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: 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) 196. “What power of the velocity is the resistance proportional to?” “...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...”

Least Squares Fitting y f ({ x } ; ⃗ v ) Test function: f ( x ;a,b )= ax + b Example: N R 2 = ∑ [ y i − a ⋅ x i − b ] 2 Error in fitting: x i Minimize the error: ∂ R 2 = 0 ∂ v i ∂ R 2 N N N b ∑ x i + a ∑ 2 = ∑ ∂ a = 0 x i y i x i i i i ∂ R 2 N N N ⋅ b + a ∑ x i = ∑ ∂ b = 0 y i i i

[ 2 ] [ a ] = [ x i y i ] 2 [ 2 ] [ x i y i ] N N N ∑ N N x i ∑ ∑ N x i ∑ y i y i [ a ] = b 1 b i i i i 2 − ( ∑ x i ) N N N N N N N N N ∑ ∑ ∑ ∑ x i ∑ x i x i ∑ x i x i ∑ i i i i i i i i 2 [ ] N N 1 2 − 1 y ∑ x ∑ x i x i y i N ¯ N ¯ [ a ] = 1 b i i N N 1 1 N ∑ 2 − ( ¯ N ∑ x i x ) x i y i − ¯ x ¯ y i i N 1 N ∑ x i y i − ¯ x ¯ y = σ ( x , y ) Covariance(x,y) i a = N σ 2 ( x ) 1 Variance(x) N ∑ 2 −( ¯ x ) 2 x i i y − ( y ) N N 1 1 N ∑ N ∑ x i y i − ¯ x ¯ 2 −( ¯ x ) 2 x i i i b = x = ¯ y − a ¯ x x ) 2 ¯ ¯ N N 1 1 N ∑ N ∑ 2 −( ¯ 2 −( ¯ x ) 2 x i x i i i

Octave Script clear; y % 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 x plot(x,y,'k*',x,a*x+b,'b'); N R 2 = 1 − ∑ ( y i − ¯ y ) Goodness of fit: i % for a noise level of +/- 3 → 94 N ∑ ( y i − f i ) i

Linearized Fit - exp(x) y A ⋅ y ln ( y )= ln ( y 0 )+ A ⋅ x y = y 0 ⋅ e 20 b A = cov ( ln ( y ) , x ) 15 ln ( y )− A ⋅ x y o = e var ( x ) 10 % Compute ly = log(ye); 5 x alfa = cov(ly,xe)/var(xe) 0 5 10 15 20 25 30 beta = exp(mean(ly)-alfa*mean(xe)) yf = beta*exp(alfa*x);

Linearized Fit – Power Laws A ln ( y )= ln ( y 0 )+ A ⋅ ln ( x ) y y = y 0 ⋅ x 6 5 A = cov ( ln ( y ) , ln ( x )) ln ( y )− A ⋅ ln ( x ) y o = e var ( ln ( x )) 4 3 % Compute ly = log(ye); 2 xl = log(xe); alfa = cov(ly,xl)/var(xl) 1 beta = exp(mean(ly)-alfa*mean(xl)) x 0 0 5 10 15 20 25 30 yf = beta*x.^alfa;

Gaussian Profile Independent random Intensity Variables. FWHM = 2 √ 2log 2 σ Σ x ∂ G = G Typically found in Density of ∂ G 0 G 0 States. 2 ( x − x 0 σ ) 2 − 1 = x − x 0 ∂ G ∂ G G ( x )= G 0 e G 0 ∂ x 0 2 ∂ G 0 σ ∂ σ = x − x 0 ∂ G ∂ G σ ∂ x 0

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. “In d , fitting the curve to a gaussian profile of the form I exp(-2 r 2 / 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 ).” Log of the intensity

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.

Gaussian Profile Example 3 Y. Zhang, B. de Boer and P. W. M. Blom, Phys. Rev. B 81 (2010) 085201. “Schematic representation of 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.”

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 = λ n + 1 −λ n 〈 λ n + 1 −λ n 〉 −π 4 s 2 p GOE =π 2 ⋅ s ⋅ e p GUE = 32 − 4 π s 2 2 ⋅ s 2 ⋅ e π − 64 18 p GSE = 2 9 π s 2 4 ⋅ e 3 6 π 3 ⋅ s

Lorentzian Profile Forced resonance on Intensity damped system. FWHM = 2 Γ x ∂ L = L Typically found in tunneling. ∂ L 0 L 0 2 Γ = 2 ( x − x 0 ) ( Γ ) 2 ∂ L L L ( x )= L 0 2 +Γ 2 ( x − x 0 ) ∂ x 0 L 0 ∂Γ =( x − x 0 ) ∂ L ∂ L Γ ∂ x 0

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).”

Lorentzian Profile Example 2 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. “(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).”

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.”

Fano Resonance Asymmetry parameter Intensity 2 ( q Γ+ x − x 0 ) V ( x )= V 0 2 + ( x − x 0 ) 2 Γ Interference between resonant state and background oscillations. x U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts”, Phys. Rev. 124 (1961) 1866.

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 of plunger gate voltage for various magnetic fields applied perpendicular to the 2DEG.”

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.”

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, t c = γ 1 and different values of γ 2 /γ 1 : (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.”

Voigt Profile Competing processes. Intensity Σ FWHM ≈ 0.5346 ⋅ F L + √ 0.2166 ⋅ F L 2 + F G 2 x V ( x )= V 0 [ L 0 ] G ( x ) ∗ L ( x ) Very rare! Most easily ∂ V = V Found in XRD, XPS, etc. ∂ V 0 V 0 G 0

Levenberg - Marquardt K. Levenberg, Quart. Appl. Math. 2 (1944) 164. D. Marquardt, J. Appl. Math. 11 (1963) 431. y Non-linear fitting with many parameters! x

− k 2 x f ({ x } ;k 1, k 2 )= sin ( k 1 x ) e Example: δ⋅∂ f p +⃗ p )+⃗ f (⃗ x; ⃗ δ)≈ f (⃗ x; ⃗ ∂ ⃗ p F J δ + [ ⋮ ] [ ≈ [ ] ∂ f ( x 1 ) ∂ f ( x 1 ) ∂ f ( x 1 ) ∂ f ( x 1 ) f ( x 1 ;k 1, k 2 )+ δ 1 + δ 2 [ ] = [ ] ∂ k 1 ∂ k 2 f ( x 1 ; [ k 1 + δ 1 ] , [ k 2 + δ 2 ]) ∂ k 1 ∂ k 2 f ( x 1 ;k 1, k 2 ) δ 2 ] δ 1 ∂ f ( x 2 ) ∂ f ( x 2 ) f ( x 2 ; [ k 1 + δ 1 ] , [ k 2 + δ 2 ]) f ( x 2 ;k 1, k 2 )+∂ f ( x 2 ) δ 1 +∂ f ( x 2 ) f ( x 2 ;k 1, k 2 ) δ 2 ⋮ ⋮ ∂ k 1 ∂ k 2 ∂ k 1 ∂ k 2 ⋮ ⋮ Jacobian Δ F ≈ J δ 2 S ≈ | Y − F − J δ | Matrix ∂ S T J )δ= J T ( Y − F ) ∂δ= 0 ( J T J +λ I )δ= J T ( Y − F ) ( J Levenberg damping term. T J +λ⋅ diag ( J T J ))δ= J T ( Y − F ) ( J Marquardt fix.