Weak Localization in Metallic Granular Media

0.22 MB·English

by  Ya. M. Blanter

Weak Localization in Metallic Granular Media
Ya.M. Blanter1,2, V.M. Vinokur3, and L.I. Glazman4
1Kavli Institute of Nanoscience, Delft University of Technology, 2628CJ Delft, the Netherlands
2Braun Center for Submicron Research and Department of Condensed Matter Physics,
Weizmann Institute of Science, Rehovot 76100, Israel
6 3Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
0 4W.I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN55455, USA
0 (Dated: January 10, 2006)
2
We investigate theinterference correction to theconductivity of a medium consisting of metallic
an grains connected by tunnel junctions. Tunneling conductance between the grains, e2gT/π~, is
J assumedtobelarge, gT≫1. Wedemonstratethattheweaklocalization correctiontoconductivity
exhibitsacrossoverattemperatureT ∼g2δ,whereδ isthemeanlevelspacinginasinglegrain. At
0 T
thecrossover, thephaserelaxation timedeterminedbytheelectron-electron interaction becomesof
3
theorderofthedwelltimeofanelectron inagrain. Belowthecrossovertemperature,thegranular

array behaves as a continuous medium, while above the crossover the weak localization effect is
]
l largely a single-junction phenomenon. We elucidate the signatures of the granular structure in the
l
a temperatureand magnetic field dependenceof theweak localization correction.
h
- PACSnumbers: 73.23.-b,72.15.Rn,73.23.Hk
s
e
m
I. INTRODUCTION closed electron trajectories, allowing for phase-coherent
t. electron motion.
a
The WL correction in a homogeneous medium origi-
m Quantumeffectsinconductionoftwo-dimensionaldis-
nates fromthe quantuminterference ofelectronsmoving
- ordered electron systems draw attention of both experi- alongself-intersectingtrajectories1andisproportionalto
d mentalistsandtheoristsfordecades. Theinterestismoti-
the return probability of an electron diffusing in a disor-
n
vatedinpartbytheinterplaybetweenseveralfundamen-
o dered medium. In one- or two-dimensional conductors
tal physical phenomena, such as quantum interference,
c this probability diverges due to the contribution com-
[ localization, superconductivity, and single-electron tun- ing from long trajectories. For a fully coherent electron
2 ntheelinpgroopcecrutrireisngofinnotrhmesael scyosntdemucst.orTs1h,2e,3i,ntneorpmlianyalalyffescuts- propagation, this divergence would lead to a divergent
v perconducting films4,5, and arrays of junctions6. Quan- WL correction. Finite phase relaxation time τφ makes
9 sufficiently long trajectories ineffective for the interfer-
tum effects become increasingly important at sheet con-
0 ence, and limits the correction. In a two-dimensional
3 ductances decreasing towards the fundamental quantum conductor, the WL correction to conductivity is δσ =
4 value of G e2/π~. The interpretation of some of the
0 most intrigQui≡ng data, however, may depend on whether −(GQ/2π)ln(τφ/τ), where τ is the electron momentum
relaxation time. There are various mechanisms of the
5 the investigated conductors are homogeneous or granu-
0 electron phase relaxation, some of them being material-
lar. While this questionhasa definite answerinthe case
/ specific8. The most generic and common for all the con-
t of an array6 of lythographically defined junctions, it is
a ductors mechanism stems from the electron-electron in-
less clear for nominally homogeneous deposited metallic
m teraction9. It yields 1/τ T(G /σ )ln(G /σ ) and
films4,5 or electron gases in semiconductor heterostruc- φ ∼ Q 0 Q 0
d- tures3. Checking the samples homogeneity traditionally pitryo,vides the temperature dependence of the conductiv-
n involvesapplicationofauxiliarytechniques,suchaslocal
o probe spectroscopy5,7. σ =σ0 (GQ/2π)ln(T∗/T) (1)
−
c
We demonstrate that information about the granular- with T∗ σ /G τ. The typical area under an electron
: 0 Q
v ity of a conductor is contained in the temperature and trajector∼y which barely preserves coherence, L2 = Dτ ,
i φ φ
X magneticfielddependenceofthe weaklocalization(WL) depends on the electron diffusion constant. Magnetic
correctionto the conductivity. The granularstructureof field B significantly affects the WL correctionif the cor-
r
a a conductor affects the correction even at high film con- responding magnetic flux through a contour of area L2
φ
ductivity, σ G . While being universalat the lowest exceeds the quantum Φ . This makes the magnetoresis-
0 Q 0
≫
temperaturesandmagneticfields,the WLcorrectionbe- tance measurement a useful tool for the investigation of
comes structure-dependent at higher values of field and the electron interference.
temperature. The corresponding crossover temperature To model a granular medium, we consider a regular
is of the order of (σ /G )2δ, where the mean-levelspac- two-dimensional array of grains of size d connected by
0 Q
ing δ in a single grain is inversely proportional to the tunnel junctions. The grains have internal disorder, but
grain volume. The field dependence of the WL correc- are characterized by conductance far exceeding the con-
tion at low temperatures exhibits two crossovers. These ductance g G of a single tunnel junction. The classi-
T Q
are associated with a significant change in structure of cal conductivity of a square array is thus σ = G g .
