Effect of electron-phonon scattering on shot noise in nanoscale junctions

Effect of Electron-Phonon Scattering on Shot Noise in Nanoscale Junctions

Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan Department of Physics, University of California, San Diego, La Jolly, California 92093-0319, USA

(Received 29 April 2005; published 12 October 2005)

We investigate the effect of electron-phonon inelastic scattering on shot noise in nanoscale junctions in the regime of quasiballistic transport. We predict that when the local thermal energy of the junction is larger than its lowest vibrational mode energy eVc, the inelastic contribution to shot noise (conductance) increases (decreases) with bias as V (p



V). The corresponding Fano factor thus increases asp



V. We also show that the inelastic contribution to the Fano factor saturates with increasing thermal current exchanged between the junction and the bulk electrodes to a value which, for V
Vc, is independent of bias. These predictions can be readily tested experimentally.

DOI:10.1103/PhysRevLett.95.166802 PACS numbers: 73.63.Nm, 68.37.Ef, 73.40.Jn

It is an established fact that for systems with dimensions much longer than the inelastic mean free path
_ph (e.g., a macroscopic sample) steady-state zero-temperature cur-rent fluctuations (shot noise) are suppressed by electron-phonon scattering [1,2]. Similarly, for metallic diffusive wires with length much smaller than
ph(and smaller than

the electron-electron scattering length), the Fano factor (i.e., the ratio between shot noise and its Poisson value, 2eI, where e is the electron charge and I is the current of the system) equals 1=3 and is not affected by inelastic processes [3]. Systems of nanoscale dimensions may not fall in either one of the above cases. In this instance each electron, on average, releases only a small fraction of its energy to the underlying atomic structure during the time it spends in the junction, making transport quasiballistic [4 – 10]. However, the current density and, consequently, the power per atom are much larger in the junction compared to the bulk. This leads to heating and inelastic features in the differential conduction which are indeed observed in experiments with metallic quantum point contacts [11–14] and molecular structures [7,9,15–17] as a direct conse-quence of the interplay between electron and phonon sta-tistics [18]. For these systems it is therefore not obvious what the effect of inelastic scattering on shot noise is.

In this Letter we show analytically that shot noise in quasiballistic nanoscale junctions is enhanced by inelastic scattering whenever electrons have enough energy to ex-cite the phonon modes of the junction. The current instead decreases. As a consequence, the Fano factor increases. We find it increases with bias asp



V when the local tempera-ture of the junction is larger than its lowest vibrational mode temperature eV_c=k_B. We also show that with increas-ing thermal current carried away from the junction to the bulk electrodes, the inelastic contribution to the Fano factor converges to a minimum value independent of bias for V
V_c. Measurements of the Fano factor and con-ductance may thus provide information on local tempera-tures and heat transport mechanisms in these systems.

Transport in a model atomic gold point contact will be used to illustrate these findings.

Since the dimensions of the junction are much smaller than
ph [and the observed inelastic features in

quasibal-listic systems are very small [11,15,16] ] first-order pertur-bation theory in the electron-phonon coupling captures the dominant contribution to inelastic scattering [19]. This is the contribution we calculate in this Letter.

Let us assume that the junction is connected to two biased bulk electrodes. The electronic states of the full system are thus described by the field operator ^
 P

E;L;RaE
Er; Kk, constructed from the

single-particle wave functions
LR_E r; Kk and annihilation

operators aLR_E corresponding to electrons propagating from the left (right) electrode at energy E. Kk is the

component of the momentum parallel to the electrode surface [20]. We also assume that the electrons rapidly thermalize into the bulk electrodes so that their statistics are given by the equilibrium Fermi-Dirac distribution, fLR_E  1=fexpE  EFLR=kBTe  1g in the left (right)

electrodes with local chemical potential E_FLR, where T_eis the electronic temperature. In the following we will assume that Te 0 K [21], and the left electrode is positively

biased so that E_FL< EFR. The stationary scattering states

LR_E r; Kk are eigenstates of an effective single-particle

Hamiltonian H_e which may be computed, e.g., using a scattering approach within the static density-functional theory of many-electron systems [20]. The combined dy-namics of electrons and phonons is described by the Hamiltonian (atomic units will be used throughout this Letter) [7]

