Interaction corrections: temperature and parallel field dependencies of the Lorentz number in two-dimensional disordered metals

0.25 MB·English

by  G. Catelani

Interaction corrections: temperature and parallel field dependencies of the Lorentz
number in two-dimensional disordered metals.
G. Catelani
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
(Dated: February 6, 2008)
7 The electron-electron interaction corrections to the transport coefficients are calculated for a
0 two-dimensional disordered metal in a parallel magnetic field via the quantum kinetic equation
0 approach. For the thermal transport, three regimes (diffusive, quasiballistic and truly ballistic)
2
can be identified as the temperature increases. For the diffusive and quasiballistic regimes, the

n Lorentznumberdependenceonthetemperatureandonthemagneticfieldisstudied. Theelectron-
a electron interactions induce deviations from the Wiedemann-Franz law, whose sign depend on the
J temperature: at low temperatures the long-range part of the Coulomb interaction gives a positive

1 correction, while at higher temperature the inelastic collisions dominate the negative correction.
3 By applying a parallel field, the Lorentz number becomes a non-monotonic function of field and
temperatureforallvaluesoftheFermi-liquidinteractionparameterinthediffusiveregime,whilein

] thequasiballistic case this is trueonly sufficiently far from the Stonerinstability.
n
n
PACSnumbers: 71.10.Ay,72.15.Gd,72.15.Eb
-
s
i
d
I. INTRODUCTION ofthe presentwork. By extending the resultsofRef.6,I
.
t analyze in detail, for two-dimensional systems, the tem-
a
m AstandardresultoftheDrude-liketheoryoftransport perature and parallel magnetic field H dependencies of
indisorderedmetalsis the Wiedemann-Franzlaw1 relat- the “generalized” Lorentz number L, defined as
-
d ingthe(Drude)thermal(κ )andelectrical(σ )conduc-
D D L(T,H) κ(T,H)/σ(T,H)T, (1.2)
n tivities via the Lorentz number L0: ≡
o
c κ π2 where, due to the electron-electron interaction correc-
[ σDDT =L0 ≡ 3e2 , (1.1) tdieopnesn,dbeontth(acosnimduilcatrivaitniaeslysairsefotremzepreor-adtiumree-nsaionndalfiseylds--
2
whereT isthetemperatureinenergyunits (k =1)and tems – open quantum dots – is presented in Ref. 11; the
v B
e is the electronic charge. “Drude-like theory” means parallel field dependence of σ is considered in Ref. 12).
4
4 that two assumptions are made in order to calculate Because of difficulties in accurately measuring the elec-
3 the transport coefficients: 1. the electrons do not in- tronic thermal conductivity, very few experiments have
0 teract with each other; 2. the scattering of the elec- beenperformedintwo-dimensionalsystemswithregards
1 tronsontotheimpuritiesiselastic.2,3 Whileitwasshown to the thermal transport – for example the Wiedemann-
6 long ago4 that the interplay of electron-electron inter- Franz law was found to hold13 in a Si MOSFET sample
0
actions and disorder leads to logarithmically divergent, withintheexperimentalaccuracy,andthevalidityofthis
/
t temperature-dependent corrections to the electrical con- law was checked for the weak localization correction.14
a
m ductivity at low temperatures T ¯h/τ (τ is the mean One of the difficulties in determining κ is in separat-
≪
free time for the impurity scattering), it is only recently ing the electronic contribution to the total thermal con-
-
d that such effects have been correctly evaluated at higher ductivity from the phonons’ contribution; however, by
n temperatures5 and for the thermal transport.6–8 In par- measuring the thermal conductivity in the presence of
o ticularearlycalculations9,10oftheinteractioncorrections a magnetic field it may be possible to extract the elec-
c
to the thermal conductivity arrived at contradictory re- tronic field-dependent part, as done e.g. for cuprate
:
v sults, due to technicaldifficulties inthe proper construc- superconductors.15Sincenewmethodsformeasuringthe
i tion of the energy current density operator (both in the thermal conductivity in thin films are being explored,16
X
diagrammatic technique and in the kinetic equation ap- the study of the field dependence of L could be experi-
r
a proach). ThisissuehasbeenresolvedinRef.6,wherethe mentally relevant.
localformofthecollisionintegralforthekineticequation The paper is organized as follows: in the next section
is also presented. I examine the temperature dependence of the Lorentz
Inderivingthequantumkineticequation,aproperde- number L to identify different regimes as a function of
scriptionofthedisorderedFermi-liquidisobtainedbyin- the dimensionless parameter Tτ/¯h and to discuss the
troducing bosonic soft modes (interacting electron-hole various approximations involved. In Sec. III I present
pairs) which contribute to the energy transport but, be- the results for the dependence of L on the parallel mag-
ing neutral, not to the charge transport. For interac- netic field. The derivation of these results is given in
tion in the triplet channelthese bosons have a total spin Sec. IV. After the conclusions, I briefly consider in Ap-
L=1;thisspindegreeoffreedomisaffectedbythemag- pendix A the field-dependent correction to the specific
neticfield: thedescriptionofsucheffectsisacentralpart heat. Appendix B contains some mathematical details,
2
and Appendix C a discussion of the electrical magneto- 0.03
conductivity in parallel field.
