Autumn 2006 Terje. G. Finstad - Universitetet i oslo...Lattice vibrations, phonons : dµ/dT

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 <10-12 S/m

S/m

Ag 6x108 S/mCu 6x108 S/mAl 3.8x108 S/m

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

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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

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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 ( ) = eir k ⋅

r r un,

r k

r r ( )

Energy bands solidsE-k diagrams

Someexamples

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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 − EF

kT

+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 <0Scattering from impurities,

€

σ =qnµ

For metals, µ(T), mobility is (considered) the most important.

Scattering centra and mechanisms:

Scattering from defects; dislocations, grain-boundaries, surfaceSemi-empirically

€

ρ =ρ0 1+k(T −T0)( ) T >T0ρ0 T <T0

€

∂ρ∂T

1ρ0

= k =∂ ln(uvib

2 )

∂T=∂ ln(1/λ)

∂T

ρ

Τ

residue

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 Bcomposition

AB 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 electronics

InP, 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

Q<0, m*>0Q<0, m*<0Q>0, m*>0

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 −EF

kT

For intrinsic semiconductor n=p=ni

The law of mass action:€

p= gv(E)(1− f (E,EF ))dE−∞

Ev∫ ≈ Nv exp −

EF −Ev

kT

€

np= NcNv exp −Eg

kT

=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 −EF

kT

Can solve with respect to EF we get for n-type

Must find EF to calculate n:€

np= NcNv exp −Eg

kT

=ni2

Charge neutrality:n + NA

- = ND+ + p

NA, ND : concentration of acceptors and donors

ND+ = ND (1-f(ED) : concentration ionized donors

NA- = 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 = σE j: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 = σE j: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 = σE j: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 well 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 O8.33, 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 research

Magneto-optical imaging UiO

magnetic vortices in type-IIsuperconductor NbSe2 at 4.3K, UiO

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