Charge carrier statistics/shot Noise

Transcription

1 Charge carrier statistics/shot Noise Sebastian Waltz Department of Physics 16. Juni 2010 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

2 Outline 1 Charge carrier statistics Short statistic review Counting of electron transfers 2 Current noise time dependence of the current General expression for noise Equilibrium noise Poisson-Value 3 Examples for noise Tunnel barriers Quantum Point contacts Metallic diffusive wires Chaotic cavities S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

3 Introduction Experiments measure the average of many readings of a measuring device current time S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

4 Introduction Experiments measure the average of many readings of a measuring device current time electron transfer is a stochastic process number of electrons traversing the nano structure during t is random S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

5 Charge carrier statistics Charge carrier statistics S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

6 Short statistic review Charge carrier statistics Short statistic review Some general statistical properties average N = 1 M N N S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

7 Short statistic review Charge carrier statistics Short statistic review Some general statistical properties average N = 1 M N N probability distribution probability P N that precisely N events occur N P N = 1 N = N NP N S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

8 Short statistic review Charge carrier statistics Short statistic review Some general statistical properties average N = 1 M N N probability distribution probability P N that precisely N events occur N P N = 1 N = N NP N characteristic function Λ(χ) = e iχn = N P N e iχn the kth cumulant = C 1 = mean C 2 = variance C 3 = skewness C 4 = sharpness k ln (Λ(χ)) χ=0 (iχ) k S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

9 Short statistic review Charge carrier statistics Short statistic review Gausssian distribution Λ(iχ) = e iµχ+1/2σ(iχ)2 (iχ) ln (Λ(χ)) χ=0 = µ 2 (iχ) 2 ln (Λ(χ)) χ=0 = σ 2 k (iχ) k ln (Λ(χ)) χ=0 = 0 for k > 2 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

10 Short statistic review Charge carrier statistics Short statistic review Gausssian distribution Λ(iχ) = e iµχ+1/2σ(iχ)2 Possian distribution ( ) Λ(iχ) = exp λ(e iχ 1) (iχ) ln (Λ(χ)) χ=0 = µ (iχ) ln (Λ(χ)) χ=0 = λ 2 (iχ) 2 ln (Λ(χ)) χ=0 = σ 2 2 (iχ) 2 ln (Λ(χ)) χ=0 = λ k (iχ) k ln (Λ(χ)) χ=0 = 0 for k > 2 k (iχ) k ln (Λ(χ)) χ=0 = λ S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

11 Counting of electron transfers Charge carrier statistics Counting of electron transfers Event an event is the transfer of an electron from one reservoir to another S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

12 Counting of electron transfers Charge carrier statistics Counting of electron transfers Event an event is the transfer of an electron from one reservoir to another we know: Q = I t there t is long enough, so that Q e S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

13 Counting of electron transfers Charge carrier statistics Counting of electron transfers Event an event is the transfer of an electron from one reservoir to another we know: Q = I t there t is long enough, so that Q e the goal: statistical properties of the random variable Q S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

14 Counting of electron transfers Charge carrier statistics Counting of electron transfers Event an event is the transfer of an electron from one reservoir to another we know: Q = I t there t is long enough, so that Q e the goal: statistical properties of the random variable Q two simple limiting cases 1 electrons are transferred only in one direction electron transfers are uncorrelated S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

15 Counting of electron transfers Charge carrier statistics Counting of electron transfers Event an event is the transfer of an electron from one reservoir to another we know: Q = I t there t is long enough, so that Q e the goal: statistical properties of the random variable Q two simple limiting cases 1 electrons are transferred only in one direction electron transfers are uncorrelated 2 ideally transmitting channel at zero temperature S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

16 Case 1 Charge carrier statistics Counting of electron transfers 1 divide t in very short intervals dt neglect two electron processes S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

17 Case 1 Charge carrier statistics Counting of electron transfers 1 divide t in very short intervals dt neglect two electron processes probability for transferring an electron during dt: Γ being the transfer rate Γdt S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