II. TEMPERATURE DEPENDENCE OF THE 0.02
LORENTZ NUMBER.
∆Ls
€€€€€€€€€€€€
L
0
This section contains the result for the temperature
dependence of the Lorentz number L. It is convenient 0.01
to separate L into a “Wiedemann-Franz law” part L
0
[Eq. (1.1)] and a “violation” part δL as follows:
L(T)=L +δL(T), (2.1) 0.00
0 10-2 10-1 100
andforthecorrectionδLtodistinguishthecontributions 2ΠTΤ(cid:144)Ñ
due to interactionsin the singletand in the triplet chan-
nels:
FIG.1: Relativecorrection totheLorentznumberasa func-
δL=δLs+δLt. (2.2) tion of temperature in the diffusive regime. Solid lines are
given by Eq. (2.3), while dashed lines by the approximate
Forclarity,the twoterms areconsideredseparately. The expression (2.5). For thick lines rs = 0.1 and for thin ones
results given below are derived in Sec. IVA within the rs = 1; the conductances are for each pair of curves (top to
bottom): g=100, g=400 and g=1000.
quantum kinetic equation approach – this is a perturba-
tive approach with 1/g as the small parameter, where
g = σ/(2e2/h) 1 is the dimensionless conductance; it
≫
assumesthevalidityoftheFermi-liquidpicture,whichin
turns requires T E , with E the Fermi energy. m/π is the 2D density of states). Compared to the
F F
≪ Altshuler-Aronov interaction correction to the electrical
conductivity,4 this logarithmic correction to the Lorentz
A. Singlet channel. number has a completely different physical origin: it
arises from the energy transported over long distances
The singlet correction δLs is, with logarithmic accu- bythe neutralbosonicsoftmodesofthe interactingelec-
tronsystem.6 Atlowtemperatures,thisadditionalchan-
racy:
nel for the energy transport (as compared to the charge
δLs 1 transport)leadstoanincreaseinthe thermalconductiv-
= g 2πTτ/¯h ln r E /T
1 s F ity over the electrical conductivity and therefore to an
L πg
0 1(cid:20) (cid:0) (cid:1) (cid:16) (cid:17) enhancement of the “generalized”Lorentz number (1.2).
g2 πTτ/¯h ln 1+(h¯/2πTτ)2 (2.3)
− 4 InFig.1IplottherelativechangeoftheLorentznum-
1 2(cid:0)πTτ 2(cid:1) (cid:16) (cid:17) ber δLs/L0, Eq. (2.3), as a function of 2πTτ/¯h for three
ln EF/T , conductances (g = 100, 400 and 1000) and for two val-
− 5 ¯h
(cid:18) (cid:19) (cid:16) (cid:17)(cid:21) ues of the interaction strength (rs = 0.1 and rs = 1);
where the functions g and g , given in Eqs. (4.4), de- for comparison the curves obtained using the approxi-
1 2
scribe the crossover form the low temperature diffusive mate expression (2.5) are also shown. From the figure
regime to the higher temperature quasiballistic one, and andthe dependence onrs in Eq.(2.3) it follows that the
r is the “gas parameter” characterizing the interaction deviationfromtheWiedemann-Franzlawgrowswiththe
s
strength: interactionstrength.20Forlowconductancesandtemper-
atures the (positive) change in the Lorentz number is of
√2e2 theorderofafew percent;unfortunatelytheuncertainty
r = (2.4)
s ε¯hv in measurements of the thermal conductivity in metallic
F
films16 is also of this magnitude, making a comparison
with vF the Fermi velocity and ε the dielectric constant. with the present theory pointless.
In the diffusive regime T ¯h/2πτ, both g and g
1 2
≪ As the temperature increases, the inelastic collisions
tend to 1 and Eq. (2.3) reduces to:
between the electrons and the bosons tend to inhibit
δLs 1 π¯hDk2 the energy transport more efficiently. The quasiballistic
d = ln , (2.5)
regime is reached in the temperature range
L 2πg 2T
0 (cid:18) (cid:19)
where D = τv2/2 is the diffusion constant and k =
F
2πνe2/ε is the 2D inverse screening radius (where ν = ¯h/2πτ T Ts , (2.6)
≪ ≪ qb
3
plotted (dashed lines). In agreementwith the above dis-
0.00
cussion, comparison of the dashed and solid lines shows
-0.01 that Eq. (2.9) deviates significantly from the full expres-
sion (2.3) at low conductance, while there is good agree-
-0.02 ment at high conductance. The intersections between
∆Ls the solid lines and the upper thin dotted curve are at
€€€€€€€€€€€€ -0.03
L0 T = 0.1Tqsb; it is evident that the region of validity of
-0.04 the quasiballistic approximation grows with the conduc-
tance. Forallconductancesthe(negative)correctioncan
-0.05 be a few percent; in Ref. 13 the Lorentz number was
measured21 in a 2DEG and found to be slightly smaller
-0.06
thanL ,inqualitativeagreementwiththepredictionsin
0
0 5 10 15 20 the present work. However the uncertainties are of the
2ΠTΤ(cid:144)Ñ same order of the calculated effect and hence no quanti-
tative comparison is possible.
