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Electric conduction title
Autumn 2006
Terje. G. Finstad
Electric conduction- outlineIntroduction What is electronic conduction It’s nature, classifications overview importance -usesClassification of materials according to conductivity Conductors/Metals Semiconductors Insulators[Classifications according to chemical bonding] Metalic/Metals - Nonmetallic - covalent bondingEnergy bands - solids Repetition Free electron model Nearly Free Electron model - bands Tight binding model - bands Fermi-Dirac statistics, distribution functionMetals Sommerfeldt-Drude model metals Fermi-surface, Temperature dependence, Alloy scattering amorphous metals, nanostructures, 2D, 1D conductorsSemiconductors Effective mass, Electrons and holes Intrinsic - carrier concentration vs T i.e. n(T) Extrinsic- Doping, n(T) mobility, µ(T) p-n junction Carrier depletion, Carrier transport
Hetero junctions,
Electric Conduction- outline 2Semiconductors Hall effect, Negative Effective Mass, Break down Ballistic transport
Superlattices, 2 D conduction, Quantum strings, Quantum dots
Wide Bandgap Semiconductors Examples ZnO, GaN, SiC Doping, applications, current research
Insulators Oxides, Polymers Insulator ->Semiconductor ->conductor
Superconductors
ConductivityDefinition
€
J =σε
€
J : electrical current density[A/m2]
€
σ : conductivity[Ω−1m-1 =S/m]
€
ε : electricfield [V/m]
Thus
€
J = qφ φ: flux density[#m-2s-1] ; q : el. charge of carrier [C]
€
J = qnv
€
J = qnµεFrom definition of mobility
€
v : drift velocity[m/s]
€
n : charge density[m-3]
€
µ :mobility[Vs/m2]
€
σ = qnµ For one type of charge carrierSo could vary conductivity by varying mobility andcarrier concentration
Both span many orders of magnitude for same material,n is mostly considered
€
σ =1/ρ
€
ρ : resistivity[Ωm]
€
= mean vel. in ε direction
The ranges of
Silicon : 1-108 x10-6 Ωm
ρ and σ
The ranges of σ
108
104
10-4
10-18
CuTi P-polyacetylene
Si
Diamond,Nylon10-12
Quartz, PolystyreneHow do you measure a conductivity of
Metals - NonmetalsM
E T
A L
SΩΩ
Conductivity - a functional property
Guiding the electrical power to your home
Electronic device - charge distributions + conductivity guide and manipulate the flow of electrons through a device
Yeah-right
€
J =σε
€
I = JA = σUlA =U A
ρ l=UR Ohm
’s law
Suitable R for heater?
Often a combination of functional properties makes a function
Light detector, humidity detector, thermistor, Pt100strain gage, voltage divider, magnetic field sensor (MR),etc..
Electronic circuits, conductors, resistors, -distributepotential and current flow
Conductivity - a functional property
Other electrical properties of materials are importantfor functional materials
For example in an electronic circuit or chip there are manyeffects - and many engineering design issues that determinethe current flow in these semiconductor circuits.
eehh. .
It is not easy to know for ’a pedestian’ - or the man in the street -whether conduvtivity is the parameter that is the interestingproperty or whether it is something else.
Historic remarks; Conductivity of SolidsJu
st fo
r you
r inf
o.
Application of the kinetic theory of gases to metals
Theory pioneered by the dude Drude in 1900 This was 3 years after Thompson's discovery of the electronModel conductivity by classical free electron gas Metal ions remain stationary, valence electrons move around. Still a fruitful picture, but with inclusion of quantum mechanicsElectrons are accellerated in external electric fieldThe resistance to current flow is collissions of the electrons.All collisions have identical probability and mean time between collisions.
Just
for y
our i
nfo.
Historic remarks; Conductivity on Solids
Drude model successOnly Classical Stat. Mech assumptions Naive model, but described several observed phenomenaCan derive Ohm's law from itPredics the Hall EffectPredicts plasma frequencyPredicts ratio of thermal/electrical conductivity well
No electrical fieldForce from electrical field
Historic remarks:Drude model success
Just
for y
our i
nfo.
