Electrical methods for the defectElectrical methods for the defect characterization in solids

1. Electrical residual resistivity in metals

2 Hall effect in semiconductors2. Hall effect in semiconductors

3. Deep Level Transient Spectroscopy - DLTS

Electrical conductivity in metals

• Ohm‘s low: current density ∝ electr. field strength: j = σE

• classical: electron velocity increases up to the scattering event, then energy loss

• scattering centers: phonons and lattice defects

• without field: all momentum components of electrons compensate each other k = -k

• with field: Fermi-sphere is shifted, because electrons exhibit resultingbecause electrons exhibit resulting momentum (free electron gas)

• due to scattering: stable state occur, i.e. ill i f h bmomentum will not increase further by

electric field

• resistivity is result of scatteringy g

4-tip measurement of electrical resistivity

• arrangement prevents voltage drop over i (i f l i imeasurement tips (important for low-resistive

samples, e.g. metals)

• specific resistivity:p y

2 K, K ... correction factorU sI

ρ = π⋅ ⋅

Correction factor

thick samples: correction often neglectable (D >> S)

Material ρ (300 Κ)

metals <10-5 Ωcm20 3 3Si:P (1020cm-3) 10-3 Ωcm

Si (undoped) 104 Ωcm

G A ( i) 1014 ΩGaAs (si) 1014 Ωcm

Van der Pauw geometry

RAB,DC = VDC/IAB RBC,AD = VAD/IBC

condition: homogeneous material; point-shaped d f lcontacts at edge of sample

ρ π= ⋅

+⋅

d R RAB,DC BC,AD fρ = ⋅ ⋅ln 2 2

f

f ti f tf ... correction factor

Phonon fraction of electrical resistance

• reduced resistance is comparable for many metals

(T Θ) Θ D b tρ(T,Θ) Θ ... Debye temp.

θ =hΩDθ =

−ΩB

D

...Debye Frequenzk

D

• is highest frequency in Debye‘s theory of specific heat

(acc. to Grüneisen)

Residual resistivity – Matthiessen‘s rule

Matthiessen‘s rule:

ρ (T) = ρR + ρP(T)P t i

ρ ( ) ρR ρP( )

ρR ... residual resistivity (lattice defects)

ρP(T) ... phonon part (scattering at acoustic

Potassium

ρP(T) ... phonon part (scattering at acoustic phonons)

• at low enough temperature: electrical i ti it d t d f t d i t

samples of different purityresistivity due to defects dominates (impurities or lattice defects)

ρR > ρP(T)ρR ρP( )

• electrons are scattered there, not any more at phonons

• possible scattering centers at low T: impurities, all point and line defects, 3D-defects (e.g. precipitates)

(Ch. Kittel, Solid state physics)

Influence of impurities on resistivity

• impurities lead to increased scattering of electrons

Ag alloys

scattering of electrons

• resistance increase distinctly

• Matthiessen’s rule remains• Matthiessen s rule remains valid for small impurity densities

(acc. to Linde)

Ordered and disordered alloys

Cu-Au Pt-Pd

disordered alloy

Cu Au Pt Pd

d d hno ordered phases

CuAuCu3Au

d d llordered alloy

(Schulze “Metallphysik”)

Quench-in of “thermal vacancies”

change of specific residual resistivity of Cu ⇒ determination of vacancy formation enthalpy

Annealing of defects

IA/C: close Frenkel pairs recombine

ID/E: more distant Frenkel pairs annihilate

II l ti f i t titi l

Cu

II: agglomeration of interstitial atoms

III: vacancy migrationIII: vacancy migration

IV: growth of defect clusters

V: annealing of these clustersV: annealing of these clusters

defects after a) quenching from 1300K b) 3 MeV-electron irradiation

k) 10% cold deformation (acc. to Bergmann, Schäfer)

Irradiated Cu – annealing in stage 1

• differentiated curves show the sensitivity of the method

• each peak: a new defect becomes mobile

• similar in most metals• similar in most metals

• however stage IA often missing

• IA-IC: very close Frenkel-pairsIA-IC: very close Frenkel-pairs (energy minimization by mutual correlation between vacancy and interstitial)vacancy and interstitial)

(Snead, 1967) • ID: several steps, but the same Frenkel pair

• IE: vacancy and interstitial belong to different Frenkel pairs

Conductivity in semiconductors

current density j:

j = j + j = (enµ + epµ ) Ej jn + jp (enµn + epµp) E

n,p ... number of free electrons and holes

µ ... mobilityµ y

• measurement of conductivity provides only product of carrierprovides only product of carrier concentration and mobility

• measurement of Hall effect id dditi llprovides n resp. p; additionally

measured conductivity: mobility can be calculated

scattering at charged defects resp. at phonons

The Hall Effect

moving electrons (holes) feel Lorentz force when magnetic field B is present

F v B= − ×e

eD( )

τv E E= μ = e

e

D D

2*

τ

τm

F E B= ×e

*( )

τm

• Lorentz force is independent of sign of carriers

• an electric field appears in y-direction: Hall-voltage UH

• trace of a single electron is rather complicated

• this leads to corrections in case of a weak magnetic field

Hall-Effect: the movement of carriers

The Hall Coefficient

I , ...Hall coefficient; I...current; d...sample thickness

... for n-semiconductors

H H HIU R B Rd

r

⊥=

⎧−⎪ ... for n semiconductorse

...correction factor... for p-semiconductors

e

Hn

R rrp

⎪⎪= ⎨⎪⎪⎩ e

= is Hall-mobility ... spH H

p

R

⎪⎩μ σ σ ecific resistivity

• r depends on scattering mechanism and field strength of magnetic field

- ionized impurities: r = 1 93ionized impurities: r 1.93

- scattering at phonons: r = 1.18

• at strong magnetic field (µ2B2 >>1): r = 1 1s o g g e c e d (µ ):

