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: Coherent Transport

Coherent Transport P. Drude (1900): N ⟨ τ⟩= 1 t 1 N ∑ t 2 t i t N i ⃗ d ⟨ ⃗ p ⟩ P dt =⃗ F − τ l mfp A. Sommerfeld (1927): = √ 1 2 E F f ( E )= ⃗ | v | − E −μ F f m k B T + 1 e l mfp = v ⟨ τ⟩ 1 E/E F

τ v F l mfp Cu: 2.7 x 10 -14 s, 1.57 x 10 8 cm/s → ~42 nm Al: 0.8 x 10 -14 s, 2.03 x 10 8 cm/s → ~16 nm Si: 1.1 x 10 -12 s, 1.1 x 10 7 cm/s → ~120 nm Good 2DEG: 1 x 10 -10 s, 3 x 10 7 cm/s → ~30 um Phase relaxation elastic e iΦ e iΦ Si: ~40 nm e iξ Good 2DEG: ~200 nm inelastic L < L φ

Schrödinger 2 ∇ 2 2 2 H =−ℏ 2 m →− ℏ d ϕ ⟩ =ε | ϕ ⟩ H | AlGaAs GaAs GaAs 2 2 m dx Finite differences E C df ( x ) dx ≈ f [ n + 1 / 2 ]− f [ n − 1 / 2 ] Δ x 2 f ( x ) d ≈ f [ n + 1 ]− 2 f [ n ]+ f [ n − 1 ] 2 2 dx Δ x E v 2 ℏ H ≈− 2 f [ n + 1 ]− 2 f [ n ]+ f [ n − 1 ] 2 qm Δ x t

2 ϕ ( x ) | | clear; N = 50; hbar = 6.62606957E-34; % [J.s] m = 9.10938215E-31; % [kg] delta = 1E-9; % [m] q = 1.602176565E-19; % [C] t = -hbar^2/(2*q*m*delta^2); H = -2*t*eye(N) + diag(ones(N-1,1),1) + diag(ones(N- 1,1),-1); [ve va] = eig(H); va = diag(va); E1 = va(1); E2 = va(2); x E3 = va(3); p1 = abs(ve(:,1)).^2; p2 = abs(ve(:,2)).^2; p3 = abs(ve(:,3)).^2; x = linspace((-N/2),(N/2),N); plot(x,E1+p1,'r',x,E2+p2,'g',x,E3+p3,'b');

Magnetic Field q ⃗ 2 H =( i ℏ ∇+ A ) GaAs + U ( z ) 2 m AlGaAs B =∇×⃗ ⃗ A z = | A z | A y | ^ ^ ^ ^ ^ x y z x y GaAs = ( ∂ z ) ∂ A z ∂ y −∂ A y ∂ ∂ ∂ ∂ ∂ ⃗ B ^ ∂ x ∂ y ∂ z ∂ x ∂ y A x A y A x z = ( ∂ z ) ^ x + ( ∂ x ) ^ y + ( ∂ y ) ^ ∂ A z ∂ y −∂ A y ∂ A x ∂ z −∂ A z ∂ A y ∂ x −∂ A x B ^ z ⃗ B ^ z ^ z y ^ ⃗ A =− By ^ x ^ x 2 m [ ( i ℏ ∂ z ] 2 H = 1 ∂ x − qBy ) ^ x + ( i ℏ ∂ ∂ y ) ^ y + ( i ℏ ∂ ∂ z ) ^ + U ( z )

2 m [ ( i ℏ ∂ z ] 2 H = 1 GaAs ∂ x − qBy ) ^ x + ( i ℏ ∂ ∂ y ) ^ y + ( i ℏ ∂ ∂ z ) ^ + U ( z ) U ( z ) ] + 1 2 2 m ( i ℏ ∂ ∂ x − qBy ) H = [ H yz + GaAs ik x x ϕ(⃗ r )=ϕ 0 χ yz e B ^ z U ( z ) ] χ yz + 1 [ H yz + 2 χ yz =εχ yz ^ z 2 m ( ℏ k x + q B y ) ^ y ( q B + y ) 2 2 ( m ) 2 ℏ k x ^ x U ( z ) ] χ yz + m q B [ H yz + χ yz =εχ yz U ( z ) ] χ yz + 1 [ H yz + 2 ( y k + y ) 2 χ yz =εχ yz 2 m ω c ω c = qB m