0 Q T
2
It corresponds10 to the effective electron diffusion con-
stant D = π−1g δd2. In the absence of phase relax-
T
ation,anelectronmaypassthroughanynumberofjunc-
tions coherently. It will result in a diverging WL cor-
rection, just like in a homogeneous conductor. The B
electron-electron interaction limits the phase relaxation
time, yielding 1/τ T/g . As long as the correspond-
φ T
∼
ing length L d g2δ/T significantly exceeds d, an
φ ∼ T
electron may return to a grain coherently after passing
p
many junctions, and the inhomogeneity of the granular
medium is irrelevant. The comparison of L with d de-
φ
fines a crossovertemperature,
T =g2δ. (2)
cr T
d
Roughly, above the crossover temperature the electron
trajectories contributing to the WL do not cross more FIG.1: Atypicaldiffusivetrajectoryinagranulararray. The
than a single junction. In this regime granular medium directed area S consists of two components. The first one
eff
behaves similarly to a single grain connected to highly is the combined contribution of separate grains, see the first
conducting leads by tunnel junctions of conductance g . term in Eq. (5). The second component is the area under
T
The WL correction at T > T comes from electron thecoarse-grained trajectory, which isthecounterpartofthe
cr
trajectories that pass throug∼h a single tunnel junction. directed area under an electron trajectory in a homogeneous
disordered sample.
Electronsmovingalonglongertrajectories,whichinclude
morejunctions,havemuchsmallerprobabilityofaphase-
coherentreturn. We find that alreadythe shortestinter-
d2 g /g and is limited by the electron dwelling time.
grain trajectories (see Fig. 2 in Section IV) provide the gr T
Atlow temperatures,T T , the number ofgrainsvis-
temperature dependence of the WL correction, p ≪ cr
ited by an electron having a typical closed trajectory, is
Tcr of the order of Tcr/T. The single-grain directed areas
δσ = A , (3)
WL have random signs, so the estimate for the full directed
− T
area is
with A being a geometry-dependent coefficient of order
one. In deriving Eq. (3), we assume that gT is much S d2 Tcrggr +d2Tcr . (5)
smaller than the number of channels in the inter-grain eff
∼ s TgT T
tunnel junction, although g 1.
T
≫
Equation(3)doesnotaccountforthephaserelaxation
Thefirsttermherecorrespondstothesumofthedirected
rate within the grains. At a sufficiently high tempera-
areasunderthe electrontrajectorywithinthe grainsvis-
ture T > T∗ the latter exceeds the electron escape rate ited by electron; the second, conventional1, term comes
g δ fro∼m a grain, which leads to a suppression of the
T fromthefactthatthe“visited”grainsformaclosedcon-
WL correction below the value Eq. (3). The charac- tour of an area L2, see Fig. 1. The characteristic level
teristic scale T∗ here depends on the intra-grain phase φ
of the field necessary to affect δσ is found from the
WL
relaxation mechanism. Assuming that it is due to the
condition S B Φ . We see now, that even within
eff φ 0
electron-electron interaction11, and that the dimension- the temperature r∼ange T < T , the granularity of the
less conductance of the grain ggr is large, ggr > gT2, we material matters. ∼ cr
find ∼
At the lowest temperatures the characteristic field co-
g incides with that of a film with the corresponding value
T∗ T gr√g . (4)
∼ crg2 T of diffusion coefficient,
T
Φ T g
In a more exotic case of a smaller grain conductance, 0 T
B , T T . (6)
gT2 ≫ ggr ≫ gT, the temperature T∗ still exceeds sig- φ ∼ d2 Tcr ≪ crggr
nificantly T , but the specific relation between the two
cr
At higher temperatures, the characteristic field is
temperature scales depends on the grain shape, and is
different for disk-like or dome-like grains.
