Lecture 3 – Moving Charges
Electric Conduction, Current, Resistance
Metals and Semiconductors
Experimentalphysik II in Englischer Sprache
27‐05‐11
1
Contents
1
J    E

• 3.1 Semiclassical picture of electrical conduction
–
–
–
–
Electrons in metals
Electric current (I) and current density (J)
Resistance (R), resistivity () and Ohms “law”
Temperature dependence of resistivity
metals
  when T 
semiconductors
  when T 
• 3.2 Drude model of conduction in solids
– Microscopic picture of conduction
– The scattering time () and electrical mobility (µ)
– Successes and failures
E
CB
• 3.3 Semiconductors
– A sneak preview of the energy band theory of solids
– Conduction due to electrons and holes
– Temperature dependence of  in semiconductors
Eg
SEMICONDUCTOR
2
3.1.1 Which elements are metallic ?
Semiconductors
Approximately two thirds of all elements are metallic !!
Why ?
Insulators
Look at ionization energy of elements
(Energy required to remove 1‐electron)
Outer (valence) electrons in simple metals seem to be very weakly bound to the nucleus (Li, Na, K, Rb, Cs)
3.1.2 Cohesion in metals
•
Metallic state favoured by most elements in the perodic table
– Mostly on left side of periodic table ‐ small numbers of weakly localised valence electrons
– Let´s take a look at the quantized energy levels of an atom of Sodium (Na)
repulsive
3s1
ATTRACTIVE
3s1
repulsive
2p6
+
+
repulsive
3s1
2s2
repulsive
1s2
In solid sodium the atoms are held together by metallic bonds
Delocalised 3s1 electrons hold the atoms together since the net electrostatic interaction is attractive
Na=[1s2,2s2,2p6,3s1]
Energy=‐eV(r)
They arrange in a periodic arrangement in space called a crystal (body centred cubic ‐ bcc)
3.1.3 Electrical conduction in metallic solids
Valence electrons (free to move)
‐
• In metals the valence electrons are delocalized ‐
+
+
+
• They are free to move and undergo random thermal ‐ +
motion with a very large average speed (~10 m/s) …
‐
‐
‐
(Similar to Maxwellian velocity distribution)
+
+
+
‐ +
• Average velocity is zero ..
‐
‐
‐‐
+
+
+
+

•
‐
‐
‐
‐
‐
+
+
+
+
6
1
 m 2
 mvx2 
 exp
f (vx )  

