Coulomb Blockade and Transport in a Chain of 1-dimensional Quantum Dots *

Transcription

1 Coulomb Blockade and Transport in a Chain of 1-dimensional 1 Quantum Dots * Thomas Nattermann Michael M.Fogler Sergey Malinin Cologne UCSD Detroit * e=h=k B =1 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 1

2 Introduction Quantum wire as a chain of quantum dots Transport in short wires Experiments Long wires: Rare events Conclusions Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 2

3 Electrons in one dimension x Breakdown of Landau s Fermi liquid picture density wave excitations (a) 1D clean wire: Luttinger liquid G = dj/dv = σ/l = (K) e 2 /h (b) single impurity : K<1 (>1): impurity (ir)relevant( ir)relevant, G (max (T,( ev)) 2/K-2 Kane & Fisher 92, Furusaki & Nagaosa 92,93 (c) Gaussian impurities: σ T 2-2K(T) 2K(T), T >> T 0 Giamarchi &Schulz 87 σ ~ exp(-t 0 /T) 1/2 T << T 0 T.N., Giamarchi,, Le Doussal 03 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 3

4 Quantum wire as a chain of quantum dots Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 4

5 Strong impurities: quantum wire as a chain of quantum dots "quantum dot" " with integer number q i of electrons x i x i+1 x x i+1 -x i =a i strong impurities,, randomly distributed Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 5

6 Luttinger liquid with strong impurities: charge density ρ(x) = 1 π (k F + x ϕ)[1 + 2 cos(2ϕ +2k F x)] S = 1 2πK v TR 0 dτ " L R 0 dx[( τ ϕ) 2 +( x ϕ) 2 ] N P i=1 w cos(2ϕ(x i )+2k F x i ) # K λ T =v/t thermal de Broglie wavelength w>>k F pinning strength x i+1 -x i =a i impurity spacing ϕ(x i )+k F x i =π N i integer number =N i+1 number q i =N i+1 -N i of electrons between impurities Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 6

7 Classical ground state K,v 0, compressibility g=k/(πv) finite S H T P j j 2T (q j Q j ) 2 q i =N i+1 -N i Δ j = 1/(ga j ) charging energy of dot Q=k F a j /π "background charge" Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 7

8 Classical ground state K,v 0, compressibility g=k/(πv) finite S H T P j j 2T (q j Q j ) 2 q i =N i+1 -N i Δ j = 1/(ga j ) charging energy of dot Q i =k F a j /π "background charge" T << Δ j q j =[Q j ] G, bifurcation dots: q j -Q j = ± ½ Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 8

9 Charged excitations ε j ± ε change number of electrons by ± 1 E j± = Δ j { 1/2 ± (q j -Q j ) } ± Coulomb blockade if E j± >> T μ E j + E j - x Bifurcation dots: a j q j -Q j = ± ½ E =0 E j+ +E j- = Δ j Example: Q=3/2, q=1, E + =0, E - =Δ Q=3/2, q=2, E + =Δ,, E - =0 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 9

10 Neutral excitations δ j = π v/a j = K Δ j S = 1 2πK NP j=0 a P h ω n ϕj+1,ω n ϕ j,ωn 2 ω n λ T sinh ω n a + ³ ϕ j,ωn 2 + ϕ j j+1,ωn 2 tanh ω na j 2 a i ω n =2πnT >> δ j, δ j+1 : ϕ(x j ) decoupled from neighbours λ T <<a j Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 10

11 Add external field: H F = -F dx ϕ(x)/ (x)/π Discrete energy levels (δ( j =KΔ j ) : energy conservation forbids tunneling except from rare dots R e α L Anderson et al (and many others) conductivity vanishes modification of the model: weak coupling to a bath Bath: (i) electrons in the gate (ohmic( ohmic) S d = η 4πλ T Pj Pω n ω n ϕ j (ω n ) 2 (ii) phonons Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 11

12 Transport in short wires Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 12

13 Transport at T=0 effective action S{ϕ(x i,τ)} on impurity sites classical metastale state ϕ i (τ) ϕ i = π ( N i + k F x i ), F>0 : Tunneling via instantons: : N N i i + 1 τ τ bifurcation dots Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 13

14 Transport at T=0 bifurcation dots: Ε + j + Ε j+m- Δ /m s(v 1,T) 2 K ln h i D max(δ,t,kv 1 ) S Y(Ε + j + Ε j+m- )/δ + m [s[ 0 + ln tanh (Y/2) - YV 1 /δ ] τ τ saddle point J Γ exp{-s inst /~} Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 14

15 Results from typical tunneling events: (short wires) Instanton hits boundary T=V T=V 1 T=V 1 K Δ δ Δ/s many impurity tunneling single impurity tunneling J/V 1 V1 2/K VRH J exp {-s({ s(δ/v 1 ) 1/2 } Δ Fa V 1 ξ loc a/s Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 15