We turn now to the discussion of the magnetic field Φ0 TgT gT
B , T T T . (7)
effect on the weak localization in the granular medium. φ ∼ d2sTcrggr crggr ≪ ≪ cr
To determine the characteristic field suppressing the in-
terferencecorrectiontoconductivity,weneedtoestimate Thehighertheappliedfield,theshorterarethetrajecto-
thedirectedareacoveredbyatypicalclosedelectrontra- ries contributing to the interference correction, and the
jectory12. For a single grain, such area is of the order smaller the correction is. Such trajectories span only a
3
single grainprovidedthe field B is ofthe orderorhigher The current through the contact is defined as Iˆ =
than eNˆ˙ = ie[Hˆ ,Nˆ ], where Nˆ is the number of par-
m T m m
− −
Φ g ticles in the grain m,
Bsg = 0 T. (8)
φ d2 g
r gr Nˆ = aˆ† aˆ .
m ασ ασ
At B Bsg, even the single-grain Cooperon is sup- ασ
≫ φ X
pressed. Consequently, Eq. (8) defines the characteristic
Calculating the average current through the barrier, we
field suppressing at T > T the WL correction Eq. (3),
cr obtain
whichstemsfromthe t∼ransitionswithintheclosestgrain
pairs.
I(t)= e Re dr dr t(r ,r )GK (r ,r ,t,t)eiVmnt
To develop a quantitative theory of the interference − 1 2 1 2 nm 2 1
Z
correction, we derive the expression for the weak local- t∗(r(cid:8),r )GK (r ,r ,t,t)e−iVmnt ,
ization correction and adapt the Cooperon equation for − 1 2 mn 2 1
granular medium. where GK is the Keldysh component of the ma(cid:9)trix
Green’s function, and the subscripts m and n are intro-
duced for convenience, in order to indicate which grain
II. CONDUCTANCE AND WEAK
points r and r belong to. We need now to calcu-
LOCALIZATION IN A GRANULAR ARRAY 1 2
late the function GK by perturbation theory in tunnel-
ing Hamiltonian. Let us first discuss the first order and
Tunneljunctionsbetweenmetallicgrainsaredescribed calculate the average conductance. Using the standard
adequatelybyamodelwithaninfinitelylargenumberof technique14, we obtain for the current in the frequency
weakly-conductingchannels. Withinthis model, onecan representation (terms which do not depend on the time
use the tunneling Hamiltonian formalism for evaluation difference would correspond to the Josephson effect and
of the conductivity of the granulararray. In this formal- thus are dropped)
ism, tunneling between the grains m and n is described
by the Hamiltonian dω
I(ω) = 2e Re dr ...dr t∗(r ,r )t(r ,r ) (12)
1 4 1 2 3 4
2π
Hˆ = t eiVmntaˆ† aˆ +h.c.= dr (9) Z
T αβ ασ βσ 1 Tr τˆ Gˆ (r ,r ,ω+eV )Gˆ (r ,r ,ω) ,
αβσ σ Zm × x m 1 3 mn n 4 2
X X
h i
×Zndr2t(r1,r2)eiVmntψˆm† σ(r1)ψˆnσ(r2)+h.c. , Gwhnner,eaτnxdiwsethuesPeathuelismtaantrdiaxrdinrtehpereKseenldtaytsihonspace,Gn ≡
where the points r and r belong to the grains m and
1 2 GR GK
n,respectively,V isthe voltageappliedto the barrier, Gˆ = .
mn 0 GA
α m and β n are exact single-particle states, and σ (cid:18) (cid:19)
∈ ∈
is the spin index. In the limit of thin barrier, the tunnel In the linear regime, it suffices to use the equilibrium
amplitude t significantly deviates from zero only if the function here, GK(E) = tanh(E/2T)(GR(E) GA(E)).
vectorsr1 andr2 refertotwoclosesttoeachotherpoints Expressing the Green’s functions in terms of−the exact
at opposite sides of the interface,
eigenfunctions,calculatingtheenergyintegrals,andsub-
t(r1,r2)=aδ(y−y′)∂z∂z′δ(z)δ(z′). (10) sIti=tuσtiVng thfoerttrhaenisnmteisrs-igornainamcuprlriteundt,esan(d10σ),=w(ee2o/bπt)agin
mn T
Here the coordinate y runs along the interface S, and for the Drude conductivity of the granular array. The
transverse coordinates z in the grain m and z′ in the introduced here dimensionless inter-graintunneling con-
grain n are defined in such a way that at the interface ductance is
z =z′ =0. (We wrote Eq. (10) for the planar geometry,
generalization to three-dimensional arrays is straightfor- g =4π2 a2 dydy′ (13)
T
| |
ward). The constant a is of the order of magnitude of ZS
2
√ /νk , where ν is the electron density of states of the
maTteriaFl of the grains, and T is the transmission coef- (cid:12)* ∂zχα(yz)∂z′χ∗α(y′z′)|z=z′=0δ(ξα)+(cid:12)
ficient of the barrier. The numerical factor in a can be (cid:12) Xα (cid:12)
(cid:12) (cid:12)
related to the measurable quantity, the barrier conduc- (cid:12)(cid:12)=4a2 dydy′ ∂z∂z′Im GR(yz;y′z′) (cid:12)(cid:12)z=z′=0 2 ,
tance g . Using Eq. (10), one may express the tunnel | | |
T ZS
amplitude in terms of the eigenfunctions χ and χ of (cid:2) (cid:10) (cid:11) (cid:3)
α β
whereξ aretheexactenergyeigenvaluesmeasuredfrom
anelectroninthe grainsm andn, respectively(see, e.g., α
the Fermi level in a grain, and angular brackets mean
Ref. 13),
impurity averaging within a grain (the eigenfunctions in
different grains are not correlated). In the last equation,
tαβ =a dy∂zχ∗α(y,z)∂z′χβ(y,z′)|z=z′=0 . (11) GR istheimpurity-averagedGreen’sfunctionevaluated
ZS h i
4
mmm n
where the subscripts of the Cooperon indicate that it
startsandendsinthegrainsmandn,respectively. Note
that due to the structure of the tunneling amplitudes
t(r,r′), point r is just across the barrier from point r
1 2
andsimilarlypointr isacrossthebarrierfrompointr .
