Electrical methods for the defect characterization in solids 1. Electrical residual resistivity in metals 2. 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 resulting momentum (free electron gas) due to scattering: stable state occur, i.e. momentum will not increase further by electric field resistivity is result of scattering
4-tip measurement of electrical resistivity arrangement prevents voltage drop over measurement tips (important for low-resistive i samples, e.g. metals) specific resistivity: U ρ= 2π s K, K... correction factor I
Correction factor thick samples: correction often neglectable (D >> S) Material ρ (300 Κ) metals <10-5 Ωcm Si:P (10 20 cm -3 ) 10-3 Ωcm Si (undoped) GA GaAs (si) 10 4 Ωcm 10 14 Ωcm
Van der Pauw geometry R AB,DC = V DC /I AB R BC,AD = V AD /I BC condition: homogeneous material; point-shaped contacts at edge of sample π d R + R ρ = AB,DC BC,AD ln 2 2 f f... correction factor
Phonon fraction of electrical resistance reduced resistance is comparable for many metals ρ(t,θ) Θ... Db Debye temp. θ = hω D kb Ω D...Debye Frequenz is highest frequency in Debye s theory of specific heat (acc. to Grüneisen)
Residual resistivity Matthiessen s rule Matthiessen s rule: Potassium samples of different purity ρ (T) = ρ R + ρ P (T) ρ R... residual resistivity (lattice defects) ρ P (T)... phonon part (scattering at acoustic phonons) at low enough temperature: electrical resistivity it due to defects dominates (impurities or lattice defects) ρ R > ρ P (T) electrons are scattered there, not any more at phonons possible scattering centers at low T: impurities, all point and line defects, 3Ddefects (e.g. precipitates) (Ch. Kittel, Solid state physics)
Influence of impurities on resistivity Ag alloys impurities lead to increased scattering of electrons resistance increase distinctly Matthiessen s s rule remains valid for small impurity densities (acc. to Linde)
Ordered and disordered alloys disordered alloy Cu-Au Pt-Pd Pd no ordered dphases Cu 3 Au CuAu ordered dalloy (Schulze Metallphysik )
Quench-in of thermal vacancies change of specific residual resistivity of Cu determination of vacancy formation enthalpy
Annealing of defects I A/C : close Frenkel pairs recombine Cu I D/E : more distant Frenkel pairs annihilate II: agglomeration of interstitial titi atoms III: vacancy migration IV: growth of defect clusters V: 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 however stage I A often missing I A -I C : very close Frenkel-pairs (energy minimization by mutual correlation between vacancy and interstitial) (Snead, 1967) I D : several steps, but the same Frenkel pair I E : vacancy and interstitial belong to different Frenkel pairs
Conductivity in semiconductors current density j: j = j n + j p = (enµ n +epµ p ) E n,p... number of free electrons and holes µ... mobility measurement of conductivity provides only product of carrier concentration and mobility measurement of Hall effect provides 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 Lorentz force is independent of sign of carriers an electric field appears in y-direction: Hall-voltage U H trace of a single electron is rather complicated this leads to corrections in case of a weak magnetic field F = e( v B) D v = μ E= e τ E D D m* 2 e τ F = E B m* ( )
Hall-Effect: the movement of carriers
The Hall Coefficient I UH = RHB, RH...Hall coefficient; I...current; d...sample thickness d r... for n-semiconductors e n RH = r...correction factor r... for p-semiconductors e p μ = σr is Hall-mobility σ... specific resistivity H H r depends on scattering mechanism and field strength of magnetic field - ionized impurities: r = 193 1.93 - scattering at phonons: r = 1.18 at strong magnetic field (µ 2 B 2 >>1): r = 1 1 RH for k different carriers (mixed conductivity): R single measurement allows hardly to conclude about defects, but measurement R H = f (T) gives more information = 1 k k
The Hall measurement practical measurements are often performed in Van-der-Pauw geometry Conditions: homogeneous material; plane-parallel p sample; dot-shaped contacts at the edge; Ohmic contacts B R R AC, BD = H d = 2 B U I BD AC ΔR AC, BD ΔU U = U ( + B ) U ( B )
Temperature dependence of carrier concentration n = N D weak compensation uncompensated n-semiconductor strong compensation intrinsic conductivity (thermal activation) defect exhaustion
Temperature-dependent Hall-effect measurements Si:B with T-dependent Hall-effect measurements: 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 as vacancies or antisite defects e.g.: in HgCdTe is Hg-vacancy dominating i acceptor [V Hg ] may be determined by Hallmeasurements at 77K
Temperature-dependent Hall-effect in Si E A = 160 mev n-si with two donors of different activation energy E A = 40 mev
Temperature-dependent Hall-effect in Hg 0.78 Cd 0.22 Te Hg vacancy is shallow acceptor carrier concentration p 77K = [V Hg ] n- p- conductivity
Interpretation of Hall-effect measurements at Hg 0.78 Cd 0.22 Te at low temperatures: holes of ionized Hg vacancies are dominating carriers at medium temperatures: intrinsic conduction starts at high temperatures: iti intrinsic i conduction dominates conduction band high temperature for intrinsic conduction (p = n): R H r 1 b μ =, with -b = ep1+ b μ n p valence band Low temp. V Hg 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 Hg 0.78 Cd 0.22 Te increasing compensation of holes (acceptor) by electrons (intrinsic conduction) Hg-vacany is shallow acceptor Carr rier con ncentrat tion n-conductivity p-conductivity p 77K = [V Hg ]
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 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 gap for wide-gap semiconductors (e.g. SiC): measurements up to 800 C
DLTS: dynamic of carriers U carrier diode in reverse direction +U injection i -U depletion layer relaxation of system i forward direction t E Fermi EF E Fermi Shift of Fermi-level by cancellation of band bending of Schottky-contact (p-njunction) during that time: change of defect charge
Increase of temperature and relaxation of system creation of depletion layer injection defect 1 Energetic position of defects defect 2 conduction band D1 D 2 valence band transient signal is slope of C(T) corresponds to thermal excitation of carriers to the conduction band edge
The DLTS signal better: periodic filling signal during slow temperature increase change of capacitance is measured in a time window corresponds to emission rate window time window and temperature defines the position in band gap peak height corresponds to concentration of defect sign of capacitance change = further information positive peak: minority carrier traps however: no structural information about defect type but: Radiotracer-DLTS (see but: Radiotracer DLTS (see below)
An example - GaAs:Cr
Radiotracer DLTS E N.Achtziger, W.Witthuhn, Phys. Rev. B 57(19), 12181 (1998) 51 Cr 51 V T 1/2 = 27.7 d SiC band gap ignal DLTS s 100 200 300 400 500 0 T (K) 20 40 60 delay time (d)
SDLTS: Scanning DLTS injection of carriers also by scanning electron beam line scans and images possible 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: very sensitive method n t /n < 10-4, absolute < 10 11 cm -3 suitable for impurity it analysis in semiconductors 0.1eV < E trap < 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) not applicable for highly doped material (carrier concentration should be < 10 17 cm -3 )