Experimental Methods in Transport Physics Prof. Carlo Requio da - - PowerPoint PPT Presentation
Experimental Methods in Transport Physics Prof. Carlo Requio da Cunha, Ph.D. unit: Coherent Transport Coherent Transport P. Drude (1900): N = 1 t 1 N t 2 t i t N i d p P dt = F l mfp A. Sommerfeld
unit: Coherent Transport
i N
t1 t2 tN
1 E/EF f
−E−μ F k BT +1
lmfp
Cu: 2.7 x 10-14 s, 1.57 x 108 cm/s → ~42 nm Al: 0.8 x 10-14 s, 2.03 x 108 cm/s → ~16 nm τ vF lmfp Good 2DEG: 1 x 10-10 s, 3 x 107 cm/s → ~30 um Si: 1.1 x 10-12 s, 1.1 x 107 cm/s → ~120 nm Phase relaxation elastic inelastic eiΦ eiΦ eiξ Good 2DEG: ~200 nm Si: ~40 nm L < Lφ
GaAs GaAs AlGaAs EC Ev
2∇ 2
2
2
2
Finite differences
2f (x)
2
2
2
2 f [n+1]−2f [n]+f [n−1]
t
2 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); 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');
GaAs AlGaAs GaAs
2
⃗ B=∇×⃗ A
⃗ B ^ z=| ^ x ^ y ^ z ∂ ∂ x ∂ ∂ y ∂ ∂ z Ax A y A z| ^ x ^ y ∂ ∂ x ∂ ∂ y Ax A y| =( ∂ Az ∂ y −∂ A y ∂ z ) ⃗ B ^ z=( ∂ Az ∂ y −∂ A y ∂ z ) ^ x+( ∂ Ax ∂ z −∂ Az ∂ x ) ^ y+( ∂ A y ∂ x −∂ Ax ∂ y ) ^ z
2
^ y ^ x ^ z
2
2
GaAs GaAs
^ y ^ x ^ z
ik xx
[H yz+
2χ yz=εχ yz
[H yz+
2
2
[H yz+
2( yk+y ) 2χ yz=εχ yz
[H yz+
2( yk+y ) 2χ yz=εχ yz
0+1
2( y k+y) 2]ψ( y)=ε y ψ( y)
0+U(z)]θ( z)=εzθ(z)
^ y ^ x ^ z
This was already solved! EC Ev EF Let us suppose EF such that there is only
E1
E2 E3
0+1
2( y k+y) 2]ψ( y)=ε y ψ( y)
Parabolic confinement!
Independent of kx! kx E No group velocity!
Circular orbits!
ikx (x+L)=e ikx x
Born von-Karman
ikx L=1
2π L
^ y ^ x ^ z
W L Total number
spin
^ y ^ x ^ z
B = 2 T B ns = 5 x 1011 cm-2
E kx EF
1
1 2 3
index B 1/B 6 2.44 0.41 5 3 0.33 4 3.71 0.27 3 4.12 0.24 2 4.78 0.21 1 5.16 0.19 1 2 3 4 5 1/B i
ns 2q h )( 1 Bi − 1 Bi+s)=s
ns = 1.06 x 1012 cm-2
EC Ev EF
E1
E2 E3
n-In0.53Al0.47As (5 nm - cap) n-In0.53Al0.47As (30 nm - doping) i-In0.53Al0.47As (10 nm - spacer) i-In0.53Ga0.47As (25 nm - QW) i-In0.53Al0.47As (10 nm - spacer) InP Substrate
*
2k BT
− π ωcτ Mobility Factor
T h e r m a l D a m p i n g Spin & Valley Degeneracies
2k BT
2k BT
ωc=qB m
*
1 K 0.5 K 0.3 K 0.1 K
−2π
2kBT
ℏ ωc
log( Δρxx ρ0T )≈C−2 π
2k B
ℏωc T log( Δρxx ρ0T )≈C−2 π
2k Bm *
(
1 B)
(
1 B)
(
1 B)
(
1 B)
2k Bm *
2kB m *
*=−
2k Bm0(1/ B)
mef (calc.) = 1.07, mef (exact) = 1.00
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');
2π L
kF
N =2 π k F
2
2π L )
2
spin
n= k F
2
2π εF= ℏ2k F
2
2m* n= mεF π ℏ2
2
* n
n(ε)= mε π ℏ2 g(ε)= ∂ n(ε) ∂ε = m πℏ2
Independent of ε!! Density of States:
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); wo = 2*pi*x(im) n = (g_s*g_v*q/(2*pi^2*hbar))*wo plot(x,z); grid on; xlabel('Frequency [T]'); ylabel('FFT Power');
2
* n
ωc=qB m
*
n (comp.) = 1.90 x 1015 cm-2 n (exact.) = 2.00 x 1015 cm-2
− π τ ωc⋅C(ωc)
ωc=qB m
*
log( Δρxx(T ,ωc) ρ0 D(T ,ωc))=C− π τωc =−πm
*
τ q 1 B
wc = q*B/(mass*m_eff); X = 2*(pi^2)*kB*T./(hbar*wc); D = X.*csch(X);
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) TD = Gamma / (pi*kB) semilogy(1./B,abs(sxx)./D); grid on; xlabel('1/B [T^{-1}]'); ylabel('log |\delta\rho_{xx}|');
τ(calc.) = 1.16 x 10-11 τ(exac.) = 1.00 x 10-11 DINGLE PLOT
84 (2011) 115429.
μ > 8.000 cm² / V.s
0+1
2( y k+y) 2]ψ( y)=ε y ψ( y)
kx E
EF Suppression of backscattering!
μL μR equilibrium equilibrium
k
k
−∞ ∞
spin
μR μL
for M modes.
ε1 ε2 ε3 ε
M
1
2 3
2