3 4
The Cooperon C can be presented in the form
π
C (r ,r ;r ,r )= Im GR(r r )
mn 1 4 2 3 ν m 1− 4
Im GR(r r ) C˜ (r (cid:10),r ) . (cid:11)
× n 2− 3 mn 1 3
(cid:10) (cid:11)
Rapiddecayoffunctions GR withthedistancebetween
h i
thecorrespondingargumentsmakespointsinpairsr ,r
1 4
andr , r inthe spatialintegralofEq.(14)to be within
2 3
the Fermi wavelength from each other. On the other
FIG. 2: Second-order correction to the conductivity. Black hand, the Cooperon C˜(r ,r ) is generally a long-range
circles represent the tunnel amplitudes, and dashed lines de- 1 3
function. Provided we are interested in times long com-
note impurity scattering inside the grains
pared to the intra-grain diffusion time, C˜ almost does
not change while its coordinates vary within respective
grains. However, C˜ (r ,r ) with m = n may dif-
at the Fermi energy. It is represented as the density of mn 1 3
6
fer significanlty from the value of single-grain Cooperon
statesν multiplied witha dimensionlessfunctionrapidly
decaying with the distance y y′. The characteristic (m = n). Substituting this coarse-grained Cooperon
length of that decay is given b−y the Fermi wavelength, C˜mn into Eq. (14) and taking into account Eqs. (10)
andtheintegralinEq.(13)isconvergingrapidly. There- and (13), we obtain
forethedimensionlessfunctionofy y′maybeevaluated
within the free-electron approxima−tion13. The precise
shape of this function is not important for our purposes. δσ = e2gTRe ∞ dtC˜ (t, t) (15)
Equation(13)thusrelatesthetunnelconductancetothe WL 2πν2 mn −
Z−∞
previously introduced constant a.
We proceed now with the evaluation of weak lo-
calization correction. The next-order contribution to with m and n being the neighboring grains. Note that
the current (Fig. 2) contains four tunnel amplitudes Eq. (15) is valid for any dimension, not just in 2D.
and four Green’s functions with the Keldysh structure
The form (15) of weak localization correction is valid
Tr (τˆ GˆGˆGˆGˆ), where Gˆ is the matrix Green’s func-
x provided the phase coherence between the grains barely
tion in the Keldysh space. The trace of a product
survives, and C˜ C˜ at m = n. This limit is real-
of several Green’s function can only have the follow- mn ≪ nn 6
ized at a sufficiently high temperature, T T . Note
ing structure: several first functions are retarded, fol- ≫ cr
also that the performed derivation, unlike the consider-
lowed by one Keldysh function and then a number of
ation of, e.g., Ref. 16 assumes the limit of large number
advanced functions. Thus, we have the combination of
of channels taken at fixed value of g .
the type GKGAGAGA+GRGKGAGA+GRGRGKGA+ T
GRGRGRGK. However, the second and third terms in To consider phase relaxation in a granular array, we
this combinationareconsiderablygreaterthanthe other derive now the proper equation for C˜ in a granular
nm
two,sincetheimpurityscatteringinsidethe grainsisthe medium.
most effective if the impurity line connect advance and
retarded, advanced and Keldysh, and Keldysh and re-
tarded Green’s functions, but not two retarded or two
advancedones. Thus,retainingonlythesetwoterms9,15,
we express the weak localization correction in terms of
the Cooperon C in the time representation, III. COOPERON IN A GRANULAR ARRAY
mn
2e2
δσ = Re dr ...dr t∗(r ,r )t(r ,r )
WL 1 4 1 2 3 4 Cooperon describes the probability amplitude of elec-
− π
∞ Z tron return and in the case of a homogeneous medium
dtC (r ,r ;r ,r ;t, t) , (14) with electron diffusion coefficient D obeys the equation
mn 1 4 3 2
× −
Z−∞
5
∂ ∂ e e 2
D i A(r, t/2) i A(r,t/2) C˜(r,r′;t,t′)=δ(r r′)δ(t t′). (16)
∂t − ∂r − c − − c − −
( (cid:20) (cid:21) )
Here the vector potential A accounts for the fluctuating gauge by the relation
electric fields representing the effect of electron-electron
interactions,andshouldbeconsideredasaGaussianclas-
sical random variable with zero average.
