 2Z 
 ( z )    2  cos

 n 
 2Za 
 ( z )    2 cos

 na 
 ( z)    2 coska
 2Z 
k 

 na 
Z  2 
k  
n a 
Vetor da rede
recíproca
Propriedades de Transporte
• SEMICONDUTORES
• METAIS
• NANOESTRUTURAS
ELECTRICAL CONDUCTION
OHM’S LAW
V  IR
Resistivity 
RA

l
ou
VA

Il
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.)
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
Resistividade elétrica 
VA

Il
Condutividade elétrica 
Condutância G
1
IL
 
 VA
Densidade de corrente J


J  
Intensidade de campo elétrico
V

l
1
G
R
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.)
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
t  0  aT
Where 0 and a are constants for
each particular metal.
Influence of impurities
i  A ci 1  ci 
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
i   V  V
Where the V’s and ’s represent
volume fractions and individual
resistivities for the respective phases.
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.)
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 the
reconstructed Si(001)
J
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
Densidade de carga - ice Ih
Bandas
Gap direto = - 6.69 eV
ELECTRON MOBILITY
Mobility

J

J
 
v
I 

Av
vd  e 
Where e is called the
electron mobility.
Electrical conductivity 
  n e e
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
I t

A l
total  t  i  d
(Matthiessen’s rule)
In which t, i, d represent the
individual thermal, impurity, and
deformation resistivity contributions,
respectively.
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.)
Lei de Ohm
V RI


I
E  j  j
A
IL
L
V  EL 
R
A
A
Se n (elétrons/volume) movem-se com velocidade :

v
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
 n e v A dt
j=-n e v
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)
t 0  0  t há colisão
v0  velocidade imediatamenteapósa colisão.
eEt
v
m
eE
eE
Na média v  
t 

m
m


como j   E
 eE 
j  n e 

 m 
n e2 

m
1
m
m
   2 ou  
 ne 
 n e2
1/ 3
definindo
 3 

rs  
 4 n 
Estudo Fundamental de um gás de elétrons
2  2
2
2  

 2  2  2  r     r 

2m  x y
z 
x  L  x 



Caixa cúbica de lado L
 
 
 
 x, y, z  L   x, y, z

Condi› es de

 x, y  L, z   x, y, z
Born  von  Karman 

 x  L, y, z   x, y, z
v v
1 i K
v
K r 
e r
V


 1

v h2 K 2
como energia  k 
2m

Note que  K

v
v
r e autoestado de p
v v
h 
v
v  r  hK  r
i r
v
v v hK
v
p  h K, v=
m


A condição de contorno (1)
e
i Kx L
e
i Ky L
e
i Kz L
1
ez  1 se z=2in, n  inteiro.
n ,n ,n int eiros
2 n y
2 n x
2 n z
=
Kx
,K y 
,K z 
L
L
L
v
N de valores primitivos de K=
0
x
y
z

2 L

3

3
onde  volume do espao K, 2 L volume de cada K

2 L
3
V

8 3
V
N de K por unidade de volume= 3
8
0
O raio da esfera de estados ocupados será KF (modelo de Fermi)
v
4
Dentro da esfera teremos o N de K permitidos, (   K 3F V),
3
0
será
4 K 3F V
K 3F
 2V
3
3 8
6
Para acomodar N-elétrons:
K 3F
K 3F
N2
V 2V
6
3
n 
K
3
F
2
3
 3 
definindo rs  
 4 n 
1/ 3


1.92
4


1/ 3
9
KF
rs
rs
3.63 1
h2
ou K F 
 onde a 0 
rs a 0
m e2
hK F
4.2
vF 

 108 (cm / s)
m
rs a 0
h2 K 2F
50.1
F 

eV
2m
rs a 0




2  rs  6 p/ metais
1.5 F  15eV
!
1% da velocidade
da luz
A energia total do estado fundamental
E2

K  KF
h2 2
K
2m
Como o volume do espaço-K permitido por K
8 3
K 
V


v
v
V
 F K  83  F K K
K
K
v
v
1
dK v
lim  F K   3 F K
r V
8
K


2
5
2
2
v
h
K
E
1
h K
1
F
  3
dK
 2
V 4 K K F
2m
 10m
K 3F
N
como
 2
V 3
A energia por elétron, E/N, no estado fundamental deve ser dividida
por N/V,
2 2
E
3 h KF 3

 F
N 10 m
5
Definindo TF (temperatura de Fermi) como
F
TF 
KB
E 3
58.2
4
 K BTF 

10
K
2
N 5
rs a 0


!
Note que um gás de elétrons clássico (Drude) a energia é (3/2)KBT,
que é zero para T=0