Experimental Methods in Transport Physics Prof. Carlo Requio da - PowerPoint PPT Presentation

Experimental Methods in Transport Physics Prof. Carlo Requião da Cunha, Ph.D. unit: Resistivity

Transmission Line Method W. Shockley (1964) → G. K. Reeves and H. B. Harrison, IEEE Elec. Dev. Let. 3 (1982) 111. d 1 d 2 d 3 d 1 d 2 r W L Kelvin method V I ij = 2 ⋅ R C + R S d ij / W -I R meas R C R S

R C R S L T = √ ρ c V(x) ρ s I √ ρ c ρ s cosh [ ( L − x )/ L T ] Gives an idea of V ( x )= spreading. Z sinh ( L / L T ) x

d 1 d 2 d 3 W L ij = 2 ⋅ R C + R S d ij / W R meas L T = R C /R S SLOPE = R S /W 2R C -2L T d 1 d 2 d 3

Resistance Measurements Drude (1900): F ⋅Δ)( 1 −Δ I ⃗ P ( t +Δ)=(⃗ P ( t )+⃗ τ ) ⃗ F = q ⃗ v ×⃗ E + q ⃗ B ⃗ B ρ= [ ρ ] ρ − B 0 / nq ~ B 0 / nq ρ= m = 1 / nq μ 2 τ nq

2-Wire Sensing Hall bar mesa structure 1 4 ρ= V 56 − V 65 ⋅ w ⋅ t w To get better statistics. t I 56 − I 65 L (anisotropy, inhomogeneity, contacts, etc.) 6 5 2 3 L Not Reliable!!! R DUT V I = R DUT + R P 1 + R P 2 R p1 R p2 R Measured = V / I = R DUT + R P 1 + R P 2 I V

4-wire Sensing R DUT V = I ⋅ R DUT R p1 R p2 R q1 R q2 R Measured = V / I = R DUT Z i → ∞ V ρ L = V 23 − V 32 ⋅ w ⋅ t I I 56 − I 65 a 1 4 ρ U = V 14 − V 41 ⋅ w ⋅ t w t I 56 − I 65 a 6 5 ρ= ρ L +ρ U 2 3 2 a

Magnetoresistivity 1 4 +B − V 32 +B )+( V 23 -B − V 32 -B ) ρ L =( V 23 ⋅ w ⋅ t w t a +B − I 65 +B )+( I 56 -B − I 65 -B ) ( I 56 6 5 2 3 +B − V 41 +B )+( V 14 -B − V 41 -B ) ρ U =( V 14 ⋅ w ⋅ t a a +B − I 65 +B )+( I 56 -B − I 65 -B ) ( I 56 ⃗ B ρ= ρ L +ρ U 2

Hall Coefficient ρ= [ ρ ] ρ − B 0 / nq ~ E y / J x = B 0 / nq 1 4 B 0 / nq w t V 12 6 V 12 / w V 12 R H = t 5 ⋅ = 1 / nq I 56 /( w ⋅ t )= B 0 / nq = B 0 / tnq 2 3 B 0 I 56 I 56 a + B − V 12 + B + V 12 - B − V 21 - B + B − V 43 + B + V 43 - B − V 34 - B R H 2 =( t / B ) V 21 R H 1 =( t / B ) V 34 ⃗ B + B − I 65 + B + I 65 - B − I 56 - B + B − I 65 + B + I 65 - B − I 56 - B I 56 I 56 R H 1 + R H 2 n = 1 / q ⋅ R H μ= 1 / nq ρ ρ= 1 / nq μ R H = 2

Magnetoresistivity ρ= [ ρ ] 2 [ ρ ] ρ − B 0 / nq ρ B 0 / nq 1 ~ ~ σ= 2 + ( B 0 / nq ) B 0 / nq − B 0 / nq ρ nq μ n i q μ i σ xx = ∑ σ xx = 2 1 +( B 0 μ) 2 1 +( B 0 μ i ) i How many different carrier species?! 2 B 0 2 B 0 σ xy = nq μ n i q μ i σ xy = ∑ 2 2 1 +( B 0 μ) 1 +( B 0 μ i ) i

Mobility Spectrum W. A. Beck and J. R. Anderson, J. Appl. Phys. 62 (1987) 541. Conductivity Density ∞ s (μ) σ xx ( B )= ∫ d μ Account for multiple carrier species in 2 1 +( B μ) −∞ multiple bands, layers, dependence of relaxation times and effective masses ∞ d μ s (μ)μ B on wavevector, anisotropies, etc. σ xy ( B )= ∫ 2 1 +( B μ) −∞ PROBLEM: − 1 {σ xx } σ xx = T { s (μ)} s (μ)= T No inverse transform. Have to fit! Original solution mathematically intense!!! Integral Transform!

Mobility Results Thermally excited carriers! J. R. Meyer, C. A. Hofgman, F . J. Bartoli, D. A. Arnold, S. Sivananthan and J. P . Faurie, Semicond. Sci. Technol. 8 (1993) 805.

Quantitative Mobility Spectrum J. Antoszewski and L. Farone, Opto-Electron. Rev. 12 (2004) 347.