In order to adapt Eq. (16) to the case of a granular C˜(r,r′;t,t′)=C(r,r′;t,t′)) (17)
medium, it is convenientto performa gaugetransforma- t/2 t′/2
tion, afterwhichthe fluctuating fieldsaredescribedby a exp ie ϕ(r,t′′)dt′′ ie ϕ(r′,t′′)dt′′
randomscalarpotentialϕ(r,t),ratherthanbythevector × ( −
Z Z
potential A(r,t), −t/2 −t′/2
+ie ϕ(r,t′′)dt′′ ie ϕ(r′,t′′)dt′′ ,
t − )
A(r,t)=c ϕ(r,t′)dt′ Z Z
r
∇
Z
(we assume there are no magnetic fields applied to the
system). DefiningtheCooperonC(r,r′;t,t′))inthenew we obtain the equation
∂ ie ie ∂2
+ ϕ(r,t/2) ϕ(r, t/2) D C(r,r′;t,t′)=δ(r r′)δ(t t′). (18)
∂t 2 − 2 − − ∂r2 − −
(cid:26) (cid:27)
Note that C˜(r,r;t, t) = C(r,r;t, t), and thus C can foratwo-dimensionalsquarelattice),andthesummation
− −
be used instead of C˜ in for evaluation of the WL correc- inthelasttermontheleft-handsiderunsoverN nearest
tion (15). neighbors k of the grain n.
Returning to the consideration of a granular array, Equations (15) and (19) provide a convenient starting
we assume that the intra-grain conductance is high, pointforevaluationoftheweaklocalizationcorrectionat
ggr gT. Then the fluctuating potential ϕ(r,t) does temperatures T T∗, see Eq. (4). At higher tempera-
≫ ≪
not vary from point to point within a single grain, while tures, the spatial dispersion of the fluctuating potentials
exhibiting random fluctuations of the inter-grain poten- and of the Cooperon inside a grain becomes important.
tial differences. This allows us to coarse-grain function The temperature domain T < T∗ is separated in
ϕ(r,t), replacingits dependence onr by the dependence two characteristic regions by the∼scale T , Eq. (2). At
cr
on the grain number n. We also can simplify the spatial T T , the dependence of Cooperon C on n n′ is
cr
dependenceoftheCooperonC(r,r′;t,t′)),incaseweare sm≪ooth,andthefinitedifferenceequation(19)can−bere-
interested in times long compared to the intra-grain dif- placed by the corresponding differential equation, which
fusiontime. Indeed,inthatcaseC doesnotchangewhile essentially returns one to the continuous-medium case,
r or r′ vary within a grain. Therefore, the dependence see Eq. (18). Weak localization corrections in this case
of the Cooperon on the coordinates may be replaced by arestudiedindetailinRefs. 9,15. Belowweconcentrate
thedependenceonthegrainnumbersnandn′ whichthe on the temperatures above the crossover.
coordinates r and r′ belong to. The resulting coarse-
grained equation for the Cooperon reads:
∂ ie ie g δ
+ ϕn(t/2) ϕn( t/2)+N T Cnn′(t,t′) IV. QUANTUM CORRECTION TO
∂t 2 − 2 − π CONDUCTIVITY ABOVE THE CROSSOVER
(cid:26) (cid:27)
g δ TEMPERATURE
T Ckn′(t,t′)=δnn′δ(t t′). (19)
− π −
k
X In the temperature regime of interest,
Here N is the number of junctions to a single grain(i.e.,
the coordination number of the lattice of grains; N = 4 T∗ T >T , (20)
cr
≫ ∼
6
as we have explained in the Introduction, electron tra- tation, we obtain
jectories are classifiedaccordingto the number of tunnel
junctionstheycross—thelongerarethetrajectories,the e2 ϕn(t)ϕn′(t′) = πTd2δ(t t′)
lesssignificantaretheir contributions. Itmeans thatthe h i g −
T
matrixCnn′ rapidlydecaysawayfromthediagonal. The dq eiq(n−n′)
biggestmatrixelementsareC ,andthemostimportant , (24)
nn × (2π)d (1 cosq d)
trajectories are those which do not leave the grain. Eq. Z a − a
(15) implies that these trajectories do not contribute to where the summation iPn the denominator is carried over
the weak localization correction, and one needs to con- all available Cartesian components a = x,y, and q are
a
sider the next-order contribution coming from trajecto- the basis vectors of square lattice of grains.