18 Case 1 Charge carrier statistics Counting of electron transfers 1 divide t in very short intervals dt neglect two electron processes probability for transferring an electron during dt: Γ being the transfer rate Γdt probability for transferring no electron during dt: 1 Γdt S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

19 Case 1 Charge carrier statistics Counting of electron transfers 1 divide t in very short intervals dt neglect two electron processes probability for transferring an electron during dt: Γ being the transfer rate Γdt probability for transferring no electron during dt: 1 Γdt 2 calculating the characteristic function for the short interval Λ dt (χ) = e iχq/e = (1 Γdt) + (Γdt)e iχ }{{}}{{} Q=0 Q=1 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

20 Case 1 Charge carrier statistics Counting of electron transfers 3 calculating the total characteristic function Λ t(χ) uncorrelated = (Λ dt (χ)) t/dt = ((1 Γdt) + (Γdt)e iχ) t/dt t dt ( ) exp Γ t(e iχ 1) ) = exp (Ñ(e (iχ) 1) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

21 Case 1 Charge carrier statistics Counting of electron transfers 3 calculating the total characteristic function Λ t(χ) uncorrelated = (Λ dt (χ)) t/dt = ((1 Γdt) + (Γdt)e iχ) t/dt t dt ( ) exp Γ t(e iχ 1) ) = exp (Ñ(e (iχ) 1) with Ñ Γ t = Q /e being the average number of transferred electrons S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

22 Case 1 Charge carrier statistics Counting of electron transfers 4 taking the inverse Fourier Transform P N = 2π 0 2π 0 = Ñ N N! e Ñ t dχ 2π Λ(χ)e iχn dχ iχ 1) 2π e iχn+ñ(e Poisson distribution hist Entries 30 Mean 9 RMS S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

23 Case 1 Charge carrier statistics Counting of electron transfers 4 taking the inverse Fourier Transform P N = 2π 0 2π 0 = Ñ N N! e Ñ t dχ 2π Λ(χ)e iχn dχ iχ 1) 2π e iχn+ñ(e Poisson distribution hist Entries 30 Mean 9 RMS uncorrelated transfer is described by Poisson statistics uncorrelated transfer occurs in tunnel junctions (1/Γ time needed to cross the junction) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

24 Case 2 Charge carrier statistics Counting of electron transfers ideally transmitting channel at zero temperature electrons are ideal wave states electron momentum in transport direction is a well defined quantum number, which does not fluctuate S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

25 Case 2 Charge carrier statistics Counting of electron transfers ideally transmitting channel at zero temperature electrons are ideal wave states electron momentum in transport direction is a well defined quantum number, which does not fluctuate = current does not fluctuate P N = δ(n Ñ) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

26 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Intermediate case 0 < T < 1 The many-channel, finite temperature result for the characteristic function is given by the Levitov formula ln(λ(χ)) =2 s t de 2π ln{1 + T n(e iχ 1)f L(E)[1 f R(E)] n + T n(e iχ 1)f R(E)[1 f L(E)]} S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

27 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Intermediate case 0 < T < 1 The many-channel, finite temperature result for the characteristic function is given by the Levitov formula ln(λ(χ)) =2 s t de 2π ln{1 + T n(e iχ 1)f L(E)[1 f R(E)] n + T n(e iχ 1)f R(E)[1 f L(E)]} Interesting properties sum over transport channels suggests that electron transfers in different channels are independent the electron transfers from the left to the right is dependent from the transfers from the right to the left S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

28 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Intermediate case 0 < T < 1 The many-channel, finite temperature result for the characteristic function is given by the Levitov formula ln(λ(χ)) =2 s t de 2π ln{1 + T n(e iχ 1)f L(E)[1 f R(E)] n + T n(e iχ 1)f R(E)[1 f L(E)]} low temperature limit (E F ev k B T ): T n(e) =const. = T n(e F ) = T n ln(λ(χ)) = 2seV t 2π n ( )] ln [1 + T n e iχ 1 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

29 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Characteristic function for a single channel 2seV t ( )] ln(λ(χ)) = ln [1 + T n e iχ 1 2π n Λ(χ) = n Λn(χ) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