FIG.2: Relativecorrection totheLorentznumberas afunc-
tion of temperature in the quasiballistic regime. Solid lines
B. Triplet channel.
are given by Eq. (2.3) with rs = 0.1, while dashed lines by
theapproximateexpression (2.9). Theconductancesare(left
to right): g = 100, 400, 1000, 1600 and 2500. Thin dotted For the triplet channel interaction correction, I con-
lines are calculated so that they intersect the solid lines at sider separately, for simplicity, the diffusive and quasi-
T =0.1Ts (top curve) and T =0.2Ts (bottom curve).
qb qb ballistic regimes; in the former case the correction is:
δLt 3 1 1
where Ts is the solution of: d = ln(1+Fσ) 1 ln(1+Fσ) ,
qb L 2πg 0 − πg − Fσ 0
0 (cid:20) 0 (cid:21)
4 Ts 2 E (2.10)
qb F
πg ln =1; (2.7) where the first term on the right hand side is again due
5 E Ts
(cid:18) F(cid:19) qb to the bosonic energy transport, and the second one
for large conductances this gives: originates from the interaction-induced energy depen-
dence of the elastic cross section. While the sign of this
5 temperature-independentcorrectionisdeterminedbythe
Tqsb ≈EFs2πgln(2πg). (2.8) sign of the Landau Fermi-liquid constant F0σ, its contri-
bution to the total correction δL [Eq. (2.2)] is generally
In this regime, the dominant contribution to the singlet small22 andthe overallpositive signof δL at low enough
correction (2.3) is: temperatures is determined by δLs. I do not plot sep-
d
arately the contribution (2.10) to δL, since its effect is
δLs 1 2πTτ 2 E
qb F simplytoshiftupwards(downwards)thecurvesinFig.1
= ln . (2.9)
L0 −5πg(cid:18) ¯h (cid:19) (cid:18) T (cid:19) for F0σ >0 (F0σ <0).
In the quasiballistic regime the correction reads:
According to condition (2.6), this expression is applica-
ble if Ts ¯h/2πτ; this can be satisfied only for large
qb ≫ Fσ 2
enough conductances. For example at g =14 I find (nu- δLt =3δLs 0
merically) Tqsb ≈ 0.11EF ≈ 10¯h/2πτ, and at g = 720, qb qb(cid:18)1+F0σ(cid:19) (2.11)
Tqsb ≈ 0.01EF ≈ 50¯h/2πτ; in the latter case, and for = 3 2πTτ 2ln EF F0σ 2
larger conductances, Eq. (2.9) can be expected to have −5πg ¯h T 1+Fσ
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) 0 (cid:19)
a sufficiently large range of validity, while in the former
(and generallyfor small conductances)there is no quasi- with δLs given in Eq. (2.9). As for the singlet chan-
qb
ballistic regime. At temperatures of the order of Ts the nel, this negative correctionoriginates from the inelastic
qb
energytransportbecomestrulyballisticinnature,asthe electron-bosoncollision,andsimilarlytothesingletchan-
dominant processes responsible for the relaxation of the nelcorrection,thevalidityofthisexpressionislimitedat
energy current are the inelastic electron-boson collisions high temperatures by Tt – the equation defining this
qb
and not the electron-impurity collisions. Although the quantity is obtained by multiplying the left hand side of
high temperature regime T >Tqsb is not consideredhere, Eq. (2.7) by 3(F0σ/1+F0σ)2. It is evident that, when
itcanbe treatedwithinthe∼kinetic equationapproach.18 the quasiballistic approximation is applicable, plotting
Fig.2showstherelativechangeoftheLorentznumber the sum of Eq. (2.11) and (2.9) would give Fig. 2 with a
δLs/L asafunctionof2πTτ/¯hforfivedifferentconduc- rescaled vertical axis, as for any value of Fσ the triplet
0 0
tances (solid lines); for the lowest and highest conduc- channelcontributionenhancesthesingletchannelcorrec-
tances considered, the approximate result (2.9) is also tion.
4
III. PARALLEL FIELD DEPENDENCE OF THE
0.8
LORENTZ NUMBER.
0.6
InthissectionIpresenttheresultsfortheparallelmag-
netic field dependence of the Lorentz number. The par-
DL 0.4
allel field H affects the electrons by shifting the energy €€€€€€€€€€d€€€€
levels by the Zeeman energy L0 0.2
E =g µ H, (3.1)
Z L B
0.0
where g is the Lande g-factor and µ the Bohr mag-
L B
neton. The Lorentz number depends on H only through -0.2
this energy and the renormalized Zeeman energy E∗:
Z 10-2 10-1 100 101 102
∗ EZ EZ(cid:144)2ΠT
E = . (3.2)
Z 1+Fσ
0
As it is the case for other transport properties (e.g. FIG.3: Relativedeviation πg∆Ld/L0 of theLorentznumber
the magneto-conductivity), it is convenient to consider from its zero-field value in the diffusive regime. Solid lines
the deviation ∆L of the Lorentz number from its zero- aregiven byEq.(3.4) with, from top tobottom: F0σ =−0.7,
field value: −0.4, −0.2, 0.2 and 0.4. For F0σ =−0.7 and 0.4 theapproxi-
mate expressions (3.5) (dashed lines) and (3.7) (dot-dashed)
∆L(T,H)=L(T,H) L(T,0). (3.3) are also shown for comparison.