ries crossing a single junction once. This leads us to Performing the averaging in Eq. (23) with the help of
Eq. (3) and also allows us to verify the existence of the Eq. (24) is cumbersome but straightforward, since for
crossovertemperature Eq. (2). Gaussian fields exp(i...) = exp( ... /2). For the
At T Tcr we expect strong fluctuations of the po- weak localizationhcorrectioni, we obta−inh i
≫
tential differences between the grains, making coherent
returns of an electron to the grain of its origin improb- δσWL gTδ
= A , (25)
able. The returns are described by the term in Eq. (19) σ − T
0
containingthesumoverk. Neglectingthatterm,wefind 1 dq 1 cosq d −1
for the diagonal component C (tt′) of the Cooperon A − a ,
nn ≡ N2V (2π)d (1 cosq d)
a (cid:18)Z a − a (cid:19)
X
g δ
Cn(0n)(t,t′)=θ(t−t′)exp −N Tπ (t−t′) whereV is the volumeofthe grPain(V =d2 in2D).Note
(cid:26) thattheinterferencecorrectionEq.(25)dependsontem-
t/2 −t/2 perature and on the type of lattice the grains form. The
ie ϕ (t′′)dt′′ ie ϕ (t′′)dt′′ .(21)
− Zt′/2 n − Z−t′/2 n ) dependence on the lattice type comes through the coef-
ficient A; for a square 2D lattice we find A = 1/4. One
The phase factors here reflect the specific gauge we used can easily generalize the evaluation of A onto the case
in Eq. (19). ofatriangularandmorecomplicatedlattices,eventually
Next, we write the Cooperon equation (19) for the even describing disordered media like ceramics.
nearest-neighbor sites m and n,
∂ ie ie g δ V. MAGNETIC FIELD EFFECT
+ ϕ (t/2) ϕ ( t/2)+N T C (t,t′)
m m mn
∂t 2 − 2 − π
(cid:26) (cid:27)
As the estimate Eq. (5) suggests, the action of the
g δ
= T C(0)(t,t′). (22) magnetic field on Cooperon is two-fold. A part of the
π nn
Cooperon suppression comes from the intra-grain elec-
The terms with Cn′,n describing the grains n′ separated tron motion, and another part stems from the magnetic
from n by two tunnel junctions, are small and can be field effect on the inter-grain coherence. Since ggr gT,
≫
omittedinthisapproximation. UsingEqs. (21)and(22), the interesting range of the fields corresponds to a small
we obtain for the neighboring grains m and n flux penetrating a grain, Bd2 Φ0. We can then con-
≪
sider the effect ofmagnetic field on the Cooperonwithin
g δ g δ t a grain perturbatively. To implement the perturbation
C (t,t′)= T θ(t t′)exp N T (t t′) dt
mn 1 theory, it is convenient to use a “tailored” to the grains
π − (cid:26)− π − (cid:27)Zt′ shape gauge of the magnetic field B. For definiteness,
t1/2 −t1/2
exp ie ϕ (t′′)dt′′ ie ϕ (t′′)dt′′ we concentrateonthe case ofa two-dimensionalarrayof
n n
× (− Zt′/2 − Z−t′/2 “flat” grains connected by point-like tunneling contacts,
see Fig. 1. We define the gauge for the points within the
t1/2 −t1/2
+ie ϕ (t′′)dt′′+ie ϕ (t′′)dt′′ . (23) grains by the relations
m m
Zt/2 Z−t/2 ) A (r)=τ ψ (r)+A , 2ψ =B, ψ (r b )=0.
φ n n n n n
×∇ ∇ ∈
This expression has to be averaged over the Gaussian (26)
fluctuations of the field ϕ. Its correlation function is de- Here τ is the normal to the plane of the grains, and
fined by the fluctuation-dissipation theorem and reads17 bn is the boundary of the n–th grain. The second and
thirdrelationsinEq.(26)fullydefinetheboundaryprob-
2T 1 lem for a scalar function ψ (r). The constants A are
e2 ϕϕ (q,ω)= Im , n n
h i − ω Π(q,ω) tuned in such way that the vector-potential is continu-
ous at the points of contact between the grains. Up to
where Π is the polarization operator, calculated for a a discrete analogue of the gradient of a scalar function,
granularmedium inRef. 10. In the space-time represen- these constants are determined fully by the solution of
7
the boundaryproblemsforallψ (r). Itis clearthatthe r,r′ belong to. In addition, we multiply the Cooperon
m
discrete version of curl applied to A must be equal B defined in Eq. (17) by yet one more gauge factor,
n
upon averaging over the array; the characteristic differ-
ence A A for two nearby junctions is of the order
n m
A A − B d. C (r,r′;t,t′)=Cφ (r,r′;t,t′)exp(iA r iA r′).
