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

d1 d2 d3 d1 d2 W L r

V

Kelvin method I

• I

Rmeas

ij =2⋅RC+RSdij/W

RS RC

• W. Shockley (1964) → G. K. Reeves and H. B. Harrison, IEEE Elec. Dev. Let. 3 (1982) 111.

RS RC V(x) x

V (x)= I √ρcρscosh[(L−x)/ LT ] Z sinh(L/LT)

LT=√ ρc ρs

Gives an idea of spreading.

d1 d2 d3 W L d1 d2 d3

Rmeas

ij =2⋅RC+RSdij/W

2RC SLOPE = RS/W

• 2LT

LT = RC/RS

Resistance Measurements

Drude (1900):

⃗ P(t +Δ)=(⃗ P(t )+⃗ F⋅Δ)(1−Δ τ ) ⃗ F=q ⃗ E+q ⃗ v×⃗ B

I

~ ρ=[ ρ −B0/nq B0/nq ρ ]

ρ= m nq

2τ

=1/nqμ

⃗ B

2-Wire Sensing

2 3 1 4 5 6 L t w

ρ= V 56−V 65 I 56−I 65 ⋅w⋅t L

To get better statistics. (anisotropy, inhomogeneity, contacts, etc.) I V Rp1 Rp2 RDUT

I=

V R DUT+RP 1+RP 2

RMeasured=V / I=RDUT +RP1+RP2

Not Reliable!!!

Hall bar mesa structure

4-wire Sensing

V Rp1 Rp2 RDUT Rq1 Rq2

V=I⋅RDUT

RMeasured=V /I=RDUT

Zi → ∞ 2 3 1 4 5 6 a t w I

ρL= V 23−V 32 I 56−I 65 ⋅w⋅t a ρU= V 14−V 41 I 56−I 65 ⋅w⋅t a ρ= ρL+ρU 2

Magnetoresistivity

2 3 1 4 5 6 a t w

ρL=(V 23

+B−V 32 +B)+(V 23

• B−V 32
• B)

(I 56

+B−I 65 +B)+(I 56

• B−I 65
• B)

⋅w⋅t a

⃗ B

ρU=(V 14

+B−V 41 +B)+(V 14

• B−V 41
• B)

(I 56

+B−I 65 +B)+(I 56

• B− I 65
• B)

⋅w⋅t a ρ= ρL+ρU 2

Hall Coefficient

2 3 1 4 5 6 a w

⃗ B

~ ρ=[ ρ −B0/nq B0/nq ρ ] E y/J x=B0/nq

t

V 12/w I 56/(w⋅t)=B0/nq V 12 I 56 =B0/tnq RH= t B0 ⋅ V 12 I 56 =1/nq RH 1=(t/B)V 34

+ B−V 43 +B+V 43

• B−V 34
• B

I 56

+ B−I 65 +B+I 65

• B−I 56
• B

RH 2=(t /B)V 21

+ B−V 12 + B+V 12

• B−V 21
• B

I 56

+ B−I 65 + B+I 65

• B−I56
• B

RH= RH 1+RH 2 2 n=1/q⋅RH

ρ=1/nqμ

μ=1/nqρ

Magnetoresistivity

~ ρ=[ ρ −B0/nq B0/nq ρ ] ~ σ= 1 ρ

2+(B0/nq) 2[

ρ B0/nq −B0/nq ρ ] σxx= nqμ 1+(B0μ)

2

σxy= nqμ

2B0

1+(B0μ)

2

σxx=∑

i

niqμi 1+(B0μi)

2

σxy=∑

i

niqμi

2B0

1+(B0μi)

2

How many different carrier species?!

Mobility Spectrum

• W. A. Beck and J. R. Anderson, J. Appl. Phys. 62 (1987) 541.

σxx(B)=∫

−∞ ∞

dμ s(μ) 1+(Bμ)

2

σxy(B)=∫

−∞ ∞

dμ s(μ)μ B 1+(Bμ)

2

Conductivity Density Account for multiple carrier species in multiple bands, layers, dependence of relaxation times and effective masses

• n wavevector, anisotropies, etc.

PROBLEM:

σxx=T {s(μ)} s(μ)=T

−1{σxx}

No inverse transform. Have to fit! Original solution mathematically intense!!! Integral Transform!