 2 k BT 
 2 k BT 
Ionic cores (fixed in lattice)
5
3.1.3 Electrical conduction in metallic solids
Valence electrons (free to move)
‐
• In metals the valence electrons are delocalized ‐
+
+
+
• They are free to move and undergo random thermal ‐ +
motion with a very large average speed (~10 m/s) …
‐
‐
‐
(Similar to Maxwellian velocity distribution)
+
+
+
‐ +
• Average velocity is zero ..
‐
‐
‐‐
+
+
+
+
• Potential difference V
‐
‐
• Generates an electric field gradient V=V/L and ‐
‐
thus an electric field E=‐V
+
+‐
+
+
6
E
Metal is equipotential
 i=dq/dt=0
Between collisions electrons are accelerated by the field (F=‐eE) and drift along the wire from the negative terminal to the positive terminal
A current i flows in the wire
Potential gradient along wire E=-V
 i≠0
i
dq
dt
Definition of electric current
6
3.1.4 A few properties of electric current
•
Current is defined as the charge flowing per unit time
– Unit =Ampere, 1A = 1C / s dq
i
dt
Formally defined as the current required to produce a force of 2x10‐7 N per meter of length between two long straight conductors separated by d=1m in a vacuum (see L4) 
•
Current is continuous
• Current is additive and scalar i
t
q   dq   idt
dq
dt
SINCE CHARGE IS A CONSERVERD QUANTITY
i0  i1  i2
0
•
The direction of currents are defined to reflect the direction in which positive charges would move (even if the current is carried by negative electrons)
7
Quiz ?
?
What is the magnitude and direction of the current i in the lower right hand wire ?
8
3.1.5 Current density (J)
•
Often we will be less interested in the total current i flowing in a particular conductor but more in the flow of charge through a cross section of the conductor at a particular point
To describe this flow we use a new vector field termed the current density J
J
The current flowing through the area element dA is given by
di  J .d A
If the current density is homogeneous then dA || J
i J A
J 
J
i
A
i   J .d A
Unit =A / m2
The current density is represented by streamlines (just like electric field lines in electrostatics)
More closely packed streamlines = higher current density
9
3.2 Current density J and drift velocity (vs)
•
Current density (J) is directly related to the average drift velocity (vd) of the charge carriers (here assumed positive)
v
v
A
I
We had that:
+ want to relate to vs
n ‐ charge density (m‐3)
L
Number of charge carriers in a length L of wire is:
 charge in a length L is:
But we know that i is defined as charge / time 
Rewriting 
i
J 
A
i
L
J 
 ne .   nev s
A
 t 
J  nev s
N  n . AL
q  eN  ne . AL
q
L
i   ne . A  
t
 t 
10
Example : Drift speed of electrons in a wire
QUESTION: What is the drift speed of the conduction electrons in a cylindrical copper wire with radius r=900µm when it has a uniform current i=17mA?
(Cu has electronic structure =[Ar.3d10.4s1 ] and assume that the current density is constant)
The main concepts to solve this problem are: (1) J=nevs, (2) J=i/A since J uniform and (3) Cu contributes only one electron per atom to the gas of conducting electrons
If the drift speed is so slow, why then does the light switch on instantly when you turn the switch ?
11
3.2.2 Resistance, resistivity and conductivity
Georg Simon Ohm (1789‐1854) (LMU !!)
Noticed experimentally that the current that flows in a material is proportional to the voltage applied across it… constant of proportionality is electrical resistance R
The resistance R depends on the geometry of the material…
I
J=I/A
A
L

A
V /L
 R.

L
i/ A
J

1


J
E
From cartoon above we have JA=i and EL=V
Scalar form of
Ohms „Law“
V
A better measure is the resistivity,  of the material, or it´s inverse the conductivity, …
E
V  iR
R
EL   JA.R
 L 
J  E

 AR 
Units of  are (m)‐1 often called the Siemen/m or Mho/m
L
A
E=V/L
Units of  are m
J  E
Vector form of Ohms law
r
â
Example : Current density and resistance
z
2cm
1cm
If  10 4  
a
J  
r


A 0.6cm thick brass washer with a 2cm inside diameter and 4cm outside diameter is cut in half across the middle
Good (zero resistance) electrical contacts were made to each cut face of the washer and a current density |J| is sent through the metal…
then calculate: (1) the total current flowing, (2) the maximum |E| at any point within the half washer
(3) The potential difference V and resistance R between the two faces
Conductivity of brass =1.5x107(m)‐1
13
r
â
z
2cm
(3) The potential difference V and resistance R between the two faces
1cm
14
L
A
L
L
R