Historic remarks; Conductivity on Solids
Sommerfeld modelmodified Drude model by Fermi-Dirac distribution (quantum mechanics)Specific Heat improvementThermal conductivity -betterThermoelectrical effect -better
Draw how the electron distribution,N(E), [density as function of energy]will differ in a classic Maxwell-Bolzman-gas and a quantummechanical Fermi-gas for metals
Student problem:
Historic remark Sommerfeld
Just
for y
our i
nfo.
Historic remarks; Conductivity on Solids
Inadequacies of Drude/Sommerfeld model
Hall coefficients should depend on T and BMegnetoresistance not explainedWiedeman-Franz law fails at intermed. TDirectional dependence of σ not explainedσ (f) poor match (AC conductivity)T3 term in low T for Cv(T) unexplained
Too simplisticA Quantum Mechanical picture includingdensity of electron - and phonon -states needed.
Historic: Drude/Sommerf.TooSimple
Energy bands of solids
Free electron model
€
E =hk( )2
2m
revisited
Justify qualitatively, why?=particle in a box = quantum gas( -> ideal gas at low el. conc.)=>quantized energy E related to wavevector k.
€
k = n πl
€
E = h2
8mn2
l2
=
h2
8mnx
2 + ny2 + nz
2
V 23
E
k
Density of states
2 states pr box
1/8-spherical shellnx
ny
nz
€
D(n)dn = 2 184π n2dn1
=> D(E)dE = V2π 2
2mh2
32
E12dE
D(E
)
Ehttp://britneyspears.ac/lasers.htm
Energy bands of solids
Nearly Free electron modelrevisited
Justify qualitatively, why?Periodicity same as xtal, Potential weak,perturbation
Electron wavelike => diffraction (Bragg) at Brilloin zone
E
kπ/a
2 π/a
Electron wave-function:
Plane wave times Bloch
Periodic potential gives bands and band-gaps
€
un, r k r r ( ) = un, r k
r r +r R ( )
€
Ψn, r k r r ( ) = ei
r k ⋅
r r un, r k
r r ( )
Energy bands solidsE-k diagrams
Someexample
s
Just
for y
our i
nfo.
Nearly Free Al
Energy bands solids
Tight binding viewrevisited
Justify qualitatively, why?
Atomic orbitals close together gives bands and gaps
Fermi-Dirac statisticsrevisited
Electrons have spin +1/2 or -1/2 ;They are Fermions and obey the Pauli exclusion principleand thus, they follow Fermi-Dirac statistics
FD-distribution function f(E) number of states occupied at energy E
€
f (E) = 1
exp E − EFkT
+1
EFEnergy
T
Energy bandsMetals Semiconductors Insulators
Can not have more than 1 electron in same quantum state With zero magnetic field this gives max 2 electrons/energy level spin up/down s=1/2 og s=-1/2
States
States.
states
Full band
1/2 full
Metal
Full band
1/2cond (insulator)
Ener
gy
Conductivity - T dependence metalsWe had
µ is limited by scattering€
J =σε =qnµεSo the temperature dependence of σ can be due to n(T) or µ(T)
Lattice vibrations, phonons : dµ/dT T0ρ0 T
T dependence metals resistivity
out
Conductivity - metal aloys
A perfect periodic lattice yields no resistanceDeviations scatter electronsFor solid solutions the resistivity increases with concentrationdue to alloy scattering.
Case not followingNorheims ’rule’
ρ
A B
Sometimes referred to as Nordheims rule
composition
ρ
A BcompositionAB A3B
Ordering:
semiconductors
The ‘main’ semiconductors and their usesSi
III-Vs
Si-GeMainstream ICT,
Wide bandgap semiconductors : SiC, GaN, ZnO, CdSe
Solar cellsMEMS
II-VIs
Narrow bandgap semiconductors: InAs, InSb, Cd1-xHgxTe
Diluted magnetic semiconductors
GaAs - High speed electronicsInP, Ga1-xAlxAs Ga1-xInxAs1-yPy Ga1-xAlxAs , GaAlInP,InSb, InAs
ZnS, CdTe, HgTe,ZnSe,Cd1–xZnxS1–ySey laser
CD laser, communication laserLEDs, car stop light, trafic light, signal indicatorsGaAsP
Photon detectors, Magnetic field sensor
Photon detectors
Power electronics, High T electronic, UV LED/Laser
IR-detectors
SiGe devices
2005 IC350 000 0000 tr.?
2Source Drain
Gate
bulk p-Si
n- Simetal
oxide65
Holes-Effective Mass
A full band gives no current
Holes have positive charge
WHATAREHOLES?
