3 =

ΩV8π3

N 0 de K por unidade de volume=

V

8π3

=

Dentro da esfera teremos o N 0 de

vK permitidos, (Ω=

43πK F

3V),

4πK F

3

3V8π3 =

K F3

6π2 V

N =2

K F3

6V =

K F3

3π2 V

∴n=

K F3

3π2 definindo r

s=

34πn

⎛

⎝⎜⎞

⎠⎟

1/3

O raio da esfera de estados ocupados será KF (modelo de Fermi)

Para acomodar N-elétrons:

será

K

F=

9π4( )

1/3

rs=1.92rs

ou K

F=

3.63rs a0

Å−1 onde a0 =h2

me2

v

F=

hK F

m=

4.2rs a0

×108(cm/ s)

∈F=

h2K F2

2m=

50.1rs a0( )

eV( )

2 < rs < 6 p/ metais

1.5 <∈F<15eV

1% da velocidade da luz!!

E =2

h2

2mK 2

K <K F

∑

ΔK =

8π3

V

F

vK( ) =

V8π3 F

vK( )ΔK

K∑

K∑

limr→ ∞

1V

FvK( ) =

dvK

8π3 FvK( )∫

K∑

A energia total do estado fundamental

Como o volume do espaço-K permitido por K

∴

EV

=14π3 d

vK

h2 K 2

2m=

1π2

h2 K F5

10mK <K F∫

como

N

V=

K F3

3π2

E

N=

310

h2K F2

m=35∈F

T

F=

∈F

K B

A energia por elétron, E/N, no estado fundamental deve ser dividida por N/V,

Definindo TF (temperatura de Fermi) como

E

N=35

K BTF =58.2

rs a0( )2 ×10

4K

Note que um gás de elétrons clássico (Drude) a energia é (3/2)KBT, que é zero para T=0

!!