H  He Hph He-ph; (1)

where Hph1₂Pi;2vib_q2i12

P

i;2vib!2iq2iis the

pho-non contribution, with q_ithe normal coordinate and !_i the normal frequency of the vibration labeled by the th component of the ith ion. H_e-ph describes the electron-PRL 95, 166802 (2005) P H Y S I C A L R E V I E W L E T T E R S 14 OCTOBER 2005week ending

phonon interaction and has the following form [7] H_e-phX ; X E1;E2 X i;j2vib










1 2!j s

A_i;jJ_E1;E2i;ay_E1a_E2

b_j by_j; (2)

where   L; R and bjis the phonon annihilation

opera-tor. fA_i;jg is the transformation matrix that relates Cartesian coordinates to normal coordinates, and J_Ei;

1;E2

is the electron-phonon coupling constant which can be directly calculated from the scattering wave functions

Ji;_E1;E2 Z drZ dKk
E1r; Kk@Vpsr; Ri


E2r; Kk;

(3)

where we have chosen to describe the electron-ion inter-action with pseudopotentials Vps_{r; R}

i for each ith ion

[20].

We use as unperturbed states of the full system (electron plus phonon) the states j
LR_E ; n_ji  j
LR_E r; Kki 

jn_ji, where nj is the occupation number of the jth

normal mode. The first-order perturbation to the wave functions is thus

jLR

E ; nji  j
LRE ; nji  j
LRE ; nji; (4)

where the first-order correction term is

j
 E; nji  lim !0 X 0L;R X m_{j0 0} Z dE0D0 E0 h
0 E0; m_j0_0jH_el-vibj
 E; njij
 0 E0; m_j0_0i "E; n_j  "E0_{; m} j00  i ; (5)

with DRL_E the partial density of states of left (right) moving electrons, and "E; nj  E  nj 1=2!jthe energy

of state j


E; nji. By applying (i) hnj 1jbjjnji 






nj

p

, and hn_j 1jby_jjn_ji p















1  nj;

(ii) ay_E1jn_Ei pf





_Ejn_E 1iEE1, and aE1jn  Ei 

1  f_E

p _jn

E 1iEE1, we observe that the nonvanishing matrix

elements h
0

E0; m_j0_0jH_el-vibj


E; nji correspond to the scattering processes shown in Fig. 1. Carrying out explicitly the

integrals in Eq. (5), the corrections to the wave function can be written as j


E; nji  Bj;1 Bj;3j
E!j; nj 1i  B



j;2 Bj;4j
E!j; nj 1i; (6) where B

j;1, Bj;2, Bj;3, and Bj;4correspond to the diagrams depicted in Fig. 1. For j
RE; nji, the coefficients are given

by: BR j;12 i X i










1 2!j s

Ai;jJi;LRE!j;ED

L E!j


















































  njfER1  fLE!j q ; (7) and BR j;34 i X i










1 2!j s

A_i;jJi;RL_E! j;ED L E!j


















































  n_jfL E1  fRE!j q ; (8)

where   1 and ‘‘’’ sign are for the scattering diagrams (a) and (c);   0 and ‘‘’’ sign for diagrams (b) and (d). Similarly, the coefficients in j
L

E; nji have the forms BLj;k BRj;kL R, where k  1; . . . ; 4; the notation L R

means interchange of labels R and L.