−
Once again I address separately the diffusive and quasi-
ballistic regimes; in both regimes the system is assumed In the opposite case E ,E∗ 2πT the approximate
to be far from the full polarization, i.e. EZ EF. The formula is: Z Z ≫
≪
derivation of the results can be found in Sec. IVB.
∆L 1 2 1
d ln(1+Fσ)+ 1 ln(1+Fσ) ,
L ≈−πg 0 3πg − Fσ 0
0 (cid:20) 0 (cid:21)
A. Diffusive regime. (3.7)
or equivalently:
For T ¯h/2πτ, the field-induced change in the
≪ 2
Lorentz number is: ∆L δLt (3.8)
d ≈−3 d
∆L 1 1 3 E E∗
d = + I Z I Z
L0 − π1g (cid:18)1F0σE 2(cid:19)(cid:20) 1E(cid:18)2πT(cid:19)− 1E(cid:18)∗2πT(cid:19)(cid:21) wasitfholδloLwtdsg:1i2veinnitnheEdqi.ff(u2s.1iv0e).reTghimiserethsueltcocrarnecbteioenxpisladionmed-
Z I Z I Z inated by processes with small energy and momentum
− πgFσ 2πT 2 2πT − 2 2πT
0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) exchange,andinthestrongfieldthebosonicmodeswith
(3.4)
non-zero spin projection become gapped with the gap
givenbytheZeemanenergy. Thereforethecontributions
with the dimensionless functions I and I defined in
1 2
Eqs.(4.16)-(4.17). SimilarlytoEq.(2.10),thetermswith due to these modes must drop out from the total correc-
tion to the Lorentz number: this is why the correction
the numerical prefactor 3/2 are due to the bosonic en-
ergytransport,whiletheremainingones,proportionalto (3.7) partially cancels the one given in Eq. (2.10), with
1/Fσ,originatefromtheenergydependenceoftheelastic the surviving contribution originating from the modes
0
crosssection. The structure ofthis expressionas the dif- with zero total spin projection.
ference of terms which depend on different energy scales In Fig. 3 the relative deviation ∆Ld/L0, multiplied
(oif.et.heEqZuaanntdumEZ∗k)inceatnicbeequtraatcioend,binackwhtiochthtehestbruocstounriec ebnytπvga,luiess polfotthteedpaarsamaeftuenrcFti0oσn. Aoft EfieZld/s2πsTuchfotrhadtifftehre-
contributions to the collision integral always appear as Zeeman energy is larger than temperature the deviation
differencesbetweenasoftmodepartanda“ghost”part.6 becomes quickly field-independent, but near EZ 2πT
∼
In the weak field limit E ,E∗ 2πT Eq. (3.4) be- all the curves are non-monotonic; the presence of peaks
comes: Z Z ≪ is due to the above discussed dependence on the two
different energies E and E∗, and through the latter
2 Z Z
∆Ld 1 f (Fσ) EZ (3.5) (and the 1/F0σ prefactors) the peaks’ positions depend
L ≈−πg d 0 2πT on Fσ. The temperature dependence of the deviation at
0 (cid:18) (cid:19) 0
fixed field can also be read from this graph by following
with
the curves from the right (low temperature) to the left
x(4+3x) (high temperature): the deviation is temperature inde-
f (x)= . (3.6)
d (1+x)2 pendent at low temperatures T EZ/2π and displays
≪
5
a power-law decay ( T−2) at high temperatures; again 0.08 0.5 0.3 0.4
the non-monotonicb∼ehaviorcharacterizesthe intermedi- 0.003 − − 0.2
ate regime. 0.001 −
0.06
-0.001
B. Quasiballistic regime. DL -0.003
€€€€€€€€€€q€€€b€€€ 0.04 10-2 10-1 100
L
0
HereIconsiderthequasiballisticregime¯h/2πτ T
Tt , with Tt defined after Eq.(2.11). In this case≪I fin≪d: 0.02 F0σ=0.2
qb qb
∆L 3 2πTτ 2 Fσ 2 E∗ Fσ
qb = 0 I Z ; 0 0.00
L −2πg ¯h 1+Fσ 3 2πT 1+Fσ
0 (cid:18) (cid:19) (cid:18) 0 (cid:19) (cid:18) 0 (cid:19) 10-2 10-1 100 101 102
(3.9)
with I given in Eq. (4.26). For E∗ 2πT the result EZ(cid:144)2ΠT
3 Z ≪
takes the form:
2 FIG. 4: Relative deviation of the Lorentz number from its
∆L 1 τE
qb f (Fσ) Z , (3.10) zero-field value as a function of the Zeeman energy in the
L0 ≈−πg qb 0 (cid:18) ¯h (cid:19) quasiballistic regime (normalized as explained in the text)
for different values of the Fermi-liquid parameter, as labeled
where (thecurvepartiallycoveredbytheinsetcorrespondstoF0σ =
−0.7). Solid lines are given by Eq. (3.9), while dashed lines
x 2 1+2x+4x2
bytheapproximateexpression(3.12). Theinsetshowsdetails
f (x)=
qb 1+x (1+2x)2 ofthelow-fieldregime,withthedashedcurvescorresponding
(cid:18) (cid:19) (cid:20) (3.11)
2x2(3+6x+4x2) x to Eq.(3.10).