A A
i
NON‐OHMIC conductor
OHMIC
Conductor

CIRCUIT SYMBOLS
(Ohmic
conductor)
V
15
3.2.3 Colour coding resistors
The coloured bands on a resistor denote the resistance, precision and sensitivity to temperature 16
3.2.3 Reading resistors
•First find the tolerance band, it will typically be gold ( 5%) and sometimes silver (10%).
•Starting from the other end, identify the first band ‐ write down the number associated with that color; in this case Blue is 6.
•Now 'read' the next color, here it is red so write down a '2' next to the six. (you should have '62' so far.)
•Now read the third or 'multiplier' band and write down that number of zeros.
•In this example it is two so we get '6200' or '6,200'. If the 'multiplier' band is Black (for zero) don't write any zeros down.
•If the 'multiplier' band is Gold move the decimal point one to the left. If the 'multiplier' band is Silver move the decimal point two places to the left. •If the resistor has one more band past the tolerance band it is a quality band.
17
3.2.3 Colour coding resistors
The coloured bands on a resistor denote the resistance, precision and sensitivity to temperature WHY ????
18
Temperature dependence of resistivity
(T) for metals and semiconductors
19
3.1.8 (T) Metals, Semiconductors, and Superconductors
i
T
For all solids  varies with T
OHMIC
Conductor
 T    T0 1   .T  T0 
(T) defined by ‐the temp coeff. of resistivity
T V
metals
  when T 
semiconductors
  when T 
superconductors
=0 for T<TC !!!
Why is  negative ?
Find out with a microscopic picture of what “causes” resistivity
20
3.2.1 Drude model of electrical conductivity
•
The first microscopic attempt to model electrical conductivity in metals was made by P. Drude (1900)
– He considered the valence electrons to be a dense classical gas ASSUMPTIONS of Drude Model
• Conduction entirely by valence ELECTRONS, positive ion cores play no role
• Electrons are confined in metal but are free to move…
• Have a random “thermal velocity”…
• Collide only with ionic cores , with an average time between collisions of …
• Between collisions travel with constant velocity when no external field exists…
• After collision, electrons have no 'memory' of previous velocity – direction randomized…
Thermal motion of one electron in the electron gas
(no net movement of charge)
3.2.2 Drude model + electric field
– Relaxation time
(average time between collisions)
Velocity
vd
t=0
<vx>=vd
t=t´
Time
E
1.
2.
Application of a voltage sets up an electric field E, results in a force F=‐eE and an acceleration ‐eE/me per electron
Electron velocity component along x increases from 0vx due to F. vx continues to increase until the electron undergoes a collision after time then vx reduces to zero on average 3.
Electrons drift with an average velocity vd ‐ drift velocity
Drude showed that the drift velocity is related to  and electric field via
eE
vd 
me
Let´s proove this !
3.2.3 Drude model of electrical conduction
•
We are going to calculate the change of momentum of a Drude electron with a momentum p moving under the influence of an external force F(t) for a time t
p(t)
p(t)
The probability that the electron undergoes a collision during time t is Pcollide ~ t/ (if >>t)
F(t)
 t 





1
P
P
1  
no collide
collide
 The probability that no collision occurs is
  
•
For these electrons that do not collide during the time t, the change of momentum p is
 p  F (t )t  O[t ]2
The average electron momentum for non‐colliding electrons is
2
O
[

t
]
~0
t
 t 
2
p (t  t )   1   p (t )  F (t )t  O[t ]
p (t  t )  p (t )  F (t )t 
p (t )
 




p (t  t )  p (t )    p    F (t )  1 p (t ) t
d p(t )
p(t )

 F (t )
dt

In the limit that t becomes infinitesimal p/t dp/dt
 Can use this expression to calculate the influence of any force (magnetic, electric..)
• If we consider the external force to be the electric field in our metallic conductor
d p(t )
dv
p(t )
 me

 e E (t )

dt
dt
F (t )  e E (t )
1
v
me
 p(t )

    eE(t )dt


This is for one electron that had an initial, randomly orientated momentum p(t)
In reality, electrons exist with a distribution of momenta in all directions
Therefore, when we want to obtain the drift velocity, we have to take the average of v
over all electrons ‐ the first term in the above integral averages to zero…

e E (t )
eE
v  vd 
dt 

me 0
me
vd 
eE 
me
The drift velocity is large when the electrons don´t make many collisions and  is long…
24
We can use this expression now to proove that metals should obey Ohms Law!
J
we have shown that  ne 2
we obtain : J  
 m
 e

 E

nev d and e E
vd 
me
OHMS LAW !!
vd e
 
E me
ne 

me
2
Drude conductivity – 
(Leitfähigkeit)
Drude mobility (µ) (Beweglichkeit)
Drift velocity per unit field
  ne
Relationship between conductivity and mobility
3.2.5 How far do electrons move between collisions?
•
We can estimate typical carrier relaxation times in the Drude model if we have the conductivity  and carrier density n…