30 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Characteristic function for a single channel 2seV t ( )] ln(λ(χ)) = ln [1 + T n e iχ 1 2π n Λ(χ) = n Λn(χ) with the single channel characteristic function Λ n = ( ) 2seV t (1 T n) + (T ne iχ 2π ) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

31 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers Characteristic function for a single channel 2seV t ( )] ln(λ(χ)) = ln [1 + T n e iχ 1 2π n Λ(χ) = n Λn(χ) with the single channel characteristic function Λ n = ( ) 2seV t (1 T n) + (T ne iχ 2π ) assuming that the number of attempts N at = 2 s tev/2π is an integer Λ n = N at N=0 ( N at N ) T N n (1 T n) N at N e inχ S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

32 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers single channel characteristic characterized by binomial distribution Λ n = N at N=0 ( N at N ) T N n (1 T n) N at N e inχ taking the inverse Fourier Transform with respect to χ ( ) P (n) N = N at Tn N (1 T n) N at N N which is the binomial distribution S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

33 Intermediate case 0 < T < 1 Charge carrier statistics Counting of electron transfers single channel characteristic characterized by binomial distribution Λ n = N at N=0 ( N at N ) T N n (1 T n) N at N e inχ taking the inverse Fourier Transform with respect to χ ( ) P (n) N = N at Tn N (1 T n) N at N N which is the binomial distribution for more than one channel, we obtain a convolution of binomial distributions corresponding to each channel S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

34 Current noise Current noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

35 time dependence of the current Current noise time dependence of the current the device what we have: two reservoirs (L, R) large enough (const. temperature T L,R and const. chemical potential µ L,R) electrons in the reservoirs are Fermi distributed ideal leads connecting the sample (quantum dod, quantum wire...) with the reservoirs S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

36 time dependence of the current Current noise time dependence of the current the device what we have: two reservoirs (L, R) large enough (const. temperature T L,R and const. chemical potential µ L,R) electrons in the reservoirs are Fermi distributed ideal leads connecting the sample (quantum dod, quantum wire...) with the reservoirs our system we deal with a large system of indistinguishable particles the wave functions have to be symmetric (bosons) or antisymmetric (fermions) different statistical properties for the scattering process different current noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

37 time dependence of the current Current noise time dependence of the current with ˆn + Ln (E, t), time-dependent occupation number operator for right moving carriers in the lead L ˆn Ln (E, t), time-dependent occupation number operator for left moving carriers in the lead L S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

38 time dependence of the current Current noise time dependence of the current with ˆn + Ln (E, t), time-dependent occupation number operator for right moving carriers in the lead L ˆn Ln (E, t), time-dependent occupation number operator for left moving carriers in the lead L the equation for the time dependent current operator Î L(t) = e 2π n de[ˆn + Ln(E, t) ˆn Ln(E, t)] S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

39 time dependence of the current Current noise time dependence of the current with ˆn + Ln (E, t), time-dependent occupation number operator for right moving carriers in the lead L ˆn Ln (E, t), time-dependent occupation number operator for left moving carriers in the lead L the equation for the time dependent current operator Î L(t) = e 2π n de[ˆn + Ln(E, t) ˆn Ln(E, t)] some plausible arguments: the integrand is just the difference of the left and right moving charge carrier the integration sums up all possible wave numbers in longitudinal direction the sum collects all open channels S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

40 time dependence of the current Current noise time dependence of the current with the Î L(t) = e 2π annihilation operator â Ln (ˆb Ln) creation operator â Ln (ˆb Ln ) n de[ˆn + Ln(E, t) ˆn Ln(E, t)] for the carriers moving to the right (left) = Î L(t) = e 2π n de[â Ln (E, t)âln(e, t) ˆb Ln (E, t)ˆb Ln(E, t)] S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

41 Current noise time dependence of the current time dependence of the current â and ˆb are related to each other by the scattering Matrix s:... ˆbj... = s... â i... as well as the creation operators â and ˆb by s S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