+ ln ,
(1+2x)3 1+x
(cid:12) (cid:12)(cid:21)
(cid:12) (cid:12)
while with logarithmic accuracy t(cid:12)(cid:12)he lar(cid:12)(cid:12)ge field limit 0.03
E∗ 2πT is:
Z ≫
0.02
∆L 2 2πTτ 2 E Fσ 2
L0qb ≈ 5πg(cid:18) ¯h (cid:19) ln(cid:18) TZ(cid:19)(cid:18)1+0F0σ(cid:19) . (3.12) €D€€€€L€€€€€q€€€b€€€ 0.01
L
0
ComparisonofEq.(3.12)with Eq.(2.11)showsthat the
0.00
partialcancelationthatwasfoundinthediffusivelimitis
alsorealizedinthequasiballisticone,butwithanimpor-
tant difference: the gapped modes still contribute to the -0.01
total correctionto the Lorentz number because the qua-
siballisticcorrectionsaredominatedbytheinelasticscat- 0.0 0.5 1.0 1.5 2.0 2.5 3.0
tering with large momentum exchange. The gap there- 2ΠT(cid:144)EZ
foreexcludesthelowenergy(E <E )contributions,but
Z
thehigherenergyones(E <E <E )arestillrelevant.
Z F
FIG. 5: Relative deviation of the Lorentz number from its
Fig. 4 shows the field dependence of ∆L , Eq. (3.9),
qb
zero-field value as a function of the temperature in the qua-
by plotting
siballistic regime (normalized as explained in the text) for
different values of the Fermi-liquid parameter. Proceeding
2
∆L 3 2πTτ
qb clockwise near the origin, starting with the steepest curve,
L0 (cid:30)2πg (cid:18) ¯h (cid:19) the parameters are: F0σ = −0.7, −0.5, −0.3, −0.2, 0.4 and
0.2
asafunctionofE /2πT;forcomparisontheapproximate
Z
result (3.12) is also plotted.23 In the inset the low-field
behaviorofEq.(3.9)iscomparedtoEq.(3.10);exceptfor For completeness, I consider in Fig. 5 the temperature
tthherecshasoeldFv0σal=ue−F0σ.7a,baolvlecuwrhviecsha∆reLnoins-amnoonno-tmonoinco.toTnhiec dependence of ∆Lqb by plotting
th qb
function can be found by requiring f (Fσ) = 0; this
gives Fσ 0.679. Although the pqrbesentht results are ∆Lqb 3 EZτ 2
not valitdhc≃los−e to the Stoner instability (see footnote22), L0 (cid:30)2πg(cid:18) ¯h (cid:19)
they suggests that as Fσ 1 the relationshipbetween
0 →−
energy and charge transport properties can be qualita- as a function of2πT/E in the low to intermediate tem-
Z
tively altered compared to the weakly interacting case. perature regime.
6
IV. DERIVATION. theasymptoticformsofthefunctions g1 andg2 givenaf-
ter Eqs. (4.4). The condition (2.7) is obtained by equat-
ing the (absolute value of the) correction (2.9) to the
This section is devoted to the calculation of the in-
non-interactingLorentznumberL [Eq.(1.1)];theresults
teraction corrections to the Lorentz number using the 0
arenot valid attemperatures higher than Ts because in
formalismofRef.6. Inthe absenceofthemagneticfield, qb
solving the kinetic equation it was assumed that the im-
one can use directly the results of that reference, while
purityscatteringisthedominantprocesscontributingto
a generalization is needed for the parallel field case, as
theenergyrelaxationrate–seethediscussionattheend
discussed in Sec. IVB. From now on, I set ¯h=1.
of Sec. 6.2 of Ref. 6; however, the kinetic equation itself
is still valid.
For the triplet channel, the correction ∆κt was calcu-
A. Temperature dependence.
lated in the diffusive and quasiballistic regimes:
The results presented in Sec. II are a straightforward T 1 T
∆κt = 1 ln(1+Fσ) + ln(1+Fσ)
consequence of the findings of Ref. 6, where it is shown d −18 − Fσ 0 12 0
(cid:20) 0 (cid:21)
thatfortwo-dimensionalsystemsthe thermalconductiv- (4.6)
ity can be written as: for Tτ 1/2π, and
≪
κ=κWF +∆κ. (4.1) ∆κt = 2π2T (Tτ)2ln EF F0σ 2 (4.7)
qb − 15 T 1+Fσ
(cid:18) (cid:19)(cid:18) 0 (cid:19)
Here
for Tτ 1/2π. Multiplying Eqs. (4.6)-(4.7) by 3 and
≫
κ =L σT (4.2) using Eq. (4.5) and the definition (2.1) of δL, the results
WF 0
(2.10)and(2.11)areobtained;thefactorof3arisesfrom
follows the Wiedemann-Franz law with L defined in the summation over the three projections of the total
0
Eq. (1.1) and the electrical conductivity σ includes the spin, which in the absence of magnetic field contribute
interactioncorrections. Theadditionalterm∆κ=∆κs+ equally to the thermal conductivity.