16 Results from typical tunneling events: (short wires) Instanton hits boundary J/V 1 T 2/K T=V T=V 1 T=V 1 K J/V 1 sinh(v 1 /2T) xexp(-s-δ/2t) Δ δ J/V 1 exp {-{ (sδ/t) /T) 1/2 } Δ/s many impurity tunneling single impurity tunneling J/V 1 V 2/K 1 VRH s(v 1,T) 2 K ln h J exp {-s({ s(δ/v 1 ) 1/2 i D max(δ,t,kv 1 ) 1/2 } Δ Fa V 1 ξ loc a/s Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 16

17 Experiments Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 17

18 Different types of conductivity : (i) variable range hopping La 3 Sr 3 Ca 8 C 24 O 41 T K σ e -( T /T)1/2 0 σ e -(E (E /E)1/2 0 Carbon-nanotubes nanotubes: : Tang et al J e -(T (T0/T)/T) 1/2 Polydiacetylen : Aleshin et al Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 18

19 Different types of conductivity : (ii) Kane-Fisher behavior MoSe Nanowires Venkataraman, PRL (2006) J /T α+1 max ( V/T, V β+1 +1 /T α+1 ) JI a 1 T + short wires (L ~ 1 μm): Voltage Temperature Temperature Exponent α is close to Voltage Exponent β Agrees with the conventional Luttinger-liquid picture with α=β=2/k-2 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 19

20 Different types of conductivity : (iii) new behavior Polymer nanofibers long wires: 10μm I a 1 T + I βb = 1.5 a TV, V< T = b+ 1 V, V > T a α= 2.8 temperature exponent exceeds voltage exponent! J /T α+1 max ( V/T, V β+1 +1 /T α+1 ) Aleshin et al., PRL (2004) Voltage Temperature Disagrees with the conventional Luttinger-liquid picture Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 20

21 Observed Power-Law Exponents L ~10 μm polymers Aleshin et al., PRL (2004) Sample α β L ~ 100 μm InSb wires Zaitsev-Zotov et al., JPCM (2000) Sample α β In long wires T-exponent exceeds V-exponent: α > β Exponents are sample-dependent Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 21

22 Long wires: Rare events Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 22

23 4. Long wires: rare events So far: typical quantum dots with a i a Now: consider regions with many narrow dots with a i << a E Break : sequence of narrow dots μ ε x ε Probablity that no charge excitation in energy interval ε P ε (ε)) =1 - (ε/δ)(1 -e -Δ/ ε ) 1 ε/δ, ε/δ<<1 ε Δ/(2ε) ε/δ>>1 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 23

24 Wire: non-overlapping overlapping breaks + low resistance connecting pieces * break resistance R 0 e u total resistance R=L P u (u) R 0 e u du P u (u) ) : density of pieces with resistance e u * Ruzin and Raikh 1988 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 24

25 (a) Ohmic break of j-k dots E + R j k R 0 e s j k e 1 T E kj R o e u E j - simplification: rectangular break s j-k & u, Ε jk / T & u ln P u (u) u 2 T/Δ s if ut<< Δ P u (u) P ε (ε) j-k P ε (ut) u/s u/s ln P u (u) u/s ln (2uT/Δ) ) if ut>> Δ Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 25

26 Infinite wire R=L P u (u) R 0 e u du saddle point solution s<< 1 : ρ exp ( sδ/4t) s Raikh Ruzin 89 s>> 1 : ρ exp (Δ( e s /4sT) new break size a b a Δ e s /(4sT) >> a break distance L b a b exp [sa[ b /a] Example: s=5, Δ T a b /a 7,4 L b /a Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 26

27 Finite wire: P u (u)(l/a b ) 1 u max st ln(l/a b ) << Δ : ln R [sδ ln (L/a) / T ] 1/2 VRH 1/2 VRH Raikh Ruzin 89 st ln(l/a b ) >> Δ : ln R s ln (L/a b ) / ln (Ts/Δ) α lnt α = s(t) ln (L/a b ) / ln 2 (Ts/Δ)] Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 27

28 Numerics: Christophe Deroulers (unpublished) 1/ln R 1/ln R R ln (Ts/Δ)/ )/ln (L/l b ) K= impurities ak F =10 3 VRH ln (T/Δ) Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 28

29 (b) Non-Ohmic breaks of m dots J j k J 0 e s j k 1 T ε kj sinh[(ζ j ζ k )/T ] assumption: largest voltage drops V b =ξ j -ξ k >>T at a few breaks constant I =I 0 e -u everywhere u s j k + ε/t V b /T u u Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 29

