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Vetor da rede recíproca
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Propriedades de Transporte

• SEMICONDUTORES
• METAIS
• NANOESTRUTURAS

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ELECTRICAL CONDUCTION
OHMS 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.)
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Condutividade elétrica
Condutividade elétrica ?
Resistividade elétrica ?
Condutância G
Densidade de corrente J
Intensidade de campo elétrico
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ENERGY BAND STRUCTURE IN SOLIDS
FIGURE 19.2 Schematic plot of electron energy
versus interatomic separation for an aggregate of
12 atoms (N12). 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.)
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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.)
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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 (gt2 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 (lt2 eV). (William D. Callister, JR.
Materials Science and Engineering an
Introduction, John Wiley Sons, Inc.)
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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 Vs and ?s represent volume fractions
and individual resistivities for the respective
phases.
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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.)
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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.)
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Band Structure of Metallic CNTs
Armchair (6,6) CNT
J
J
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Band Structure of the Si(001) Surface
Dangling bonds of the reconstructed Si(001)
J
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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
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Densidade de carga - ice Ih
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Bandas
Gap direto - 6.69 eV
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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
(Matthiessens rule)
In which ?t, ?i, ?d represent the individual
thermal, impurity, and deformation resistivity
contributions, respectively.
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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.)
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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
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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)
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Estudo Fundamental de um gás de elétrons
Caixa cúbica de lado L
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A condição de contorno (1)
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O raio da esfera de estados ocupados será KF
(modelo de Fermi)
será
Para acomodar N-elétrons
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!
1 da velocidade da luz
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A energia total do estado fundamental
Como o volume do espaço-K permitido por K
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A energia por elétron, E/N, no estado fundamental
deve ser dividida por N/V,
Definindo TF (temperatura de Fermi) como
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!
Note que um gás de elétrons clássico (Drude) a
energia é (3/2)KBT, que é zero para T0