n− m ∼ · mn mn m· − n·
In the definition of the Cooperon, it is convenient to (27)
presentagainthe coordinatesaspairs r,n and r′,n′ Inthesenewnotations,theequationforCooperoninthe
{ } { }
whichpointexplicitlytothelabelofgrainsthetwopoints absence of tunneling has the form
2
∂ ie ie ∂ e
+ ϕ(r,t/2) ϕ(r, t/2) D i τ ψ (r) Cφ (r,r′;t,t′)=δ δ(r r′)δ(t t′), (28)
(∂t 2 − 2 − − gr(cid:18)∂r − c ×∇ m (cid:19)) mn mn − −
where D d2g δ is the diffusion coefficient within the with the unit quantum, Bd2 Φ , we may treat the
gr gr 0
∼ ≪
grain (here g δ is the Thouless energy for the electron effect of a magnetic field within a grain perturbatively.
gr
motionwithinagrain). WiththedefinedgaugeEq.(26), Considering the low-energy limit, T g δ, and taking
gr
the normal to the boundary component of A(r) is zero. into account the boundary condition≪s for Cφ, we start
Thus, the magnetic field does not affect the boundary perturbations from r–independent Cooperon Cφ (t,t′).
mn
conditions for Cooperon, i.e. the normal component of In the presence of inter-grain tunneling, the correspond-
∂Cφ/∂r at the boundary is zero. ing generalization of Eq. (19) reads
Aslongasthefluxpiercingonegrainissmallcompared
∂ Bd2 2 ie ie g δ
+αg δ + ϕ (t/2) ϕ ( t/2)+N T Cφ (t,t′)
(∂t gr (cid:18) Φ0 (cid:19) 2 m − 2 m − π ) mn
g δ
T eirkm·(Ak−Am)Cφ (t,t′)=δ δ(t t′). (29)
− π kn mn −
k
X
Here the magnetic field dependence sponding coarse-grainedcoordinate R, we find
∂ Bd2 2 ie 2
+αg δ D A(R) (30)
gr R
Bd2 2 D e2 (∂t (cid:18) Φ0 (cid:19) − (cid:20)∇ − c (cid:21)
αg δ = gr d2r ψ (r)2
gr (cid:18) Φ0 (cid:19) d2 c2 Zgrain |∇ n | +ieϕ(R,t/2) ieϕ(R, t/2) Cφ(R,R′;t,t′)
2 − 2 −
(cid:27)
=δ(R R′)δ(t t′).
− −
comesfromtheψ –dependentterminEq.(28)integrated
n The second term in the left-hand side here reflects the
over the volume of a single grain; α 1 is the dimen-
∼ suppression of interference by the magnetic flux pene-
sionless coefficient depending on the grains shapes. The
trating the grains. Apart from that term and from the
vector rkn points to the junction between grains k and value of the effective diffusion constant D = π−1g δd2,
T
n.
which reflects the granularity of the medium, this equa-
tionisidenticaltothatofahomogeneousthinfilm. Using
The discreteness of the medium is adequately ac-
the known results1 for the films, we find the magnetore-
counted for by the structure of Eq. (29). However,
sistance of a granular array,
the discreteness is not important in the domain of low
temperatures and relatively low fields, T T and
cr δσ (B,T)=δσ (B,T) δσ (0,T) (31)
B Bsg. There we can replace the left-h≪and side of MR WL − WL
≪ φ e2 T g Bd2 2 Bd2 T
Eq. (29) by its gradient expansion. After the expansion = ln + gr + ln
and replacement of the grain number n by the corre- 2~( "Tcr gT (cid:18) Φ0 (cid:19) Φ0 #− Tcr)
8
f G
6
4
g
2
T
y T −1
-3 -2 1 2 3
-2 ln T
2
-4
-6
1
FIG.3: Magneticfielddependence(31)oftheconductivityat
low temperatures, T ≪ TcrgT/gcr. The horizontal axis here Tcr T
isproportionaltothelogarithmoftheappliedmagneticfield,
y =ln(Bd2/Φ0)+ln(ggr/gT); the vertical axis is the magne- FIG. 4: Sketch of the temperature dependence of the con-
toresistanceindimensionlessunits,δσMR =(e2/2~)f(y)with ductance in various magnetic fields: B = 0 (1); B ≪ Bφsg
f(y) = y+ln(1+ey), see Eq. (31). The crossover between (2). Thetemperaturesat whichcurve2departsconsiderably
lnB and 2lnB dependencesis clearly seen. from curve1dependontheapplied field;thesetemperatures
are of the order of Tcr(Bd2/Φ0) and Tcr(Bd2/Φ0)2(ggr/gT)
for B ≪ (Φ0/d2)(ggr/gT) and B ≫ (Φ0/d2)(ggr/gT), respec-
(we dispensedwith the factorα 1 here). The two field tively.