me
9.110 31 4 107 
 2 
 8 fs
19 2
29
1.8110 1.602 10 
ne
Relaxation time in Aluminium
  vth
How far do the electrons move in this time ?  given by mean free path 
2
If the electron gas is an ideal classical gas 
me vth
3k T
 B
2
2
vth 
3k BT
me
For Al, the mean free path predicted by Drude model seems in excellent aggreement with typical lattice constants – a~0.2‐0.8nm , c.f. =0.5nm !!!!
Seems to be consistent with Drude picture of electrons colliding with ionic cores only !
3.2.6 Successes and failures of Drude´s Model
•
SUCCESSES ‐ we have seen some quite convincing evidence that Paul Drude´s simple model of electrical conduction in solids is correct
– Consistent with Ohm´s Law
– Qualitatively explains electrical resistivity due to electron scattering
– Gives fairly good values for the conductivity
•
FAILURES – unfortunately many experiments can´t be explained by Drude´s model
ne 2
We know   
me
So 

and  
3kT m e
ne 2 

3kTme
 should be similar for many metals since the atoms are packed closely together
Therefore we expect that 
a=0.42nm
n

T
0.37nm
0.32nm
0.40nm
0.49nm
27
n

T
 The electrical conductivity is certainly NOT linearly proportional to the free electron density !
The resistivity does NOT vary according to T1/2, rather T for higher temperatures and constant at low temperatures for most metals…
It also depends how pure the materials are 
We should not be too surprised that Drude´s theory does not give us a complete picture of electrical conduction in solids…
We ignored : Coulomb interactions between electrons : e‐e and e‐ions
and quantum mechanics 28
3.2.7 A look into the future !
•
In the 5th semester you will revisit conduction in solids and find that many of the equations we developed in the Drude model are actually OK
– The biggest problem is Drude´s interpretation of  – the scattering time
A quantum mechanical treatment of an electron moving in a periodic Coulomb potential in a solid was first performed by Felix Bloch in 1925
(He got the Nobel Prize for this in 1952)
He showed that an electron moving in a perfect +
crystal (periodic V(r)) propagates as a wave It never ever sees the ions and  !!!!!
+
Any “defect” that breaks the perfect symmetry of the potential causes the electron to scatter
This “defect” can be due to  An impurity atom
 A missing atom (vacancy)
 Vibrations of the crystal (phonons)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
e‐
+
+
+
+
+
+
+
+
+
2+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
29
The equations we have developed are also valid in a fully quantum picture
•
DRUDE CONDUCTIVITY
ne 2
 neµ

me
But, the relaxation time 
must be calculated using quantum mechanics
Also,  is not simply the average time between collisions, it depends on what the collision process does to the direction (momentum) of the electron •
B
A
•This then explains, e.g. why for metals:
• (T) varies linearly with temp • (T) saturates at low temperatures at (T0K)= sat …
• sat depends strongly on the purity of the materials…
It does not explain the increase of (T) for semiconductors with reducing temperature
30
3.3 Semiconductors ?
Great online guide to semiconductor physics from Britney Spears http://britneyspears.ac/lasers.htm
3.3.1 Another look ahead – energy bands in solids
•
Individual atoms have discrete “allowed” energy levels
2p
2s
1s
When two atoms approach each other, the potential around each atom V(r) is perturbed
 this modifies energy levels and each level splits into two energetically distinct levels

(r)
A
(r)
e(r)(r)
Charge distribution
 
B
r
r
1s
Eg

r
r0
N=2 atoms  2N=4 states
r
32
3.3.2 Example ‐ 2 carbon atoms
Electronic structure of carbon = [1s2] 2s2 2p2
sp3 hybridization