Brian' Method D. Chrastina, J. P . Hague and D. R. Leadley, J. Appl. Phys. 94 (2003) 6583.

Restriction ∞ s (μ) σ xx ( B )= ∫ d μ 2 1 +( B μ) −∞ Solution indeterminate! ∞ d μ s (μ)μ B σ xy ( B )= ∫ Similar Problem 2 1 +( B μ) −∞ Implanted channel d 1 /ρ= ∫ 0 σ( z ) dz In order to be an d σ≈μ q ∫ 0 inverse, these have n ( z ) dz to be orthogonal functions!

Van der Pauw L. J. van der Pauw, Phil. Tech. Rev. 20 (1958) 220. I δ r B ⃗ J = 2 I J = ρ I ⃗ V AB =− ∫ A E ⋅⃗ 2 π r δ ^ E =ρ⃗ ⃗ r π r δ ^ r dr B dr V AB =−ρ I V AB =−ρ I πδ ∫ A πδ ln ( B / A ) r

Van der Pauw I M N O P a b c V PO = V PO ( I M )+ V PO (− I N ) δ V AB =−ρ I πδ ln ( a + b ) V PO ( I M )=−ρ I a + b + c πδ ln ( B / A ) πδ ln ( b ) V PO (− I N )=ρ I b + c Case #1: -I I π δ ln ( b + c ) V PO =− ρ I a + b + c b M N O P a + b a b c V- V+ a + b + c b −π R MNOP /ρ s b + c = e a + b

Van der Pauw I M N O P a b c V MP = V MP ( I N )+ V MP (− I O ) δ V AB =−ρ I π δ ln ( a ) V MP ( I N )=ρ I b + c πδ ln ( B / A ) πδ ln ( a + b ) V MP (− I O )=−ρ I c Case #2: I -I πδ ln ( b + c ) V MP =−ρ I c a M N O P a + b a b c V+ V- c a −π R NOPM /ρ s b + c = e a + b

a + b + c b −π R MNOP /ρ s b + c = e a + b −π R MNOP /ρ s + e −π R NOPM /ρ s 1 = e + c a −π R NOPM /ρ s b + c = e Nested intervals! a + b R nopm R mnop Hall I V+ V- V+ -I m p m p m p V+ n n n o o o V- V- -I -I I I

Python Code # Calculate error in guessing a sheet resistance value def erro(val): global Rv, Rh return abs(1 - np.exp(-Rv*np.pi/val) - np.exp(-Rh*np.pi/val)) # Start nested intervals Li = 1E1 La = 1E6 −π R MNOP /ρ s + e −π R NOPM /ρ s 1 = e while La-Li > 1: ma = Li + (La-Li)/4 mb = La - (La-Li)/4 Era = erro(ma) Erb = erro(mb) if Era < Erb: La = (Li+La)/2 elif Erb < Era: Li = (Li+La)/2 Ps = (Li+La)/2

Reciprocity Theorem Problem: Parasitics! e.g. Seebeck effect. V+ V- -I ∆V ∆V I V + -V - = V+∆V V + -V - = V-∆V I V- V+ -I

H. Lorentz (1896) i 2 i 1 = [ Z 22 ] [ [ V 2 ] i 2 ] + + Z 11 Z 12 V 1 i 1 v 1 v 2 Z Z 21 - - V 1 V 1 Z 11 = Z 11 = i 1 i 1 i 2 = 0 i 2 = 0 V 1 V 1 Z 11 = Z 11 = i 1 i 1 i 2 = 0 i 2 = 0

CASE #1: 0 ] = [ Z 21 Z 22 ] [ − i 2 ] [ Z 11 Z 12 V 1 i 1 i -i 2 + Z 11 Z 22 − Z 12 Z 21 [ Z 11 ] [ 0 ] v 1 [ − i 2 ] = 2 Z = Z 11 Z 22 − Z 12 − Z 12 V 1 1 Z 22 V 1 i - i 2 Z 12 − Z 21 CASE #2: V 2 ] = [ Z 22 ] [ i' ] [ Z 11 Z 12 − i 1 0 i' -i 1 Z 21 + Z 11 Z 22 − Z 12 Z 21 [ Z 11 ] [ [ i ] = V 2 ] v 2 2 Z V 1 = Z 11 Z 22 − Z 12 − Z 12 1 Z 22 − i 1 0 - i 2 Z 12 − Z 21 R ABCD = R CDAB

R nopm R pmno -I I V+ V- m p m p 2 ⋅ R H = ( R nopm + R pmno ) n o n o V+ V- -I I R mnop R opmn I I m p m p V+ V+ 2 ⋅ R V = ( R mnop + R opmn ) n n o o V- V- -I -I

V V V V V V V V 4 ⋅ R V 4 ⋅ R H −π R V /ρ s + e −π R H /ρ s = 1 e

2-Lockin Technique G. T. Kim, J. G. Park, Y . W. Park, C. Müller-Schwanneke, M. Wagenhals and S. Roth, Rev. Sci. Inst. 70 (1999) 2177. 18 Hz V/I 120 Hz V/I

Bipolar Voltage/Current Converter C 10k R I = V/R V 10k OPA404 10k