30 (b) Non-Ohmic breaks prob. distribution P V (V b ) P_u(uT+V b /2) P u (u) ) exp[-v b /s T] V=L dv b P V (V b ) = V (u(j( u(j)) J(V,T) Different regimes (i) V b, ut<< Δ Fogler&Kelly (ii) ut>> >>Δ >>V b >>T (iii) ut>> >>V b >>Δ>>T >>T Fogler&Kelly (2005) u=-ln (J/J 0 )= s ln (2T/V 1 )/ ln(tu/δ) J V β, β= = s/ ln (Ts/Δ), α/β = ln(l/a b ) / ln(ts/δ)>1 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 30

31 Regime diagram T D ohmic regime non-ohmic ohmic regime s = log of tunneling transparency Δ = typical charging energy J T α J V β VI Δ/s < ln R> q s ln( /V 1 )/T Δ/s lnl VRH q < ln R> 2 2 /V 1 Δ/lnL s Δ e a/l Δ L l V 1 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 31

32 R= N j=1 R 0 exp u j R j l b x p(r j ) Rj 1 κ, κ = s 1 ln[ R TR ] j power law R j Levy random walk P(R) Levy distribution, κ << 1: Frechet distribution P R (R) R 1 κ exp [ (R /R) κ ] κ ln(r /R 0 )=ln(l b /κ ), κ = κ(r ) Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 32

33 Weak pinning: w,a 0, ξ 0 =(a/w 2 ) 1/3 fixed weak pinning a = ξ 3 0 w 2, strong pinning Giamarchi,, 2004: "Quantum Physics in One Dimension" T>T 1,cr : single impurity weak T>T 1,cr T<T 2 : collective effects weak strong Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 33

34 Conclusions: linear and non-linear conductivity, field and temperature cross-over over between single and many impurity tunneling low field and temperature: Mott-Shklovskii Shklovskii-VRH larger V,T: Kane-Fisher - power law behavior Kane-Fisher single-dominant-barrier theory is not valid in long wires that contains many (> 100?) impurities true power-law exponents exceed the single-barrier ones by a large log-factor (perhaps, by 2 or 3 in practice)! resistance is controlled by breaks dense clusters of impurities global weak/strong pinning regime diagram Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 34

35 Model: Bosonization: displacement from perfect order; Charge density ρ(x) = n δ(x x n ) x =n π/k n F -φ(x n )/k F ρ(x) =Ψ = + Ψ= = (k( F + x φ) n δ(k F x + φ(x) - π n) = π -1 (k F + x φ) - e 2mi(k F x+φ) Ψ + F(x)= [π[ -1 (k F + x φ)] )] 1/2 m odde im(k F x+φ ) e iθ(x (x) ; 2[ x θ(x) ),φ(x'), ] =iπ δ(x-x') H=(~ 2 /2m) dx Ψ 2 + 1/2 dx dy V(x-y) y)ρ(x) (x)ρ(y) H=(~/2 /2π) dx {v J ( x θ) 2 +v N ( x φ) 2 } v J =~k F /m Kv v N =(~π κ) -1 v/k K=1 : free electrons K CDWs, SDWs S/~ =(2π K) - 1 dx dτ {v - 1 ( τ φ) 2 +v( x φ) 2 +2KF φ(x) } Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 35

36 Large Voltage/temperature: power laws ε μ a x Larkin and Lee '78 Tunneling action dominated by spreading of charge: S tun dτ E(τ) dτ (gx(τ)) -1 -K -1 ln (E final /E initial ) s s eff s +2K -1 ln [min(1, KΔ/Τ, Δ /Fa)] T > Δ : J k,k+1 (max(t,v)/δ) 2/K Kane-Fisher 1992 Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 36

37 Strong and weak pinning Giamarchi,, 2004: "Quantum Physics in One Dimension" T>T 1,cr : single impurity weak T>T 1,cr T<T 2 : collective effects weak strong u<k F : SCHA u u u eff k k F (ak F ) K-1 u u F c KΔa/ξ = T strong e VRH -c T TΔ/s a/s T 2 T KΔ strong pinning weak pinning Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 37 loc T 2-2K(T) a T 2 K -2 T T 1,cr 2-2K weak F T

38 Experiments: sample dependence Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 38

39 Tunneling rate Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 39

40 Quantum creep: Tunneling in applied field A) Classical creep nucleation = formation of a critical droplet of free energy F droplet (f) ) = σ A surface - f V droplet creep velocity τ-1 e -F cd cd (f)/t F droplet (f) F cd (f) B) Quantum creep (see Coleman "fate of the false vacuum") nucleation in space-time = formation of an instanton of action f S instanton = σ A surface f V instanton creep velocity τ - 1 e -S inst inst(f)/ )/~ Hangzhou 2007 Sino-German Workshop on Disordered and Interacting Quantum Systems 40