y k =ℏ k x U ( z ) ] χ yz + 1 [ H yz + 2 ( y k + y ) 2 χ yz =εχ yz 2 m ω c q B χ yz =ψ( y )θ( z ) ε yz =ε y +ε z [ H y 2 ] ψ( y )=ε y ψ ( y ) 0 + 1 2 ( y k + y ) 2 m ω c [ H z 0 + U ( z ) ] θ( z )=ε z θ( z ) This was already solved! B ^ z ^ z ^ y E C x ^ E 3 Let us suppose E F such E 2 that there is only E F one subband occupied. E 1 E v

[ H y 2 ] ψ( y )=ε y ψ ( y ) 0 + 1 2 ( y k + y ) 2 m ω c Parabolic confinement! ε y =( n + 1 / 2 )ℏω c ε=ε 1 + ( n + 1 / 2 ) ℏω c ε z =ε 1 Independent of k x ! E No group velocity! Circular orbits! ∂ E 1 = 0 ℏ ∂ k x k x

y k =ℏ k x Δ y k = ℏΔ k x q B q B L Born von-Karman W k x = v ⋅ 2 π ik x ( x + L ) = e ik x x ik x L = 1 e e L 2 π spin B ^ z L ^ z Δ y k = ℏ 2 π N = W = 2 W q L B ^ y Total number ^ x q LB of levels: Δ y k ℏ 2 π n = q B ℏ π

Shubnikov – de Haas n Landau = n s B = 2 T n = q B = 5.2 n s = 5 x 10 11 cm -2 q B ℏ π ℏ π E n s n s − = 1 E F q B i qB i + 1 ℏ π ℏ π k x ( h ) ( B i + s ) = s n s 1 − 1 B ^ z 2 q B i 3 ^ y 2 x ^ 1 1 B L. Shubnikov and W. de Haas, Leiden Comm. 207a (1930) 3.

Real Data index B 1/B 1 6 2.44 0.41 2 3 4 5 3 0.33 5 4 3.71 0.27 3 4.12 0.24 2 4.78 0.21 i 1 5.16 0.19 ( h ) ( B i + s ) = s n s 1 − 1 2 q B i n s = 1.06 x 10 12 cm -2 1/B

Multiple Subbands E C E 3 E F E 2 E 1 E v N. Aoki, C. R. da Cunha, R. Akis, D. K. Ferry and Y. Ochiai, J. Phys.: Condens. Matter 26 (2014) 193202. n -In 0.53 Al 0.47 As (5 nm - cap) n -In 0.53 Al 0.47 As (30 nm - doping) i -In 0.53 Al 0.47 As (10 nm - spacer) i -In 0.53 Ga 0.47 As (25 nm - QW) i -In 0.53 Al 0.47 As (10 nm - spacer) InP Substrate

T. Ando, A. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. Mobility Factor ω c = qB l a m g r n e * h i p m T m a D = D ( T , ω c )⋅ M (ω c )⋅ cos ( ℏ ω c ⋅ g s ⋅ g v ) Δρ xx ( T , ω c ) 2 π E F ρ 0 Spin & Valley Degeneracies − π ω c τ M (ω c )= e D ( T , ω c )= X ( T )⋅ csch ( X [ T ]) 2 k B T X ( T )= 2 π ℏ ω c

Effective Mass ⋅ csch ( ℏ ω c ) ⋅ M (ω c )⋅ C (ω c ) 2 k B T 2 k B T Δρ xx ( T , ω c ) = 2 π 2 π ω c = qB ρ 0 ℏω c * m ( B ) ( B ) 1 1 1 K 0.5 K ≈ 1 2 k B T − 2 π Δρ xx ℏ ω c ρ 0 T ≈ C ⋅ e 2 k B log ( Δρ xx ρ 0 T ) ≈ C − 2 π T ( B ) ( B ) 1 1 ℏω c 0.1 K 0.3 K 2 k B m * ( B ) T log ( Δρ xx ρ 0 T ) ≈ C − 2 π 1 ℏ q P. T. Coleridge, M. Hayne, P. Zawadzki and A. S. Sachrajda, Surf. Sci. 361/362 (1996) 560.