At Te  0 K the first-order correction to the current is thus:

I  iZEFR EFL dEZ dRZ dKkI~RRE;E  1 X j hjBR j;1j2i  hjBRj;2j2i  ; (9) where ~I_E;E  
 E _@ z
 E  @z
E _
 E and hi

indi-cates the ensemble average over phonon states. Here we assume that the ions of the junction reach thermal equilib-rium with a well-defined local temperature Tw such that

ensemble averages of phonon states are hnji 

1=exp!j=kBTw  1 and hn2ji  exp!j=kBTw 

1=exp!j=kBTw  12 [7,9]. Equation (9) has been

simplified by using (i) ~IRR_E!j;E!j’ ~IRR_E;E, valid for ener-gies close to the chemical potentials; and (ii) ~IRR_E;E  ~ILL_E;E,

a direct consequence of time-reversal symmetry. The cur-rent is therefore reduced by inelastic effects.

Let us now calculate the corresponding correction to shot noise. We have previously shown that shot noise can be written in terms of single-particle scattering states as [22,23] S ZEFR EFL dE         Z dRZ dK~ILR E;E         2 ; (10)

PRL 95, 166802 (2005) P H Y S I C A L R E V I E W L E T T E R S 14 OCTOBER 2005week ending

which reduces to the well-known formula S /P_iTi1 

Ti when the eigenchannels transmission probabilities Ti

are extracted from the single-particle states with indepen-dent transverse momenta [1,22,23]. Replacing (4) into (10) we get S ZEFR EFL dE         Z dRZ dK~ILR E;E         2  1  X j;k1;2 hjBR j;kBLj;kj2i  : (11)

Since the summation over vibrational modes contains only positive terms, shot noise is enhanced by electron-phonon inelastic effects in the quasiballistic regime. Therefore, the Fano factor F normalized to the correspond-ing value in the absence of electron-phonon interactions (F0_{) is} F=F0 _ REFR EFLdE1  P j;k1;2 hjBR j;kBLj;kj2i REFR EFLdE1  P j;k1;2 hjBR j;kj2i ; (12)

which increases with electron-phonon scattering.

Note that due to the orthogonality of phonon states, the absolute value of the correction to shot noise is smaller than that to the current [cf. Eqs. (9) and (11)]. Note also that conservation of energy and the Pauli exclusion principle play an important role. The former dictates an onset bias V_cfor inelastic contributions; the latter prohibits the scattering processes depicted in Figs. 1(c) and 1(d) at T_e  0 K.

These results are illustrated in Fig. 2 where the inelastic contribution to the conductance and Fano factor are plotted for a gold atom placed in the middle of two bulk gold electrodes (represented with ideal metals, jellium model, rs 3). Details of the calculations can be found in

Refs. [7,20]. In the absence of electron-phonon interac-tions, the unperturbed differential conductance G0_{is about}

1.1 (in units of 2e2_{=h) and the Fano factor is F}0 _{’ 0:14 [22]}

in the bias range of Fig. 2. Inelastic effects cause a dis-continuity in the conductance, and a variation of the Fano factor ratio [Eq. (12)], at a bias V_c  11 mV, correspond-ing to the energy of the lowest longitudinal mode of the system. In addition, the above inelastic corrections depend on the local temperature of the junction Tw [see Eqs. (7)

and (8)] which, in turn, is the result of the competition between the rate of heat generated locally in the nano-structure and the thermal current Ithcarried away into the

bulk electrodes [4 –7,9,10]. The latter has a temperature dependence of Ith AthT4w [24], where the constant Ath

depends on the details of the coupling between the local modes of the junction and the modes of the bulk electrodes. At steady state this thermal current has to balance the power generated in the nanostructure, which is a small fraction of the total power of the circuit V_R2(V is the bias, Ris the resistance) [4,7].

The larger A_th, the larger the heat dissipated into the bulk and, thus, the lower the local temperature T_w[25]. In the limit of infinite A_th, i.e., T_w 0, at any given bias larger than V_c, electrons can only emit phonons [n_j 0 in Eqs. (7) and (8)]. The inelastic contribution to the con-ductance and Fano factor, therefore, saturate to a specific value (see Fig. 2). We can derive both the bias dependence and this saturation value, to first order in the bias, as follows.

