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

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

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Experimental Methods in Transport Physics Prof. Carlo Requio da Cunha, Ph.D. unit: Review of Statistics H. Cavendish, An Account of Some Attempts to Imitate the Effects of the Torpedo by Electricity, Phil. Trans. R. Soc. Lond. 66 (1776)

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SLIDE 1

Experimental Methods in Transport Physics

  • Prof. Carlo Requião da Cunha, Ph.D.

unit: Review of Statistics

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SLIDE 2

“...let a person take some yards of very fine wire, holding the end in each hand, and let him discharge the jar by touching the outside with one end of the wire, and the inside with the other; he will feel a shock...”

  • H. Cavendish, “An Account of Some Attempts to Imitate the Effects of

the Torpedo by Electricity”, Phil. Trans. R. Soc. Lond. 66 (1776) 196.

“What power of the velocity is the resistance proportional to?”

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SLIDE 3

Least Squares Fitting

x y Test function:

f ({x};⃗ v)

Example:

f (x ;a,b)=ax+ b

Error in fitting:

R2=∑

i N

[ yi−a⋅xi−b]2

Minimize the error: ∂ R2

∂ vi =0 ∂ R2 ∂ a =0 ∂ R2 ∂ b =0 b∑

i N

xi+a∑

i N

xi

2=∑ i N

yi xi N⋅b+a∑

i N

xi=∑

i N

yi

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SLIDE 4

[

N

∑

i N

xi

∑

i N

xi ∑

i N

xi

2][

b a]=[

∑

i N

yi

∑

i N

xi y i]

[

b a]= 1 N ∑

i N

xi

2−(∑ i N

xi)

2[

N

∑

i N

xi

∑

i N

xi

∑

i N

xi

2][

∑

i N

yi

∑

i N

xi yi]

[

b a]= 1 1 N ∑

i N

xi

2−(¯

x )

2[

1 N ¯ y∑

i N

xi

2− 1

N ¯ x∑

i N

xi yi 1 N ∑

i N

xi yi−¯ x ¯ y

]

a= 1 N ∑

i N

xi yi−¯ x ¯ y 1 N ∑

i N

xi

2−(¯

x)2 = σ (x , y) σ2( x)

b= 1 N ∑

i N

xi

2−(¯

x)2 1 N ∑

i N

xi

2−(¯

x)2 ¯ y−( 1 N ∑

i N

xi yi−¯ x ¯ y) 1 N ∑

i N

xi

2−(¯

x)2 ¯ x=¯ y−a ¯ x

Covariance(x,y) Variance(x)

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SLIDE 5

Octave Script

clear; % Load File d = load('file.dat'); x = d(:,1); y = d(:,2); % Compute a = cov(x,y)/var(x); b = mean(y) – a*mean(x); % Plot plot(x,y,'k*',x,a*x+b,'b'); x y

R2=1−∑

i N

( yi−¯ y)

∑

i N

( yi−f i) → 94

Goodness of fit: % for a noise level of +/- 3

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SLIDE 6

5 10 15 20 5 10 15 20 25 30

Linearized Fit - exp(x)

% Compute ly = log(ye); alfa = cov(ly,xe)/var(xe) beta = exp(mean(ly)-alfa*mean(xe)) yf = beta*exp(alfa*x); x y

y= y0⋅e

A⋅y

ln(y)=ln(y0)+A ⋅x

b

A=cov(ln( y), x) var(x)

yo=e

ln( y)−A⋅x

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SLIDE 7

1 2 3 4 5 6 5 10 15 20 25 30

Linearized Fit – Power Laws

% Compute ly = log(ye); xl = log(xe); alfa = cov(ly,xl)/var(xl) beta = exp(mean(ly)-alfa*mean(xl)) yf = beta*x.^alfa; x y

y= y0⋅x

A

ln(y)=ln(y0)+A ⋅ln(x) A=cov(ln( y),ln(x)) var(ln (x))

yo=e

ln( y)−A⋅ln(x)

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SLIDE 8

Gaussian Profile

Σ Independent random Variables. Intensity x

∂G ∂G0 = G G0

FWHM=2√2log 2σ

∂ G ∂ x0 = x−x0 σ

2

G0 ∂G ∂ G0

G(x)=G0e

−1 2( x−x0 σ )

2

∂G ∂ σ = x−x0 σ ∂ G ∂ x0

Typically found in Density of States.

