characterization in solids
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1 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
2 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
3 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
4 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 Ωcm
5 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
6 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)
7 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)
8 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)
9 Ordered and disordered alloys disordered alloy Cu-Au Pt-Pd Pd no ordered dphases Cu 3 Au CuAu ordered dalloy (Schulze Metallphysik )
10 Quench-in of thermal vacancies change of specific residual resistivity of Cu determination of vacancy formation enthalpy
11 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)
12 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
13 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
14 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* ( )
15 Hall-Effect: the movement of carriers
16 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 = 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
17 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 )
18 Temperature dependence of carrier concentration n = N D weak compensation uncompensated n-semiconductor strong compensation intrinsic conductivity (thermal activation) defect exhaustion
19 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
20 Temperature-dependent Hall-effect in Si E A = 160 mev n-si with two donors of different activation energy E A = 40 mev
21 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
22 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
23 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 ]
24 Deep levels: important defects in semiconductors
25 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
26 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
27 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
28 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)
29 An example - GaAs:Cr
30 Radiotracer DLTS E N.Achtziger, W.Witthuhn, Phys. Rev. B 57(19), (1998) 51 Cr 51 V T 1/2 = 27.7 d SiC band gap ignal DLTS s T (K) delay time (d)
31 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
32 Advantages and drawbacks of DLTS-technique Advantages: very sensitive method n t /n < 10-4, absolute < 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 < cm -3 )
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