3∆κtisgivenbythesumofthesingletandtripletchannel
corrections. The former was calculated with logarithmic
accuracy:24 B. Parallel field dependence.
T vFk To obtain the results of Sec. III I give here an exten-
∆κ = g (2πTτ)ln
s 1
6 T sion of the calculations of Ref. 6. As in Eq. (4.1), I sep-
(cid:18) (cid:19)
arate the thermal conductivity in a part which follows
T 1
− 24g2(πTτ)ln 1+ (2πTτ)2 (4.3) the Wiedemann-Franz law and a correction; both term
(cid:18) (cid:19) now depend on the applied parallel magnetic field H, or
2π2 2 EF more precisely on the Zeeman splitting, Eq. (3.1). The
T (Tτ) ln
− 15 T termκ (T,H)isstraightforwardlycalculatedusingthe
(cid:18) (cid:19) WF
results of Ref. 12, so one needs to consider only the cor-
with the functions g1 and g2 given by: rection ∆κ(T,H). As discussed in e.g. Refs. 12 and 17,
the singlet and the triplet L =0 contributions to κ are
z
3 1
g1(x)= x2 x 2ψ′ 1/x −x2 −2 , (4.4a) nthoettaoffteacltsepdinbyalothnegpthaerafilleelldfideilrdec[Ltiozni]s.tThheeperffoejeccttoiofnthoef
(cid:26) h (cid:0) (cid:1) i (cid:27) fieldontheremainingL = 1componentsofthetriplet
where ψ′ is the derivative of the digamma function, and z ±
channel correction is to shift the frequency of the inter-
actionpropagators;inthelangaugeofRef.6,thebosonic
26 8 5
g2(x)= x2+ g1(x) . (4.4b) propagators σ(ω,q;n1,n2;Lz) are:
15 3 − 3 L
σ(n ,n ;L )=Ω δ(n n ) (n ;L )+
1 2 z 2 1 2 0 1 z
Note that g (0)=g (0)=1, and g (x) 3/x for x 1. L L
ordUesriningt1h/eg1:definitio2ns(1.2),(4.1)a1nd(≃4.2)Ifindat≪first L0(n1;Lz)L0(n2;Lz) (cid:16)d(L−i)ω1+FF0σ0σiω+ τ1F(cid:17)0σC(+Lz1) , (4.8)
C z − − 1+F0σ τ
L(T) ∆κ (cid:16) (cid:17)
=1+ (4.5) and the corresponding (triplet) “ghost” propagators
L σ T
0 D g(ω,q;n ,n ;L ) are:25
1 2 z
L
Then from the definition (2.1) and Eq. (4.3) one arrives g(n ,n ;L )=Ω δ(n n ) (n ;L )+
1 2 z 2 1 2 0 1 z
atEq.(2.3). The limiting expressionsgiveninEqs.(2.5) L L
and(2.9)arefoundbykeepingtheleadingordercontribu- (n ;L ) (n ;L ) τ1C(Lz) , (4.9)
tions to δLs for small andlargeTτ respectively by using L0 1 z L0 d2 z (L ) 1
C z − τ
7
where and
1 e2 ω2 E2
L0(ni;Lz)= −i(ω−LzEZ∗)+ivi·q+1/τ , (4.10) ∆B0 = σD2π2ω ln(cid:12)ω2−−EZ∗Z2(cid:12) . (4.14b)
(cid:12) (cid:12)
C(Lz)= −i(ω−LzEZ∗)+1/τ 2+ vFq 2. Asdiscussedabove,theseke(cid:12)(cid:12)rnelsarefo(cid:12)(cid:12)undbysubstitut-
q ing the expressions (4.8)-(4.9) for the propagators into
(cid:0) (cid:1) (cid:0) (cid:1)
In the above formulas I dropped the variables ω, q for thedefinitionsof and 0giveninEqs.(6.9)and(6.36d)
compactness, vi = vFni and EZ∗ is the Zeeman energy ofRef.6.26 SubstEitutingBEqs.(4.14)intoEqs.(4.13)[and
renormalized by the interactions, Eq. (3.2). by a change of variable ω 2πTω] I get:
Thecalculationofthefield-dependentthermalconduc- →
tivity proceeds now as in Ref. 6: the evaluation of the T 1 E E∗
δκ = I Z I Z
transport coefficients can be reduced to integrals over el,m − 6 Fσ 1 2πT − 1 2πT
thefrequencyωwhoseintegrandsconsistofadistribution E 0 (cid:26) (cid:18)E (cid:19) (cid:18)E∗ (cid:19) (4.15a)
functionparttimesakernelpartK(ω);thelatterisfound + Z I Z I Z
2 2
2πT 2πT − 2πT
after integration over the momentum q and summation (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:27)
over the total spin projections. The field-dependent ker-
and
nelscanbefoundusingtheexpressions(4.8)-(4.9)instead
of their zero-field counterparts; below I calculate explic- T E E∗
itly the correction ∆κm to the thermal magnetoconduc- κσm−κgm =−4 I1 2πZT −I1 2πZT , (4.15b)
tivity κ (T,H) = κ(T,H) κ(T,0). In other words, I (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
m
−
want to write κ(T,H) in the form (4.1) and define ∆κm where I introduced the dimensionless functions:
as:
ω2 E2
I (E)=π dω ln 1 (4.16)