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SLIDE 9

Gaussian Profile Example 1

  • T. Schwartz, G. Bartal, S. Fishman and M. Segev, “Transport and Anderson

Localization in Disordered Two-Dimensional Photonic Lattices”, Nature 446 (2007) 52. Log of the intensity

“In d, fitting the curve to a gaussian profile of the form I exp(-2r2/2) yields the value = 92 m. In e, the fitted curve corresponds to an intensity profile of the form , where is the distance from the centre of the beam, and = 64 m is the localization length as determined by the exponential fit. In terms of FWHM, the width of the fitted profile of e is 44 m, compared to 108 m FWHM for the gaussian fit in the diffusive case of d, and it is also three times narrower than the diffraction pattern observed in the absence of disorder: 120 m (c). The transition from the gaussian curve of d to the exponentially decaying curve of e displays the crossover from diffusive transport (d) to Anderson localization (e).”

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SLIDE 10

Gaussian Profile Example 2

  • H. Bässler, “Charge Transport in Disordered Organic Photoconductors”, Phys.
  • Stat. Sol. b 175 (1993) 15.

Density of states is a Gaussian distribution.

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SLIDE 11

Gaussian Profile Example 3

  • Y. Zhang, B. de Boer and P. W. M. Blom, Phys. Rev. B 81 (2010) 085201.

“Schematic representation

  • f

the energy-level alignment of a Gaussian DOS (LUMO) for the transport of free electrons and an exponential DOS for electron traps that are separated by an energy

E.”

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SLIDE 12

Gaussian Ensembles

  • T. Kriecherbauer, J. Marklof and A. Soshnikov, “Random matrices and quantum

chaos” Proc. Nat. Acad. Sci. Un. St. 98 (2001) 10531. “Level spacing distribution for the energy spectrum of a quantum particle in the chaotic heart-shaped region of Fig. 1 vs. the level spacing distribution for Gaussian Unitary Ensemble, Gaussian Orthogonal Ensemble, and Poisson, respectively.”

s= λn+1−λn

〈λn+1−λn 〉

pGOE=π 2⋅s⋅e

−π 4 s2

pGUE=32 π

2⋅s2⋅e − 4 π s2

pGSE= 2

18

36π3⋅s

4⋅e − 64 9 π s2

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SLIDE 13

Lorentzian Profile

Forced resonance on damped system. Intensity x

L(x)=L0 Γ

2

(x−x0)

2+Γ 2

FWHM=2Γ

∂ L ∂L0 = L L0 ∂L ∂x0 =2(x−x0) L0

(

L Γ)

2

∂L ∂Γ =(x−x0) Γ ∂ L ∂ x0

Typically found in tunneling.

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SLIDE 14

Lorentzian Profile Example 1

  • M. Bockrath, D. H. Cobden, P. L. McEuen,
  • N. G. Chopra, A. Zettl, A. Thess and R. E.

Smalley, “Single-Electron Transport in Ropes of Carbon Nanotubes”, Science 275 (1997) 5308.

“...Fitting the peak shapes reveals that they are approximately Lorentzian, as expected for resonant tunneling through a single quantum level (1).”

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SLIDE 15

Lorentzian Profile Example 2

“(b) Enlarged resonance measured in a device of the design shown in Fig. 6(a), using a bias voltage of 400 μeV. The data points (black dots) are fit to a Lorentzian line shape (solid line). For comparison we plot a thermally broadened resonance with a fitted temperature T ~34mK (dashed line) (from van der Vaart et al., 1995).”

  • W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha

and L. P. Kouwenhoven, “Electron Transport Through Double Quantum Dots”

  • Rev. Mod. Phys. 75 (2002) 1.
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SLIDE 16

Lorentzian Profile Example 3

  • M. Araidai and M. Tsukada, “Theoretical Calculation of Electron Transport in

Molecular Junctions: Inflection Behavior in Fowler-Nordheim Plot and its Origins”, Phys. Rev. B 81 (2010) 235114.

“Results obtained from the ab initio calculations of Ph-

  • SH. (a) and (b) are the F-N

plot of the current-voltage characteristics and the transmission function around the HOMO level at each bias

  • voltage. The solid line in (a)

is a guide for the eyes. In (b), the origin of energy is set to the averaged value of the Fermi energies between the left and right electrodes, and the longer dotted bar is the bias window at 0.4 V and the shorter one is that at 0.3 V.”

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SLIDE 17

Fano Resonance

Intensity x

  • U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts”,
  • Phys. Rev. 124 (1961) 1866.

