Pis'ma v ZhETF, vol.88, iss.ll, pp.800-804
© 2008 December 10
Inelastic cotunneling through a long diffusive wire
M. V. Feigel'man, A. S. Ioselevich L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
Moscow Institute of Physics and Technology, 141700 Moscow, Russia Submitted 30 October 2008
We show that electron transport through a long multichannel wire, connected to leads by tunnel junctions, at low temperatures T and voltages V is dominated by inelastic cotunnelling. This mechanism results in experimentally observed power-law dependence of conductance on T and V, in the diffusive regime where usual Coulomb anomaly theory leads to exponentially low conductance. The power-law exponent a* is proportional to the distance between contacts L.
PACS: 73.23.Hk, 73.63.^b
Electronic transport through nanowires was intensively studied by many groups in the past years. In particular, conductance of multi-channel diffusive nanowires with relatively poor contacts to metal terminals was measured, cf. e.g. [1]. The Coulomb phenomena play an important role in transport, provided the contacts between the wire and the leads are weak. The mechanism of the Coulomb blockade, as well as the Coulomb anomaly due to tunneling spreading of charge, are presently well understood. In the ballistic regime (at relatively high temperature T and/or bias voltage V) the Coulomb effects lead to the power-law temperature and voltage dependence of the conductance:
G = dl/dV oc Va (low T), G oc T° (high T), (1)
characteristic for Luttinger Liquid, while at low T and V - in diffusive regime - an exponential dependence (see (4) below) should be observed. The puzzle is that the power law (1) is found in almost all experiments, even in those where the conditions for the diffusive regime seem to be fulfilled.
The existing theories (see, e.g., [2-4]) considered the Coulomb effects at each of contacts separately. However, if both contacts are taken into account simultaneously, then some analog of cotunneling becomes possible and at low temperatures this mechanism should dominate. The standard theory of cotunneling deals with small grains or quantum dots, while a long wire is an extended object: internal dynamics of charge within it may be important. In the present letter we develop a theory for such an extended cotunneling and show that in the diffusive regime the resulting cotunneling conductance still obeys the law (1), though with different exponent a*, depending on the separation L between the contacts.
Consider a multichannel metallic wire (it may be a multiwall nanotube) of length L0 and diameter a. The
wire is connected to massive metallic leads through two weak tunnel contacts A and B with identical dimension-less conductances g ■C 1 (see Figure). A voltage, applied
1 « - ¿o >
0 )
t S 8
ft—1-iSi
Electrons tunnel between a wire of length Lq and diameter a and two leads A and B, placed symmetrically with respect to the center of the wire, at distance L from each other
between the contacts is V. The classic dimensionless resistance of the piece of wire between the leads is assumed to be not very small: R(L)/(h/e2) = L/£ > 1, where £ ~ NciJ is the localization length, I is the mean free path, and 1 is the number of channels.
In this paper the relevant energy scales will be assumed so low, that the motion of electrons in the wire is diffusive. On the other hand, we will neglect the localization effects. As long as usual conductivity in a wire is concerned, the condition of "no localization" reads
T » TLoc ~ DC2 ~ vP/N*hl (2)
It is not evident that inequlity (2) is in fact necessary when the under-the-barrier spreading process is considered; however, it is certainly the sufficient one, and we will assume it is fulfilled below. This requirement is consistent at iVCh 1 with the diffusive dynamics of charge spreading.
If the temperature T is not very low, the diffusive transport between the two leads proceeds in a "single-particle mode": At first one electron (one hole) tunnels
800
IlHCbMa b ?K3T<D tom 88 Bbin.11-12 2008
into the wire from one of the leads and is accommodated in the wire, then one hole (one electron) tunnels from another lead. Because of the (thermoactivated) tunneling character of the accommodation process, the corresponding conductance G^g is exponentially suppressed
G« ~gexp{^Sl(T,V)}.
(3)
At temperatures T > Tc(1) = E2C(L0)/EC(Q the accommodation proceeds according to the semiclassic scenario [2] (the Coulomb zero-bias-anomaly regime, see also [3, 5, 6]), and the accommodation action
Si(T,V)
0.76^1
Ec(0 T :
for eV « y/Ec{0T,
(4)
Ec(0/eV, for eV » у/ЕсЦ)Т.
Here Ec(x) = e2 In(x/a)/ex is the charging energy of a piece of wire of length x a.
At T -C Tc ^ the single-particle accomodation proceeds according the "orthodox" Coulomb blockade scenario (see, e.g., [7]):
S1(T,V)^EC(L0)/T.
(5)
The abovementioned independent single particle processes should be less effective than some correlated cotunneling process, in which any charged states of the wire would only enter as virtual intermediate states. The theory of such processes is well developed for transitions via small grains, where the intergrain charge transfer processes are the bottlenecks for the transport, while the intragrain charge transfer is easy (see [8]). In our case, however, the charge spreading within the wire is a crucial ingredient of the process, so that the standard perturba-tional description of the cotunneling is inapplicable.