• for k different carriers (mixed conductivity):

• single measurement allows hardly to

RRH

kk

= −∑1

1

g yconclude about defects, but measurement RH = f (T) gives more information

The Hall measurement

practical measurements are often performed in Van-der-Pauw geometry

Conditions: homogeneous material; plane-parallel sample; dot-shaped p p p ; pcontacts at the edge; Ohmic contacts

BR

UI

d

AC BDBD

AC, =

B

R d R

U U U

H AC BD,

( ) ( )

=

= +

2 BB B

Δ

ΔU U U( ) ( )= + − −B BΔ

Temperature dependence of carrier concentration

weak compensationn = ND uncompensated

n-semiconductor

p

strong compensation

intrinsic conductivity(thermal activation)

defect exhaustion

Temperature-dependent Hall-effect measurements

• with T-dependent Hall-effect

Si:B

pmeasurements: concentration and energetic level of electrically active defects in band gap may be determined

• these are: dopant- or impurity atoms, but also intrinsic defects such asbut also intrinsic defects such as vacancies or antisite defects

• e.g.: in HgCdTe is Hg-vacancy d i idominating acceptor

• [VHg] may be determined by Hall-measurements at 77K

Temperature-dependent Hall-effect in Si

n-Si with two donors of different activation energy

EA = 160 meV

gy

EA = 40 meV

Temperature-dependent Hall-effect in Hg0.78Cd0.22Te

Hg vacancy is shallowHg vacancy is shallow acceptor

ratio

nco

ncen

t

p77K = [VHg]

carr

ier c

n- p- conductivity

c

Interpretation of Hall-effect measurements at Hg0.78Cd0.22Te

• at low temperatures: holes of ionized Hg vacancies are dominating carriers

• at medium temperatures:at medium temperatures:intrinsic conduction starts

• at high temperatures:i t i i d ti d i t

conduction band

intrinsic conduction dominateshigh temperature

Low tempVHg

Low temp.

• for intrinsic conduction (p = n):

valence bandn

p

1 , with -e 1Hr bR bp b

μ−= =

+ μ

• carriers with higher mobility determine the conduction type

• general: electrons more mobile ⇒ n-conduction at high temperatures for p-doped samples

Temperature-dependent Hall-measurements at Hg0.78Cd0.22Te

increasing compensation of holes (acceptor) by electrons (intrinsic conduction)

Hg-vacany is shallowHg-vacany is shallow acceptor

tion

ncen

trat

p77K = [VHg]

rier c

on

n-conductivity p-conductivityCar

r

Deep levels: important defects in semiconductors

DLTS: Deep-level transient spectroscopy

• asymmetrical p-n-junction or Schottky-contact required

• in reverse direction: depletion zone

• exhibits capacitance in pF range

h t l i f d• short pulse in forward direction at low temperature: electrons or holes are captured in deep levels

• capacitance changes

• during increase of temperature: captured carriers are thermally liberated

• temperature is measure for energetic position in band gaptemperature is measure for energetic position in band gap

• for wide-gap semiconductors (e.g. SiC): measurements up to 800°C

DLTS: dynamic of carriers

diode in reverse directionU carrier injection+Ui

d l ti l l ti f t

t-Ui

forward direction

depletion layer relaxation of system

EFermi EF iFermi EFermi

• Shift of Fermi-level by cancellation of band bending of Schottky-contact (p-n-junction)

• during that time: change of defect charge

Increase of temperature and relaxation of system

injectioncreation of depletion layer

defect 1

Energetic position of defects

defect 2

conduction bandD 1D 1D 2

• transient signal is slope of C(T)

valence band • corresponds to thermal excitation of carriers to the conduction band edge

• better: periodic filling signal during slow temperature increase

The DLTS signal

• change of capacitance is measured in a time window

• corresponds to emission rate window

• time window and temperaturetime window and temperature defines the position in band gap

• peak height corresponds to concentration of defectconcentration of defect

• sign of capacitance change = further information

• positive peak: minority carrier traps

• however: no structural information about defect type

• but: Radiotracer-DLTS (seebut: Radiotracer DLTS (see below)

An example - GaAs:Cr

Radiotracer DLTS

N.Achtziger, W.Witthuhn,Phys. Rev. B 57(19), 12181 (1998)

51Cr → 51V T1/2 = 27.7 d

y ( ), ( )

SiCE

40

60ig

nal

e (d)d ga

p

100 200 300 400 500 0

20

40D

LTS

s

delay tim

e (

band

500T (K)

SDLTS: Scanning DLTS

• injection of carriers also by scanning electron beam

• line scans and images possibleg p

• for T=const. and defined time window: distribution of a certain defect can be visualized

Example: EL2-signal in GaAs as scan over a grown-in dislocation line

Advantages and drawbacks of DLTS-technique

Advantages:

g

Advantages:

• very sensitive method nt/n < 10-4, absolute < 1011 cm-3

it bl f i it l i i i d t• suitable for impurity analysis in semiconductors

• 0.1eV < Etrap < 2eV (but problems for the detection of shallow traps, i.e. dopants, but they can be studied by Hall-effect)

• lateral resolution possible by injection of carriers by scanning electron beam (Scanning DLTS)

Drawbacks:

• no structural information on the carrier traps (radiotracer DLTS sometimes applicable; not suitable for intrinsic defects, e.g. vacancies)applicable; not suitable for intrinsic defects, e.g. vacancies)

• not applicable for highly doped material (carrier concentration should be < 1017 cm-3)