2 k B m * 2 k B m * log ( ρ 0 T ) ≈ C − 2 π ( B ) T slope =− 2 π ( B ) Δρ xx 1 1 ℏ q ℏ q ℏ q m ef (calc.) = 1.07, m ef (exact) = 1.00 * =− m slope 2 k B m 0 ( 1 / B ) 2 π T = [1 0.5 0.3 0.1 0.05]; A = [0.0053 0.1375 0.41 0.81 0.88]; lA = log(A./T); S = cov(T,lA)/var(T); m_eff = -S*hbar*q/(2*pi^2*0.53*mass*kB) semilogy(T,A./T,'*'); grid on; xlabel('Temperature [K]'); ylabel('Ln A/T');

Electron Density: 2DEG spin 2 2 ε F = ℏ 2 k F 2 N = 2 π k F n = k F ( L ) 2 π 2 2 m * 2 π k F n = m ε F π ℏ 2 2 π 2 ε F = πℏ L * n m Density of States: g (ε)= ∂ n (ε) n (ε)= m ε = m Independent of ε!! ∂ε πℏ 2 π ℏ 2

= D ( T , ω c )⋅ M (ω c )⋅ cos ( ℏ ω c ⋅ g s ⋅ g v ) Δρ xx ( T , ω c ) 2 π E F 2 ε F = πℏ ω c = qB * n ρ 0 * m m NFFT = 128; x_axis = linspace(min(1./B),max(1./B),L); y_axis = interp1(1./B,sxx,x_axis); fs = 1/(x_axis(2)-x_axis(1)); z = fft(y_axis,NFFT); z = z.*conj(z)/(NFFT*L); z = abs(z(1:NFFT/2)); x = fs*(0:(NFFT/2-1))/NFFT; [m im] = max(z); n (comp.) = 1.90 x 10 15 cm -2 wo = 2*pi*x(im) n (exact.) = 2.00 x 10 15 cm -2 n = (g_s*g_v*q/(2*pi^2*hbar))*wo plot(x,z); grid on; xlabel('Frequency [T]'); ylabel('FFT Power');

Mobility ω c = qB * m Δρ xx ( T , ω c ) − π τ ω c ⋅ C (ω c ) log ( ρ 0 D ( T , ω c ) ) = C − π = D ( T , ω c )⋅ e Δρ xx ( T , ω c ) * τω c =−π m 1 ρ 0 τ q B DINGLE PLOT wc = q*B/(mass*m_eff); X = 2*(pi^2)*kB*T./(hbar*wc); D = X.*csch(X); oB = [0.536 0.574 0.613 0.652 0.690]; m = [0.437 0.401 0.378 0.357 0.337]; lm = log(m); S = cov(oB,lm)/var(oB); tau = -pi*mass*m_eff/(q*S) Gamma = hbar/(2*tau*q) τ(calc.) = 1.16 x 10 -11 TD = Gamma / (pi*kB) τ(exac.) = 1.00 x 10 -11 semilogy(1./B,abs(sxx)./D); grid on; xlabel('1/B [T^{-1}]'); ylabel('log |\delta\rho_{xx}|');

Example: Graphene Z. Tan, C. Tan. L. Ma, G. T. Liu and C. L. Yang, Phys. Rev. B 84 (2011) 115429. μ > 8.000 cm² / V.s

What is happening? [ H y 2 ] ψ( y )=ε y ψ ( y ) y k =ℏ k x 0 + 1 2 ( y k + y ) 2 m ω c q B E E F k x Suppression of backscattering!

Quantum Hall Effect equilibrium ∂ε k I = q f (ε[ k ]) v = q f (ε[ k ]) 1 L ∑ L ∑ ℏ ∂ k k k spin ∞ dk f (ε) ∂ε k I = 2 × L 1 q L ∫ μ L μ R 2 π ℏ ∂ k −∞ μ L for M modes. I = 2 q h ∫ d ε M (ε) f (ε) μ R equilibrium 2 (μ L −μ R ) I = 2 q M M 3 h q 2 1 K. v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. ε 1 ε 2 ε 3 ε

Example: Graphene Y. Zhang, Y.-W. Tan, H. L. Stormer and P. Kim, Nature 438 (2005) 201.

Scanning Gate N. Aoki, C. R. da Cunha, R. Akis, D. K. Ferry and Y. Ochiai, J. Phys.: Condens. Matter 26 (2014) 193202.