In the present Letter we propose a modification of the approach [2], which allows for description of charge spreading effects under the two-particle cotunneling conditions. Our main result reads as follows:
rW ~ n2 AB ~ 9
/max{LT/C, (eV)}
EC(L)
a = (6)
In the case L « L0 the crossover from one-particle tunneling (in the Coulomb blockade mode) to the two-particle one takes place at
T(2)
e2£
In(L0/a)
<T«. (7)
The last inequality becomes strong for very low conductance of contacts, g ■C 1; in this situation a sequence of
crossovers may be seen with the temperature decrease: from the Coulomb anomaly mode (4) to the Coulomb blockade mode (5) at T = and then to the inelastic cotunneling regime (6) at T = TC(2).
In the case L -C L0 the Coulomb blockade regime is absent, and the crossover from the Coulomb anomaly to the inelastic cotunneling takes place at
T(2)
e2£
ln(£/a)
eL2
In
\eL2Tj + L
НУ9)
(8)
In the nonlinear regime the crossover between the single-particle Coulomb anomaly and the inelastic cotunneling takes place at
eV ~ (eV)c
eL
In (C/a).
(9)
Thus, at low enough temperatures, the cotunneling scenario always dominates. On the other hand, the condition (2) of "no localization" should also be fulfilled for applicability of the formula (6). The necessary temperature range only exists if
< I «
Iec(0
V Пас
In (C/a)
Nch
1/2
(10)
i.e. the condition iVCh 1 is necessary.
The method of Levitov and Shytov [2] is based on classic equations of motion for the electron density p(x,r) and current j(x,r) in imaginary time r = —it. In a case of wire one can write
3 =
= -D
d/r+ilLx=JM>
e\x — x'\'
(И) (12)
where a = e2£/27rft is effective one-dimensional conductivity, c is the effective dielectric constant, and D is a diffusion constant. The instanton is chosen in a form of a symmetric bounce, so that the source J in the continuity equation (11) corresponds to the injection of one extra electron into the system at time t\ = —¿To at point x = —L/2 with its subsequent elimination at the same point at moment ¿2 = itq: Ji = [¿(t + t0) - S(t - t0)]5(a; + L/2).
The crucial point of our approach is that we describe cotunnelling through a diffusive wire by the same semi-classical Eqs. (11), (12), but with modified source
J2 = [5(T + t0) - 5(t - t0)][¿(a; + L/2) - 5(x - L/2)},
(13)
е
which describes simultaneous tunnelling of an electron and a hole via both contacts.
The density p(x,r) should be even with respect to t —y —t, while the current j(x,r) should be odd. p(x,t) and j(x,r) are defined on the interval —1/2t < t < 1/2t and obey periodic boundary conditions. Expanding p(x,r) and j(x,r) in Fourier series, we get p(x,r) = p(x, w) cos(o;t) and j(x,r) = = j{x,w) sin(o;t), where the Matsubara frequency summation, as usual for Bose excitations, runs over even frequencies w = 2irTn, with n = 0,1,2,____
If the wire is very long (L0 —t oo), the system of Eqs. (11), (12) can be solved by the spatial Fourier transformation. Then, proceeding in the full analogy with [2], we obtain G^ « g2 ex.p{—S2(T, F)}, where
S2(T, V) = S2(T, Tq ) — 2BVtq , дё2(Т,т0)/дт0 =2eV.
e »—ч
s2(T,T0)=-y:
\j2(wq)\2u,
Q
uj,q
{u + Dq>){u + aq2Uqy
J2(w,q) = -Aism(wT0)sm(qLl2), f!
j-ооФГ e
giq® 2 1
-—-dx = - In—. ^Ы f qa
(14)
(15)
(16)
(17)
(18)
The semiclassical method, used above, is applicable, if S2 > 1. From (16,17) it is clear, that dS2/dr0 = 0 for To = 1 ¡AT. Therefore we conclude that
то (T, V —»■ 0) = 1/4T,
(19)
so that in the low-voltage case the summation runs only over the odd n = 2k + 1:
00 « , f
S2(T,V^ 0) «
We2Tsin2(qL/2)Uq
^ J 2tt (2irT(2k + 1) + Dq2)(2<jrT(2k + 1) + aq2Uq)'
There are three different temperature ranges: T ^max! ^'rriin «T« Wmax! and T -C wmin, where
a>n
«W = aL^UiL-1) ~ (t/L)Ec(L) DL-2
Nc h
In (L/a)
•C
(20) (21)
we consider these three ranges separately.
1. T wmax. Here the sum is dominated by k ~ 1, u] ~ T; it can be shown that S2(T,V -»■ 0) « 2S\(T,V —¥ 0), which means that in this temperature range the two-particle process looses a competition with the one-particle Coulomb anomaly one.
2. Wmin -C T -C wmax. Here the integral over q is dominated by q ~ L~x, while the logarithmical sum over k is dominated by an interval 0 < k -C wmaX/T, so that
(22)
and, with the help of (20), we arrive at the final expression (6).
3. T -C wmin. In this range presumably the elastic cotunneling should dominate. However, since wmin -C •C Iloc,
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