2p
2s
1s
2p
E
2s
1s
33
3.3.3 Atoms  molecules  crystals
conduction bands
(empty)
2p
Eg=5.5eV
2s
Bandgap
valence bands (filled)
1s
core bands (filled)
1 atom
2 atoms
N atoms
(Crystal)
2N
34
3.3.4 Properties of Semiconductors ?
Material with electrical resistivity in range =1  105 m
•
1010
1015
Polystyrene
105
Sulphur
100
C (dia)
Glass (SiO2)
10‐5
GaAs
Silicon
Germanium
Zn
Cu / Ag
10‐10
1020
 [m]
METALS
SEMICONDUCTORS
INSULATORS
• Has a number of completely filled bands and an electronic bandgap Eg<3eV
E
CB
CB
Eg
METAL
SEMICONDUCTOR
Eg
INSULATOR
• Resistivity increases with reducing temperature (negative ) 3.3.5 Semiconductors in the periodic table
Each band of allowed states can accommodate a total of 2N electrons, where N is the number of atoms in the crystal
[He] 2s2 2p2
[Ar].3d10.4s2.4p1
[Ne] 3s2 3p2
[Ar].3d10.4s2.4p3
[Ar].3d10.4s2
[Ar].3d10.4s2.4p4
• Insulators & semiconductors
 Even number of valence electrons per unit cell of crystal  Completely filled bands  insulator or semiconductor
• Metals Odd number of valence electrons per unit cell of crystal
 Partially filled bands  metallic
36
3.3.6 Electrical conductivity of semiconductors ‐ (T)
•
•
T=0  Intrinsic semiconductors insulating since no carriers in CB T≠0  Electrons thermally excited across bandgap:
– Conduction due to free electrons in conduction band and holes in valence band
T>0
T>>0
T=0
E
  ne ee  nh e h
 Eg 

ne (T )  nh (T )  C exp 
 k BT 
Eg
Intrinsic (pure) semiconductor
n(T) explains why
 when T 
For semiconductors !!
3.3.7 Sensitivity to Optical Excitation
E

Ef
Ef

n~1022 -1023 cm-3
Metal
Semiconductors
partially filled bands
completely filled bands
• Conductivity of metals ‐ not photosensitive
• Conductivity of semiconductors ‐ highly photosensitive
THIS IS WHY SEMICONDUCTORS ARE USED AS OPTICAL DETECTORS
3.3.8 Sensitivity to Impurities
•
Doping ‐ controlled incorporation of impurities in semiconductors in order to manipulate free carrier density (doped semiconductors are called extrinsic)
Si=[Ne] 3s2 3p2 , As=[Ar 3d10] 4s2 4p3 , B=[He] 2s2 2p1
E
CB
‐
‐
‐
‐
‐
Eg
‐
‐
‐
‐
‐
Donor
Acceptor
e.g. group V atom in group IV Semiconductor
extra free electrons
e.g. group III atom in group IV Semiconductor
extra free holes
3.3.9 Extrinsic semiconductor – ne/h(T)
T>>0
T=0
E
‐
Eg
‐
‐
‐
‐
n(T)
  Ec  E f 
n  N c exp 

k
T
B


  E f  Ev 
p  N v exp 

k BT


40
Summary – Basic Semiconductor Properties
•
Intermediate electrical resistivity (=1  105 cm)
•
Intrinsic (pure) semiconductors (better than 1 in 1012 purity)
– Resistivity increases exponentially with temperature since n(T) decreases
– Resistivity can be lowered by illumination E>Eg
 Eg 

ne (T )  nh (T )  C exp 
 k BT 
•
Extrinsic (doped) semiconductors (1 in ~10‐10  10‐5 impurities)
– Resistivity depends strongly on impurity concentration via n
•
Charge transport in semiconductors occurs due to negatively charged electrons and / or positively charged “missing electrons” or holes
  ne ee  nh e h
Summary of Lecture 3
•
Ohmic conductors are those for which V=i R (OHMS LAW)
– Not really a law, since many solids exist that are non‐ohmic conductors…
– The vector equivalent of Ohm´s law is J=E, where J is the current density vector field…
– Resistivity () and it´s reciprocal (conductivity – ) are material specific, resistance R is not…
– (T) depends on whether material is metallic or semiconducting…
•
The Drude model provides a good microscopic picture of the origin of electrical resistance
– Due to scattering of conducting electrons from defects in the crystal (impurities, vacancies or phonons),
– Characterised by  – the momentum relaxation time
ne 2

•
Semiconductors
me
 neµ
– Materials with intermediate conductivity and a conduction carrier density that varies exponentially with temperature and is zero at T=0
42
Next week ‐ Lecture 4
Magnetic forces on moving charges
Generating magnetic fields
Experimentalphysik II in Englischer Sprache
!!! 10‐06‐11 !!!
43