Mobility Results

• J. R. Meyer, C. A. Hofgman, F

. J. Bartoli, D. A. Arnold, S. Sivananthan and J. P . Faurie, Semicond.

• Sci. Technol. 8 (1993) 805.

Thermally excited carriers!

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

σxx(B)=∫

−∞ ∞

dμ s(μ) 1+(Bμ)

2

σxy(B)=∫

−∞ ∞

dμ s(μ)μ B 1+(Bμ)

2

In order to be an inverse, these have to be orthogonal functions! Solution indeterminate!

1/ρ=∫0

d

σ(z)dz

Similar Problem Implanted channel

σ≈μ q∫0

d

n(z)dz

Van der Pauw

I

• L. J. van der Pauw, Phil. Tech. Rev. 20 (1958) 220.

r

⃗ J = 2 I 2 πr δ ^ r ⃗ E=ρ⃗ J= ρI πr δ ^ r

δ

V AB=−∫A

B ⃗

E⋅⃗ dr V AB=−ρI πδ∫A

B dr

r V AB=−ρ I πδ ln(B/ A)

Van der Pauw

I M N O P a b c δ

V AB=−ρ I πδ ln(B/ A)

Case #1: M N O P a b c I

• I

V PO(I M)=−ρI πδ ln( a+b+c a+b ) V PO(−I N)=ρ I πδ ln( b+c b )

V+ V-

V PO=V PO(I M)+V PO(−I N) V PO=− ρI π δ ln( a+b+c a+b b b+c)

a+b+c a+b b b+c =e

−π RMNOP /ρs

Van der Pauw

I M N O P a b c δ

V AB=−ρ I πδ ln(B/ A)

Case #2: M N O P a b c I

• I

V MP(I N)=ρ I π δ ln( b+c a ) V MP(−I O)=−ρ I πδ ln( c a+b)

V+ V-

V MP=V MP(I N)+V MP(−I O) V MP=−ρI πδ ln( c a+b a b+c)

c a+b a b+c =e

−π R NOPM /ρs

a+b+c a+b b b+c =e

−π RMNOP /ρs

c a+b a b+c =e

−π R NOPM /ρs

+

1=e

−π R MNOP/ρs+e −π R NOPM/ρs

Nested intervals! m n p

• I
• I

V+ V- m n p

• I
• I

V+ V- Rnopm Rmnop m n p

• I
• I

V+ V- Hall

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 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

1=e

−π R MNOP/ρs+e −π R NOPM/ρs

Reciprocity Theorem

I

• I

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

• I

V- V+ V+-V- = V-∆V ∆V

• H. Lorentz (1896)

Z +

• i1

v1 +

• v2

i2

[

V 1 V 2] =[ Z11 Z12 Z21 Z22][ i1 i2]

Z11= V 1 i1

i2=0

Z11= V 1 i1

i2=0

Z11= V 1 i1

i2=0

Z11= V 1 i1

i2=0

CASE #1:

Z +

• i

v1

• i2

[

V 1 0 ]=[ Z11 Z12 Z21 Z22][ i1 −i2]

[

i −i2]= 1 Z11 Z22−Z12Z 21[ Z22 −Z12 −Z21 Z11 ][ V 1 0 ] V 1 i2 = Z11Z22−Z12

2

Z12

CASE #2:

Z

• i1

i'

[

V 2]=[ Z11 Z12 Z21 Z22][ −i1 i' ]

+

• v2

[

−i1 i ]= 1 Z11 Z22−Z12Z21[ Z22 −Z12 −Z21 Z11 ][ V 2] V 1 i2 = Z11Z22−Z12

2

Z12

RABCD = RCDAB

m n p

• I
• I

V+ V- m n p

• I
• I

V+ V- Rnopm Rmnop m n p

• I
• I

V+ V- m n p

• I
• I

V+ V- Rpmno Ropmn

2⋅RV=(Rmnop+Ropmn) 2⋅RH=(Rnopm+R pmno)

4⋅RV 4⋅RH

V V V V V V V V

e

−π R V/ρs+e −π R H/ρs=1

2-Lockin Technique

V/I V/I

• 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 120 Hz

Bipolar Voltage/Current Converter

V I = V/R 10k 10k 10k R C OPA404