42 Time dependence of the current Current noise time dependence of the current average of the ladder operators for calculating the average current at equilibrium we need: â αm(e)â βn (E ) = δ αβ δ mnδ(e E )f α(e) with this we get for the average current: ÎL(t) = e 2π n de T n(e)(f L(E) f R(E)) T n(e) = transmission probability S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

43 noise power Current noise General expression for noise the second cumulant the auto correlation function S(t t ) (Î(t) Î(t )) 2 Î2 = can be used as an measurement of the noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

44 noise power Current noise General expression for noise the second cumulant the auto correlation function S(t t ) (Î(t) Î(t )) 2 Î2 = can be used as an measurement of the noise for t = t = S(0) = (Î(t) Î(t))2 is nothing else than the second cumulant (variance) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

45 noise power Current noise General expression for noise the second cumulant the auto correlation function S(t t ) (Î(t) Î(t )) 2 Î2 = can be used as an measurement of the noise for t = t = S(0) = (Î(t) Î(t))2 is nothing else than the second cumulant (variance) the Fourier transform of this correlation function πδ(ω + ω )S(ω) = Î(ω) Î(ω ) is often called the noise power S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

46 noise power Current noise General expression for noise from now on we are only interested in the zero-frequency noise S = S(0) using the properties of the scattering matrix s we obtain after some algebra for the noise power S = e2 de[t n(e) {f L(E) (1 f L(E)) + f R(E) (1 f R(E))} π n ± T n(e)[1 T n(e)](f L(E) f R(E)) 2 ] S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

47 noise power Current noise General expression for noise from now on we are only interested in the zero-frequency noise S = S(0) using the properties of the scattering matrix s we obtain after some algebra for the noise power S = e2 de[t n(e) {f L(E) (1 f L(E)) + f R(E) (1 f R(E))} π n ± T n(e)[1 T n(e)](f L(E) f R(E)) 2 ] there the upper sign stands for the Fermi statistic and the lower sign for the Bose statistic f(e) is understood also for Fermi or Bose statistic S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

48 Equilibrium noise Current noise Equilibrium noise Nyquist-Johnson noise at equilibrium f L(E) = f R(E) = f(e) f(1 f) = k BT f E S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

49 Equilibrium noise Current noise Equilibrium noise Nyquist-Johnson noise at equilibrium f L(E) = f R(E) = f(e) f(1 f) = k BT f E = S = 2e2 k BT π n ( de f ) T n(e) = 4k BT G E with G being the conductance S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

50 Equilibrium noise Current noise Equilibrium noise Nyquist-Johnson noise at equilibrium f L(E) = f R(E) = f(e) f(1 f) = k BT f E = S = 2e2 k BT π n ( de f ) T n(e) = 4k BT G E with G being the conductance the equilibrium noise is due to fluctuations of the occupation number in the reservoirs S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

51 Equilibrium noise Current noise Equilibrium noise Nyquist-Johnson noise at equilibrium f L(E) = f R(E) = f(e) f(1 f) = k BT f E = S = 2e2 k BT π n ( de f ) T n(e) = 4k BT G E with G being the conductance the equilibrium noise is due to fluctuations of the occupation number in the reservoirs no further information of the system beyond that already known from conductance measurement S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

52 noise power Current noise Equilibrium noise contributions to the noise S = e2 de[t n(e) {f L(E)(1 f L(E)) + f R(E)(1 f R(E))} π n ± T n(e)[1 T n(e)](f L(E) f R(E)) 2 ] 1 equilibrium noise (fl(e) = f R(E)) 2 non equilibrium or shot noise different sign for bosons or fermions can be neglected at high energies S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

53 Current noise noise power practically important case for fermions T n(e) T n const. T n(e F ) Equilibrium noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

54 noise power Current noise practically important case for fermions T n(e) T n const. T n(e F ) n Equilibrium noise S = e2 de[t n(e) {f L(E)(1 f L(E)) + f R(E)(1 f R(E))} π + T n(e)[1 T n(e)](f L(E) f R(E)) 2 ] [ e2 2k ( ) ] BT Tn 2 ev + ev coth T n(1 T n) π 2k n BT n S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