∆κm(T,H)=∆κ(T,H) ∆κ(T,0) , (4.11) 1 sinh2πω − ω2
− Z (cid:12) (cid:12)
(cid:12) (cid:12)
where ∆κ(T,0)is the correctionconsideredin the previ- and (cid:12)(cid:12) (cid:12)(cid:12)
oussection. Tocalculate∆κ Iintroduceforeachkernel
m
ω 1+ω/E
Kthecorrespondingkerneldifference∆K=K(H) K(0)
I (E)=π dω ln . (4.17)
betweenthekernelcalculatedwithandwithoutth−efield. 2 sinh2πω 1 ω/E
Z (cid:12) − (cid:12)
Asintheprecedingsection,Iconsiderseparatelythedif- (cid:12) (cid:12)
(cid:12) (cid:12)
fusive and quasiballistic regimes. Eq. (4.16) can be identically rewr(cid:12)itten as: (cid:12)
∞
I (E)=2E2 4 n ln 1+ E/n 2 (E/n)2
1. Diffusive regime. 1 − − 1+(E/n)2
nX=1 (cid:20) (cid:16) (cid:0) (cid:1) (cid:17) (4.18(cid:21))
In the diffusive regime, the two main contributions to which is useful to obtain the E 1 expansion, and as:
≪
∆κcomefromthe energydependenceoftheelasticcross
section and from the bosonic energy transport. Writing 2 ω2 ω2
I (E)= ln E +C+π dω ln 1
1 3 | | sinh2πω − E2
∆κm,d =δκel,m+(κσm−κgm), (4.12) Z (cid:12)(cid:12)(cid:12) (4.(cid:12)(cid:12)(cid:12)19)
whichgivestheE 1asymptoticbehavior;t(cid:12)heconst(cid:12)ant
the two terms are given by [cf. Eqs. (6.11b) and (6.36b) C appearing above≫is:
of Ref. 6]:
ω2
δκel,m =−2eσ2DT dω∆E(ω) ω123∂∂NωP (4.13a) C ≡−22πZ dω sinh2π3ω ln|ω4| (4.20)
Z (cid:20) (cid:21) ′
= γ+ln2π ζ (2) 0.99,
3 − 2 − π2 ≃
and (cid:18) (cid:19)
σ ω3∂N whereγ 0.577istheEulerconstantandζ′(2) 0.938
κσm−κgm = e2DT dω∆B0(ω) 4 ∂ωP (4.13b) is the de≃rivative of the zeta function evaluated≃at−2. As
Z (cid:20) (cid:21) for Eq.(4.17),in the givenformthe largeE limit canbe
readily obtained [I (E) = 2/3E+...], while to find the
with the kernels 2
small E limit I rewrite it as:
e2 1 ω2 E2
∆E =−σD2π2ω2F0σ(cid:20)ωln(cid:12)ω2−−EZ∗Z2(cid:12) (4.14a) I2(E)=πs∞gnE−4E
+EZln (cid:12)(cid:12)(cid:12)ωω+EEZ · ωω(cid:12)(cid:12)(cid:12) −+EEZ∗∗ +4 Im ln 1+iEn − 1 1iE/n . (4.21)
(cid:12) − Z Z(cid:12)(cid:21) nX=1 (cid:20) (cid:18) (cid:19) − (cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
Note also the identity [primes indicate derivatives]: V. CONCLUSIONS.
′ ′
I (E)+EI (E)=0 (4.22)
1 2 The quantum kinetic equation approach is a powerful
method to investigate the effects of the electron-electron
which enables to verify that the correction δκ of
Eq. (4.15a) vanishes in the limit Fσ 0, as expeeclt,emd. interactions on transport in disordered metals5,6,12 and
0 → open quantum dots.11 Using this approach I considered
Using Eqs. (4.15), together with the definitions (4.1),
the temperature and parallel magnetic field dependence
(4.11) and (3.3), and dropping terms of higher order in
of the Lorentz number in two-dimensional disordered
1/g, one arrives at Eq. (3.4). The approximate expres-
metals.
sions (3.5)-(3.7) follow from Eqs. (4.18)-(4.21).
Three regimes can be distinguished as the tempera-
ture increases: diffusive, quasiballistic and truly ballis-
2. Quasiballistic regime. tic. In the low-temperaturediffusive regime,the Lorentz
number is enhanced above its Drude value due to the
energy transported by neutral bosonic modes that de-
In the quasiballistic regime, the correction ∆κ is de-
scribe the interacting electron-hole pairs, see Eq. (2.5)
termined by the inelastic electron-bosoncollisions and it
and Fig. 1. At intermediate temperatures (the quasibal-
can be written as [cf. Eq. (6.39a) of Ref. 6]:
listic regime) the Lorentz number is suppressed by the
σ ω3∂N inelastic electron-boson collision, Eq. (2.9) and Fig. 2,
∆κm,qb = e2DT dω∆B1(ω) 4 ∂ωP (4.23) withthe crossoverbetweenthe two regimesdescribedby
Z (cid:20) (cid:21) Eq.(2.3). Theeffectoftheinteractioninthetripletchan-
with nelisgiveninEqs.(2.10)and(2.11)forthediffusiveand
quasiballistic regimes respectively.