V (x)=V 0

(qΓ+x−x0)

2

Γ

2+(x−x0) 2

Asymmetry parameter

Interference between resonant state and background oscillations.

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SLIDE 18

Fano Example 1

  • J. Göres, D. Goldhaber-Gordon, S. Heemeyer, M. a. Kastner, H. Shtrikman, D.

Mahalu and U. Meirav, “Fano Resonances in Electronic Transport through a single-electron transistor”, Phys. Rev. B 62 (2000) 2188.

“Conductance as a function

  • f plunger gate voltage for

various magnetic fields applied perpendicular to the 2DEG.”

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SLIDE 19

Fano Example 2

  • T. A. Papadopoulos, I. M. Grace and C. J. Lambert, “Control of Electron

Transport Through Fano Resonances in Molecular Wires”, Phys. Rev. B 74 (2006) 193306.

“Solid line: Electron transmission coefficient versus energy for the molecule having attached an oxygen atom as a side group. Dashed line: Oxygen bonds removed from the Hamiltonian.”

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SLIDE 20

Fano Example 3

  • M. L. Ladrón de Guevara, F. Claro and P. A. Orellana, “Ghost Fano Resonance

in a Double Quantum Dot Molecule Attached to Leads”, Phys. Rev. B 67 (2003) 195335. “Conductance as a function of the Fermi energy, for Δε = 0, tc = γ1 and different values of γ2/γ1: (a) 0, (b) 0.3, (c) 0.6, and (d) 1. The dash and dotted lines in (b) correspond, respectively, to Breit- Wigner and Fano line shapes.”

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SLIDE 21

Voigt Profile

Competing processes. Intensity x

V (x)=V 0[ G (x) G0 ∗ L( x) L0 ]

FWHM≈0.5346⋅FL+√0.2166⋅FL

2+FG 2

∂V ∂V 0 = V V 0

Σ

Very rare! Most easily Found in XRD, XPS, etc.

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SLIDE 22

Levenberg - Marquardt

  • K. Levenberg, Quart. Appl. Math. 2 (1944) 164.
  • D. Marquardt, J. Appl. Math. 11 (1963) 431.

x y Non-linear fitting with many parameters!

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SLIDE 23

[

f (x1;[k1+ δ1],[k2+ δ2]) f (x2;[k1+ δ1],[k2+ δ2]) ⋮

]

≈[ f (x1;k 1,k2)+ ∂f (x1) ∂k1 δ1+ ∂f (x1) ∂k 2 δ2 f (x2;k 1,k2)+∂f (x2) ∂k1 δ1+∂f (x2) ∂k 2 δ2 ⋮

]

=[ f (x1;k 1,k2) f (x2;k 1,k2) ⋮

]

+[ ∂f (x1) ∂k1 ∂f (x1) ∂k2 ∂f (x2) ∂k1 ∂f (x2) ∂k2 ⋮ ⋮ ][ δ1 δ2]

f ({x};k 1,k2)=sin(k1x)e

−k2 x

Example:

Δ F≈J δ

J

S≈|Y −F−J δ|

2

∂S ∂δ=0 (J

T J)δ=J T(Y−F)

(J

T J+λ I)δ=J T(Y −F)

F δ Levenberg damping term.

(J

T J+λ⋅diag(J T J))δ=J T(Y −F)

Marquardt fix.

f (⃗ x; ⃗ p+⃗ δ)≈ f (⃗ x; ⃗ p)+⃗ δ⋅∂ f ∂ ⃗ p

Jacobian Matrix

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SLIDE 24

1 2 3 4

x1 = 0.1; x2 = -0.3; s = 0.1; for it = 1:4 F = exp(-0.5*((x-x1)/s).^2)+exp(-0.5*((x-x2)/s).^2); J1 = ((x-x1)/(s*s)).*exp(-0.5*((x-x1)/s).^2); J2 = ((x-x2)/(s*s)).*exp(-0.5*((x-x2)/s).^2); J3 = (((x-x1).^2).*exp(-0.5*((x-x1)/s).^2)+((x-x2).^2).*exp(-0.5*((x-x2)/s).^2))/(s^3); J = [J1 J2 J3]; JT = transpose(J); delta = inv(JT*J+L*diag(diag(JT*J)))*JT*(Y-F); x1 += delta(1); x2 += delta(2); s += delta(3); plot(x,Y,'*b',x,F,'r'); sleep(1E-3); endfor;