55 noise power Current noise practically important case for fermions T n(e) T n const. T n(e F ) n Equilibrium noise S = e2 de[t n(e) {f L(E)(1 f L(E)) + f R(E)(1 f R(E))} π + T n(e)[1 T n(e)](f L(E) f R(E)) 2 ] [ e2 2k ( ) ] BT Tn 2 ev + ev coth T n(1 T n) π 2k n BT n ev 0 S = e2 π [2kBT n ev 0 2e2 k BT π ( ) Tn 2 ev + ev coth 2k BT }{{} ev 0 2k B T T n = Nyquist Noise n T n(1 T n)] n S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

56 Poisson value Current noise Poisson-Value Poisson value (T 0) S = e2 π [2kBT n T 0 e3 V π ( ) Tn 2 ev + ev coth 2k BT }{{} T n(1 T n) n T 0 e V T n(1 T n)] n S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

57 Poisson value Current noise Poisson-Value Poisson value (T 0) S = e2 π [2kBT n T 0 e3 V π ( ) Tn 2 ev + ev coth 2k BT }{{} T n(1 T n) n T 0 e V T n(1 T n)] neither closed (T n = 0) nor open (T n = 1) channels contribute to shot noise n S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

58 Poisson value Current noise Poisson-Value Poisson value (T 0) S = e2 π [2kBT n T 0 e3 V π ( ) Tn 2 ev + ev coth 2k BT }{{} T n(1 T n) n T 0 e V T n(1 T n)] neither closed (T n = 0) nor open (T n = 1) channels contribute to shot noise n In the low transparency limit T n 1 we obtain the Poisson value S P = e3 V π T n = 2e I n S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

59 Current noise General expression for the Poisson value Poisson-Value Poisson value (T 0) for arbitrary carrier charge (e ) we can make the same derivation: S P = 2e I measure for carrier charge fractional quantum Hall effect = e = 1/3e superconductors = Cooper pairs e = 2e S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

60 Fano factor F Current noise Poisson-Value Fano factor A measure for the sub-poisson shot noise is the Fano factor F F = S 2e I S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

61 Fano factor F Current noise Poisson-Value Fano factor A measure for the sub-poisson shot noise is the Fano factor F F = S 2e I F assumes values between 0 all channels are transparent (T n = 1) 1 Poissonian noise (T n 1) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

62 Examples for noise Examples for noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

63 Examples for noise Tunnel barriers Tunnel barriers Tunnel barriers all transmission coefficients T n are small (T n 1) neglect quadratic terms T 2 n S = e3 V π [ 2k BT n ( ) ] ev coth T n = e3 V 2k BT π n n ( ) e V coth S p 2k BT S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

64 Examples for noise Tunnel barriers Tunnel barriers Tunnel barriers all transmission coefficients T n are small (T n 1) neglect quadratic terms T 2 n S = e3 V π [ 2k BT n ( ) ] ev coth T n = e3 V 2k BT π n n ( ) e V coth S p 2k BT At a given temperature it describes the crossover from thermal noise at voltages e V k BT to shot noise at voltages e V k BT S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

65 Quantum point contact Examples for noise Quantum Point contacts a quantum point contact is defined as a constriction between two metallic reservoirs the constriction is assumed to be shorter than the mean free path transport is ballistic S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

66 Quantum point contact Examples for noise Quantum Point contacts a quantum point contact is defined as a constriction between two metallic reservoirs the constriction is assumed to be shorter than the mean free path transport is ballistic S = e3 V π T n(1 T n) n at the plateaus we have T n = 1 for open channels and T n = 0 for closed channels no shot noise then the Fermi energy lies close to the top of the potential barrier the transmission coefficient for this channel increases from 0 to 1 shot noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

67 Quantum point contact Examples for noise Quantum Point contacts a quantum point contact is defined as a constriction between two metallic reservoirs the constriction is assumed to be shorter than the mean free path transport is ballistic S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