E
k
EC
EV
EF
Filled electron stateempty state w. negative curvature
€
j = jiFullband∑ = 0 = + jk − jk + ji
Fullband∑
€
( jiFullband∑ ) − jk = − jk
So current from one empty stateis equal to current from thatstate filled with opposite charge
revisited
Effective massWant to write F=m*a
€
vg =1h
∂E∂k
€
a =∂vg∂t
=1h
∂∂t∂E∂k
€
=1h
∂∂k∂E∂t
=1h
∂∂k∂E∂k
∂k∂t
€
=1h
∂ 2E∂k 2
∂k∂t
€
FvgΔt = ΔE
€
=ΔEΔk
Δk ≈ hvgΔk
€
⇒ hΔkΔt
= F
€
=1h2∂ 2E∂k 2
F = m*( )−1F
F acts during time ∆t
€
1m*
=1h2∂ 2E∂k 2
ε
Q0Q0
mp*>0
€
ji = −qvki
simlified
=Group velocity
Holes-Summary
A representationdescribing vacantenergy stateswhen electrons inthese states havenegative effectivemass
WHAT
ARE
HOLES?Holes havepositive effectivemass,
And they havepositive charge
Re-revisited
summary
SemiconductivityWe had
µ can also vary much for one material varies with doping, varies with crystal defects, amorphous structure yields very different mobility comp single xtal€
J =σε =qnµε
Many other parameters than conductivity or resistivity areinteresting for semiconductors
€
σ =qnµFor semiconductors n can vary by very many orders ofmagnitude. For single xtal Si: 1012 - 1020 cm-3 at room temp
Let us first consider the importance of n
Semiconductorscarrier concentration n, p
e + h nil
€
n= gc(E) f (E,EF )dEEc
∞∫ ≈ Nc exp −
Ec −EFkT
For intrinsic semiconductor n=p=ni
The law of mass action:€
p= gv(E)(1− f (E,EF ))dE−∞
Ev∫ ≈ Nv exp −
EF −EvkT
€
np= NcNv exp −EgkT
=ni2 Eg :bandgap:
Ec
E
EF
gc(E)
f(E,EF)
g :density of states in conduction band
Nc, Nv :effective density of states for conduction and valence band respectively and
gv :density of states in valence band
gv(E)
Ec
1-f(E,EF)
g, f
Doping of semiconductor
Covalent bonding
Si
Conduction band
Valence band
Donor group V (P,As,Sb)
Acceptor group III
Example Si
Si group IV 4 nearest neighbours
n=1, 2 el
Z=14n=2, 8 el
n=3s
6 levels
2 levels
Doped semiconductorscarrier concentration n(T)
€
n= g(E) f (E,EF )dEEc
∞∫ ≈ Nc exp −
Ec −EFkT
Can solve with respect to EF we get for n-type
Must find EF to calculate n: €
np= NcNv exp −EgkT
=ni2
Charge neutrality:n + NA- = ND+ + p
NA, ND : concentration of acceptors and donors
ND+ = ND (1-f(ED) : concentration ionized donorsNA- = ND f(EA) : concentration ionized acceptors
Ev
EcND
and p-type
E
T
Electrical (semi)conduction
low temperature
j=je+jh=q(nµe+pµp)E = σEj:current density, q: electron chargeje,jp:electron or hole current density µe,µp:mobility of electrons and holes, E:electrical field
Concuction band
Valence band
Thermal exitation Ionisation energy=smallSo # electrons increase strongly w. T
1/T
ln(n
)
Electrical resistance in semiconductors
Electrical (semi)conduction
j=je+jh=q(nµe+pµp)E = σEj:current density, q: electron chargeje,jp:electron or hole current density µe,µp:mobility of electrons and holes, E:electrical field
1/T
ln(σ
)
Electrical resistance in semiconductors
intermediate temperatureAll donors ionized ,n=ND(donor cons)Electron density is constant n>>pconductivity changes dominated by mobility -scattering mechanism typical T 1/2 ..T-1/2
Electrical (semi)conduction
j=je+jh=q(nµe+pµp)E = σEj:current density, q: electron chargeje,jp:electron or hole current density µe,µp:mobility of electrons and holes, E:electrical field
1/T
ln(n
)
