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Vetor da rede rec íproca. Propriedades de Transporte. SEMICONDUTORES METAIS NANOESTRUTURAS. ELECTRICAL CONDUCTION. OHM’S LAW. Resistivity .

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slide1

Vetor da rede

recíproca

propriedades de transporte
Propriedades de Transporte
  • SEMICONDUTORES
  • METAIS
  • NANOESTRUTURAS
slide5

ELECTRICAL CONDUCTION

OHM’S LAW

Resistivity 

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.

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.)

slide6

Condutividade elétrica

Resistividade elétrica 

Condutividade elétrica 

Condutância G

Densidade de corrente J

Intensidade de campo elétrico

slide7

ENERGY BAND STRUCTURE IN SOLIDS

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.)

slide8

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.)

slide9

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.)

slide10

Influence of temperature

Where 0 and a are constants for each particular metal.

Influence of impurities

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

Rule-of-mixtures expression

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

slide11

CONDUCTION IN TERMS OF BANDS AND ATOMIC BONDING MODELS

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.)

slide12

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.)

slide13

Singlewall Nanotube

Bethune et al. Nature 367, 605 (1993)

slide15

Band Structure of Metallic CNTs

Armchair (6,6) CNT

J’

J

slide16

Band Structure of the Si(001) Surface

Dangling bonds of the

reconstructed Si(001)

J

c lula com 8 mol culas de h 2 o
Célula com 8 moléculas de H2O

DO-O = 2.67 Å

ALAT = 4.418921 Å

B/A = 0.980298 Å

C/A = 1.618664 Å

DO-H (H2O)= 1.00 Å

DO-H = 1.67 Å

bandas
Bandas

Gap direto = - 6.69 eV

slide20

ELECTRON MOBILITY

Mobility

Where e is called the electron mobility.

Electrical conductivity 

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.

Electric resistivity of metals

(Matthiessen’s rule)

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

slide23

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.)

slide24

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

j=-n e v

slide25

Modelo de Drude

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.

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)

slide31

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

será

Para acomodar N-elétrons:

slide32

!

1% da velocidade da luz

slide33

A energia total do estado fundamental

Como o volume do espaço-K permitido por K

slide34

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

Definindo TF (temperatura de Fermi) como

slide35

!

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