68 1/3 suppression Examples for noise Metallic diffusive wires Diffusive wires: multi-channel diffusive wire, length L l =mean free path N transverse channels with N 1 Random walk y x S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

69 1/3 suppression Examples for noise Metallic diffusive wires Diffusive wires: multi-channel diffusive wire, length L l =mean free path N transverse channels with N 1 The Drude-Sommerfeld formula for conductance and the Landauer formula G = 2e2 nτω πml G = e2 2π n T n yields the average transmission coefficient T = l/l 1 = F 1 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

70 1/3 suppression Examples for noise Metallic diffusive wires F 1 experiments (F ex = 1/3) it is known that in the metallic regime, for any energy open channel (T 1) exists a closed channel (T 1) need of a distribution function P (T ) for the transmission coefficients S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

71 1/3 suppression Examples for noise Metallic diffusive wires F 1 experiments (F ex = 1/3) it is known that in the metallic regime, for any energy open channel (T 1) exists a closed channel (T 1) need of a distribution function P (T ) for the transmission coefficients This distribution function can be shown to be P (T ) = l 1 2L T 1 T = T (1 T ) = l 3L, for T min < T < 1 T min = 4 exp ( 2L/l) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

72 1/3 suppression Examples for noise Metallic diffusive wires F 1 experiments (F ex = 1/3) it is known that in the metallic regime, for any energy open channel (T 1) exists a closed channel (T 1) need of a distribution function P (T ) for the transmission coefficients This distribution function can be shown to be P (T ) = l 1 2L T 1 T = T (1 T ) = l 3L with this we get for the shot noise, for T min < T < 1 T min = 4 exp ( 2L/l) S = e3 V N l 3π L = 1 SP F = 1/3 3 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

73 1/3 suppression Examples for noise Metallic diffusive wires F 1 experiments (F ex = 1/3) it is known that in the metallic regime, for any energy open channel (T 1) exists a closed channel (T 1) need of a distribution function P (T ) for the transmission coefficients This distribution function can be shown to be P (T ) = l 1 2L T 1 T = T (1 T ) = l 3L with this we get for the shot noise, for T min < T < 1 T min = 4 exp ( 2L/l) S = e3 V N l 3π L = 1 SP F = 1/3 3 this result is universal as long as l L L ξ (L ξ = localization length) for L L ξ F becomes 1 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

74 1/4 suppression Examples for noise Chaotic cavities Chaotic cavities are quantum systems which would in the classical limit exhibit chaotic motion without any disorder inside the cavity chaotic nature of classical motion is a consequence of the shape of the cavity following results are understood as averages over ensemble of cavities P (T ) = 1 π T (1 T ) F = 1/4 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

75 1/4 suppression Examples for noise Chaotic cavities Chaotic cavities are quantum systems which would in the classical limit exhibit chaotic motion without any disorder inside the cavity chaotic nature of classical motion is a consequence of the shape of the cavity following results are understood as averages over ensemble of cavities P (T ) = 1 π T (1 T ) F = 1/4 this result is universal for chaotic cavities S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

76 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

77 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

78 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

79 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise Tunnel barrier = crossover S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

80 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise Tunnel barrier = crossover Quantum point contact = oscillating noise S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

81 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise Tunnel barrier = crossover Quantum point contact = oscillating noise Diffusive wires = 1/3-suppression S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

82 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise Tunnel barrier = crossover Quantum point contact = oscillating noise Diffusive wires = 1/3-suppression Chaotic wires = 1/4-suppression S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36

83 Summary Examples for noise Chaotic cavities Charge carrier statistics Uncorrelated transfer = Poisson statistic Ideally transmitting channels = current does not fluctuate 0 < T < 1 = each channel is characterized by binomial distribution second cumulant as measurement for noise differs for Bosons and Fermions Fano factor as measurement of sub Poisson noise Tunnel barrier = crossover Quantum point contact = oscillating noise Diffusive wires = 1/3-suppression Chaotic wires = 1/4-suppression Reference: Ya. M. Blanter and M. Büttiker, Shot Noise in Mesoscopic Conductance (Oct 18, 1999) S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni / 36