Electrical resistance in semiconductors
conduction
valence
Thermal excitation from valence band. When n>>ND (donor cons) ,we will have.. n=p and dσ/dT given by dn/dT and dp/dT Den til n og p
‘Chemica lreaction: e + h = nil Law of mass actionn*p =n2= k1*exp(-∆H)=k1exp(-Eg/kT)
p-n junctions
Spatial band diagram :
n-type p-type
ND NA
Ec
Ev
EF
After making junction :
Ec
Ev
EF
n-type p-type
-
Conduction in p-n junctionsAfter making junction :
Ec
Ev
EF
I
V
IV characteristic :
€
I = I0 expqVkT
−1
n-type p-type
-
Zero
bia
s
Conduction in p-n junctionsAfter making junction :
Ec
Ev
EF
I
V
IV characteristic :
+
Forward
€
I = I0 expqVkT
−1
n-type p-type
-
forw
ard
bias
Conduction in p-n junctionsAfter making junction :
Ec
Ev
EF
I
V
IV characteristic :
-
Reverse
€
I = I0 expqVkT
−1
n-type p-type
-
reve
rse
bias
Hetro junctions
Spatial band diagram :
Eg1
Ec
Ev
EF
Band offset ∆Ec
Eg2
Band offset ∆Ev
Common heterojunction system : Si / Si1-xGex GaAlAs/GaAs CdTe/CdHgTe Al2O3/Si
Made by MOCVD, MBE
Used in high mobility transistors = fast transistors Laser diodes and optical waveguides
Superlattices
=One lattice interwoven in another latticeOne lattice with larger lattice parameter than the otherCommon AlGaAs/GaAs/AlGaAs/GaAs/……… AlGaAs/GaAs…
If the smallest bandgap layer thickness is small, the electrons are confined and theseparation between levels becomes larger than kT. If the layer thickness of the largebandgap material is also small in comparison with the deBoglie wavelength ofelectrons, then we have superlattice, if it is larger we have a multiple quantum well
AlGaAs and GaAs have nearly the same lattice constant other combinations have different lattice constant, that creates stress,In a superlattice this stress is ‘alternating’ between layers, can have highly large mismatch
band band
π/a 2π/aπ/L
Lmini-band
Hetero-junction coupledquantum wells, superlattices
Lattice mismatch, strain ,
IR detector w. increased 1/2 transition
Multiple quantum wellsuperlattice
Ec
EV
wavefunction
engineering
3D,2D,1D (0D) conduction
The density of states varies with dimension
Systems beeing confined in 0, 1, 2 or 3 dimensions
2D conduction
Example Epitaxial metallic film Down to 4 ML
Exam
ple:
thin
met
alic f
ilm
-100 20-40-160 80TEMPERATURE (°C)
0.95
1.0
1.075
1.05
1 nm
VSiSi
TEMPERATURE (°C)-100 20-40-160 80
150 nm
0.6
0.8
1.0
1.2
5 nm
VSiSi
Size m
atte
rs
Salomonsen et al.
Palmstrøm et al.
2D conductionExample:
multiple quantum wel
l QW
GaAsAlGaAs 8 nm lag
p n
voltage
curr
ent
Normal pn diode
Helgesen et al.
Isolatingpolymers
insulating polymers
ConductivityOfpolymers
sConductingpolyacetylene
Conducting polyacetylene
cis-polyacetylene
conjugated system C=C-C=C-C=C-C
trans-polyacetylene
Acetylene C2H2
ρ =103 Ωm ( 105 Ωcm)
ρ =10 Ωm ( 103 Ωcm)
These are at first surprising;y high values
Naive (historic) band picture:
1 p electron pr C atom can contribute to conduction, (the rest in sp2 orbital in lower
energy band) Could from that expect a metallic conductor -
How many pz electrons are there pr (1D) unit cell? What kind of 1D band would U expect?
Conductingpolymers Nobel
Conducting polymers
xxxxx
Alan MacDiarmid shares the 2000 Nobel Prize in Chemistry for the discovery and development of conductive polymers."