∼
scales introduced in Eqs. (6) and (7) can be obtained
from a comparison[in the argumentof logarithmic func-
tionEq.(31)]ofthedephasingtermT/T withthelinear the electronmotion. This correspondsto magnetic fields
cr
and quadratic in B terms, respectively. At lowest tem- B > (Φ /d2)(τδ)−1, with τ being the momentum re-
0
peratures, there is a clear crossover in the δσ vs. B laxation time in a grain. Since τδ 1 (conditions for
WL
≪
dependence from δσ lnB to δσ 2lnB. Note metallic diffusive behavior), such fields are well outside
WL WL
∝ ∝
that the crossover occurs in the 2D regime, where the our consideration range.
typical closed path for a coherent electron motion spans Apart from the weak localization correction, there is
manygrains. Itisremarkablethateveninthe2Dregime onemoretemperaturedependentcontributiontothecon-
there is a clear difference in the magnetoresistance of a ductance — interaction correction. For granular media,
granular system from that of a homogeneous film, see it was calculated for all temperatures in Ref. 18. It
Fig. 3. crossesoverfromlow-tohigh-temperatureregimeatthe
temperature g δ, which is different (much lower) than
T
T . For a two-dimensional array, this correction is log-
cr
VI. DISCUSSION arithmic at any temperatures; for T g δ, one has
T
δσ/σ g−1ln(g E /T), where E is≫the charging en-
0 ∼ T T C C
Let us now discuss the temperature dependence of ergy in a single grain. The temperature dependence of
the conductance in various magnetic fields (Fig. 4). In the interaction correction is featureless at T T , and
cr
∼
zero field, the weak localization correction behaves as thereforeitshouldnotmaskthecrossoverinthetemper-
δσ /σ g−1lnT at T < T and then crosses over ature dependence of the WL correction. The interaction
WL 0 ∼ T cr
to the power-law behavior, δσ /σ T /(g T), at correctionisalsoindependentofthemagneticfield. Thus
WL 0 cr T
∼
higher temperatures. Finite magnetic field leads to sup- it does not affect the crossover in the magnetic field de-
pression of the WL correction even at the lowest tem- pendence of the conductance, which is induced by the
perature. Thus, at B (Φ /d2)(g /g ) the WL cor- granular structure. The measurements of the conduc-
0 T gr
≪
rection becomes temperature-independent. (Note that tance therefore can be used to characterize the medium.
(Φ0/d2)(gT/ggr) ≪ Bφsg.) At higher temperatures, the Letusfinally givesomeestimates. We considermetal-
dimensionless conductivity δσ /σ has the same tem- lic grains of a size of 500 nm, which can be easily pro-
WL 0
perature dependence as at B = 0. In higher fields, duced lythographycally6. They have the level spacing
(Φ /d2)(g /g ) B Bsg, the same low-temperature of order δ/k 20mK. Choosing g = 10, we obtain
0 T gr ≪ ≪ φ B ∼ T
saturation occurs at T = (Bd2/Φ0)2(ggr/gT)Tcr, see the crossover temperature Tcr = 2K, that can be easily
Fig. 4. In the highest fields, B Bsg, the WL correc- observed experimentally.
≫ φ
tion is suppressed for all trajectories — even those lying This work was supported by NSF Grants DMR02-
within a single grain, and the WL correction disappears 37296, DMR04-39026, and EIA02-10736 (University of
at all temperatures. Minnesota),by the U.S. DepartmentofEnergy,Office of
Note that all magnetic fields which we have discussed Science via the contract No. W-31-109-ENG-38, by the
above are too small to change orbital dynamics of elec- MinervaEinsteinCenter(BMBF), andby Transnational
trons. Indeed, the cyclotronradius,r =mv c/eB must Axis Program RITA-CT-2003-506095 at the Weizmann
c F
be smaller than the mean free path l in order to affect Institute of Science. We acknowledge useful discussions
9
with Igor Aleiner, Yuval Gefen, and Alex Kamenev. a different method, in the work by Biagini et al very
Note added: After completing this work, we noticed recently19. We are grateful to Andrei Varlamov for the
that a formula similar to our Eq. (31) was derived, with discussion of relation between the two works.
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