By equating the thermal current Ithto the power

gener-ated in the junction, it is easy to show that T_w p



V [6,26], where the constant  depends on the details of the thermal contacts between the junction and electrodes. Let us assume for simplicity a single phonon mode of FIG. 2 (color online). Top panel: ratio of the total conductance Gof an atomic gold point contact and its value in the absence of inelastic effects G0_{as a function of bias for different values of} thermal current coefficient (see text): Ath 1019 (dotted line), 1017(dot-dashed line), 1015_{(dashed line), and 1 (solid line)} dyn=sK4_{. Bottom panel: corresponding Fano factor ratio.} FIG. 1 (color online). Feynman diagrams and corresponding

amplitudes (see text) of the main electron-phonon scattering mechanisms contributing to the correction of the current and noise.

PRL 95, 166802 (2005) P H Y S I C A L R E V I E W L E T T E R S 14 OCTOBER 2005week ending

frequency !. For T_w> !=k_B, we expand hn_ji  k_BT_w=! in Eq. (9). We then get

G G0 ’ 1   3 2 k_B ! IV  Vc



V p ; (13)

where V  V_c is the Heaviside function; I jdI 

I0_=dV=dI0_{=dVj is the relative change in conductance}

due to inelastic effects at Vc[its value is about 1% for the

specific case, in agreement with experiments on similar systems [7,11] ]. The inelastic contribution to the conduc-tance thus decreases with bias as p



V. This square-root dependence is clear in Fig. 2 for A_th< 1015 dyn=sK4_

which corresponds to temperatures for which the condition Tw> !=kB is satisfied.

The same analysis can be applied to shot noise. In Eq. (11), for Tw> !=kB we expand hn2ji  2kBTw=!2

which leads to S S0 ’ 1  2 2kB ! _{2 s}V  V_cV  V_c; (14) where _S jdS  S0_=dV=dS0_{=dVj is the relative}

change of shot noise due to inelastic effects at V  V_c(it is about 0.04% for the specific gold junction). The inelastic correction to shot noise thus increases linearly with bias for Tw> !=kB. Consequently, F=F0/

V p

as it is also evi-dent from Fig. 2.

In the opposite limit of perfect heat dissipation in the bulk electrodes, i.e., for T_w! 0 [see Fig. 2, Ath!

1 dyn=sK4_{], then from Eqs. (7) and (8) it is easy to}

prove that I=I₀  1  V  Vc IV  Vc=V and

S=S₀  1  SV  Vc=VV  Vc. Therefore,

F=F0 1  SV  Vc=VV  Vc

1  _IV  V_c=VV  V_c; (15) which tends to the constant value F=F0 _{! 1 }

S=1 

_I as V
Vc. The predictions reported in this Letter

should be readily tested experimentally.

We acknowledge partial support from the NSF Grants No. DMR-01-33075 and No. ECS-04-38018. We also thank M. Bu¨ttiker and M. Zwolak for useful discussions.

*Electronic address: [email protected] †_{Electronic address: [email protected]}

[1] For a recent review, see, e.g., Ya. M. Blanter and M. Bu¨ttiker, Phys. Rep. 336, 1 (2000).

[2] A. Shimizu and M. Ueda, Phys. Rev. Lett. 69, 1403 (1992).

[3] C. W. J. Beenakker and M. Bu¨ttiker, Phys. Rev. B 46, R1889 (1992).

[4] T. N. Todorov, Philos. Mag. B 77, 965 (1998).

[5] M. J. Montgomery, T. N. Todorov, and A. P. Sutton, J. Phys. Condens. Matter 14, 5377 (2002).

[6] M. J. Montgomery, J. Hoekstra, T. N. Todorov, and A. Sutton, J. Phys. Condens. Matter 15, 731 (2003).