Often conjugated systems C=C-C=C-C=C-C
superconductors
Super-conductors
The ambient pressure record is 138K. It is held byHg(1-x)Tlx Ba2 Ca2 Cu3 O{8.33}, x=0.2
World record Tc 160 K( at 300 000 atm, Hg 1223)
Common High Tc superconductor: YBCO
Regular metal
Superconductor
TTc
R
Naming convention,Naming Scheme that allows for identification andcomparison. The scheme chosen uses four numbers. The firstdenotes the number of insulating layers between adjacentconducting blocks. The second represents the number ofspacing layers between identical CuO2 blocks. The third givesthe number of layers that separate adjacent CuO2 planeswithin the conducting block. And, the fourth is the number ofCuO2planes within a conducting block.
Example Using the TlBa2Ca2Cu3O9molecule depicted at left as anexample, there is 1 insulatingTlO layer, 2 spacing BaOlayers, 2 separating Ca layers,and 3 conducting CuO2planes - making it a "1223"type.
(Y0.5Lu0.5)Ba2Cu3O7 Tc=107 KYbBa2Cu3O7, Tc= 89 KTl2Ba2Ca2Cu3O10, Tc= 128KLa1.85Ba.15)CuO4 , Tc= 30 K 1st HiTcMgB2 , Tc= 39K
Superconductors BCS theory
Super-conductorsBCS mechanismElectron attracts atoms as it movesCreates a positive charge in spaceAttracts another electron=> Cooper pairs bound to each other
Eg=7/2 kTc.
current popular model(s) for high Tcsuper-conductors : d-wave pairing..
Theory- still to be developed
UPd2A13 heavy fermion superconductorNew type superconductor
Sperconductor MgB2 current research
Super-conductors
MgB2 Theory Confirmed ( 1 May 2003 ) Physicists in Japan and the US have confirmed that magnesium diboride contains twosuperconducting energy gaps. Although theorists had predicted that the material had two gaps, they had never been observed inan experiment. The gaps are thought to be responsible for the relatively high superconducting transition temperature of 39 Kobserved in the material (S Souma et al. 2003 Nature 423 65).Magnesium diboride consists of hexagonal planes of boron atomsseparated by planes of magnesium atoms, with the magnesium centred above and below the boron hexagons (see figure). Thisstructure is very similar to that of graphite: each carbon atom - which has four valence electrons - is bonded to three others andoccupies all planar bonding states (the sigma bands). The remaining electron moves in orbitals above and below the plane toform pi bands. Boron atoms have fewer valence electrons than carbon so not all of the sigma bands are occupied. This meansthat lattice vibrations in the planes are much larger, which results in the formation of strong electron pairs.Most superconductorshave only one energy gap but in 2002, theorists predicted magnesium diboride might have more than one. These gaps developsimultaneously at the superconducting transition temperature, Tc. Now, Takashi Takahashi from Tohoku University andcolleagues have used high-resolution photoemission spectroscopy (ARPES) to directly observe the two gaps by resolving thesigma and pi bands.Tahakashi and co-workers measured ARPES spectra at two temperatures below and above Tc (17 K and 45K). They found that the sigma bands have a large gap of 6-7 meV, whereas the pi band has a smaller gap of 1-2 meV. Theseresults agree with previous reports. As the superconductivity of magnesium diboride is bulk in nature, the researchers concludethat the sigma band is dominant. NOTE: Industrial prospects for Magnesium Diboride appear now to be more encouragingfollowing the demonstration that introducing internal structural defects can improve its performance in practical applications suchas hospital Magnetic Resonance Imaging (MRI) scanners. In superconductors the ability to carry current without dissipation isenhanced if the material contains impurities or defects of suitable characteristics in order to "pin" the magnetic flux. Recentstudies of magnesium-diboride thin films by Bracinni, et al, found Hc2 > 50T when the boron sites in MgB2 were doped withcarbon. An increase in Jc to 106 A/cm2 at 10 K was realized when MgB2 was doped with SiC, YB4, TiB2 or MgSi2
Recent/current research
Super conductors current research
Super-conductorsRecent/current researchMagneto-optical imaging UiO
magnetic vortices in type-IIsuperconductor NbSe2 at 4.3K, UiO
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