[7] Y.-C. Chen, M. Zwolak, and M. Di Ventra, Nano Lett. 3, 1691 (2003); 4, 1709 (2004); , 5, 813 (2005).

[8] A. Troisi, M. A. Ratner, and A. Nitzan, J. Chem. Phys. 118, 6072 (2003).

[9] Y.-C. Chen, M. Zwolak, and M. Di Ventra, Nano Lett. 5, 621 (2005).

[10] Z. Yang, M. Chshiev, M. Zwolak, Y.-C. Chen, and M. Di Ventra, Phys. Rev. B 71, 041402(R) (2005).

[11] N. Agraı¨t, C. Untiedt, G. Rubio-Bollinger, and S. Vieira, Phys. Rev. Lett. 88, 216803 (2002).

[12] J. I. Mizobata, A. Fujii, S. Kurokawa, and A. Sakai, Phys. Rev. B 68, 155428 (2003).

[13] T. Frederiksen, M. Brandbyge, N. Lorente, and A.-P. Jauho, Phys. Rev. Lett. 93, 256601 (2004).

[14] R. H. M. Smit, C. Untiedt, and J. M. van Ruitenbeek, Nanotechnology 15, S472 (2004).

[15] J. G. Kushmerick, J. Lazorcik, C. H. Patterson, R. Shashidhar, D. S. Seferos, and G. C. Bazan, Nano Lett. 4, 639 (2004).

[16] W. Wang, T. Lee, I. Kretzachmar, and M. A. Reed, Nano Lett. 4, 643 (2004).

[17] L. H. Yu, Z. K. Keane, J. W. Ciszek, L. Cheng, M. P. Stewart, J. M. Tour, and D. Natelson, Phys. Rev. Lett. 93, 266802 (2004).

[18] H. Fo¨rster, S. Pilgram, and M. Bu¨ttiker, Phys. Rev. B 72, 075301 (2005).

[19] Second-order scattering mechanisms have negligible contributions to both the current and noise due to the competition between electron and phonon statistics. [20] N. D. Lang, Phys. Rev. B 52, 5335 (1995); M. Di Ventra,

S. T. Pantelides, and N. D. Lang, Phys. Rev. Lett. 84, 979 (2000); M. Di Ventra and N. D. Lang, Phys. Rev. B 65, 045402 (2002); Z. Yang, A. Tackett, and M. Di Ventra, Phys. Rev. B 66, 041405(R) (2002).

[21] At this temperature, and under quasiballistic assumptions, thermal noise is negligible.

[22] J. Lagerqvist, Y.-C. Chen, and M. Di Ventra, Nanotechnology 15, S459 (2004).

[23] Y.-C. Chen and M. Di Ventra, Phys. Rev. B 67, 153304 (2003).

[24] See, e.g., K. R. Patton and M. R. Geller, Phys. Rev. B 64, 155320 (2001).

[25] The coefficient Ath can be estimated assuming the junc-tion as a weak thermal link with a given stiffness [24]. In this case, Ath’ 3:7 1015dyn=sK4 [7]. The cor-responding local temperature is estimated to be Tw 37 K at V  70 mV [7]. For the other thermal coef-ficients reported in Fig. 2, the local temperature at V  70 mV is Tw 426 K [Ath 1019 dyn=sK4], Tw 157 K [Ath 1017 dyn=sK4], and Tw 51 K [Ath 1015 dyn=sK4].

[26] The thermal current scales as l2 with wire length l [Ref. [24] ]. On the other hand, the power dissipated in the junction has weak length dependence for metallic wires [6,7,10]; instead it scales as expl for insulating junctions, where  is a characteristic inverse length of the wire [9]. The local temperature of the wire thus scales with length asp

l[4,14] and expl₄ [7,9] for metallic and insulating wires, respectively.

PRL 95, 166802 (2005) P H Y S I C A L R E V I E W L E T T E R S 14 OCTOBER 2005week ending