∆ 1 = e2 τ2ω F0σ 2J EZ∗ ; F0σ , If a magnetic field is applied parallel to the two-
B σ 2π2 1+Fσ ω 1+Fσ dimensionalmetal,theZeemansplittingoftheelectronic
D (cid:18) 0 (cid:19) (cid:18) 0 (cid:19)
(4.24a) energies affects the transport properties; in particular in
the diffusive regime the deviationof the Lorentznumber
(1 LzE)2 fromitszero-fieldvaluedisplaysanon-monotonicdepen-
J(E;F)= 2 ln 1 L E −
− | − z |(1 L E)2 F2 dence on the ratio between the Zeeman energy and the
LzX=±1(cid:26) − z − temperature, see Eq. (3.4) and Fig. 3. Finally, as dis-
1 1
ln F F2 . cussedinSec.IIIB,inthe quasiballisticregimethe devi-
− | | (cid:20)(1−LzE)2−F2 − 1−F2(cid:21)(cid:27) ation can be either a monotonic or non-monotonic func-
(4.24b) tion of both temperature and Zeeman energy depending
on the value of the Fermi-liquid parameter.
Some details on the derivation of this kernelare givenin
Appendix B. Substitution of Eqs. (4.24) into Eq. (4.23)
results in Acknowledgments
T Fσ 2 E∗ Fσ
∆κm,qb =−4(2πTτ)2 1+0Fσ I3 2πZT;1+0Fσ IamgratefultoI.AleinerandY.Ahmadianforreading
(cid:18) 0 (cid:19) (cid:18) 0 (cid:19) and commenting on the manuscript. Financial support
(4.25)
through the J.A. Krumhansl Postdoctoral Fellowship is
with the dimensionless function I defined as:
3
acknowledged.
ω4 E
I (E;F)=π dω J ;F . (4.26)
3 sinh2πω ω
Z (cid:18) (cid:19) APPENDIX A: PARALLEL FIELD
The largeandsmallE limits ofI canbe foundbykeep- DEPENDENCE OF THE SPECIFIC HEAT.
3
ing the leading orderterms in the expansionof the func-
tion J(E;F).27 In this way I obtain: In this appendix I calculate for completeness the cor-
rection δCV to the specific heat in the presence of a par-
I3(E;F) 4 ln E (4.27) allel magnetic field; the general expression for δCV is:6
≈−15 | |
∂
for E ≫1, and δCV = ∂T uσ−ug , (A1)
(cid:16) (cid:17)
2 1+3F2 F2(3+F2) where the energy densities uα are given by:
I (E;F) E2 + lnF2
3 ≈ 3 (1 F2)2 (1 F2)3
(cid:20) − − (4(cid:21).28) uα = dωωbα(ω)NP(ω). (A2)
forE 1. Knowing∆κm,qbandtheapproximateformu- Z
las(4.≪27)-(4.28),proceedingasintheprevioussubsection Here N (ω)is the Planckdistribution andbα(ω)are the
P
finally leads to Eq. (3.9), (3.10) and (3.12). bosonicdensitiesofstates. Inthepresenceoftheparallel
9
field I find: APPENDIX B: DERIVATION OF THE
KERNEL ∆B1.
bσ(ω;H) bg(ω;H)=
−
Re d2q Fσ 1 1
0 In this appendix I briefly outline how to derive the
− 2πiX(Lωz ZL(2Eπ)∗2)(cid:20)+1+1/Fτ0σ(cid:18)C(L1z)−b − C(Lz1)(cid:19) okferSneecl.∆6B.21ogfivRenefi.n6E. qT.h(4e.r2e4,at)hsetaerxtaincgt f(raotmlintheaerreosrudletsr
− − z Z in T) solution of the kinetic equation was given; this
− (L ) (L ) b − (L ) 1/τ ∇
C z (cid:18)C z − C z − (cid:19)(cid:21) solution is unaffected by the parallel field. In the quasi-
(A3)
ballisticregime,themaincorrectionstothethermalcon-
ductivity were found to originate from the term in the
with defined in Eq. (4.10) and
C bosonic distribution function defined as δN1 and which
b= iω F0σ + 1 (A4) can be neglected in the diffusive limit; these corrections
− 1+Fσ τ are given in Eqs. (6.35c) and (6.36c) of Ref. 6. Perform-
0
ing the angularintegrations in those equations results in
Performing the integration and the summation I arrive
Eqs.(6.38a)and(6.38b)respectively;byrepeating those
at:
calculations using the propagators in Eqs. (4.8)-(4.9) a
bσ(ω;H) bg(ω;H) bσ(ω;0) bg(ω;0) = similar result is obtained in the presence of the parallel
− − − field. The explicit expression is simply found by redefin-
(cid:16) 1 1 (cid:17) (cid:16)E2 E∗2(cid:17) ing some of the quantities appearing on the right hand
ln 1 Z ln 1 Z
− 8π2D 1+Fσ − ω2 − − ω2 sidesofEqs.(6.38)asfollows: forthefunction ,Isubsti-
(cid:20) 0 (cid:12) (cid:12) (cid:12) (cid:12) C
(cid:12) (cid:12) (cid:12) (cid:12) (A5) tutethefunction (Lz)giveninEq.(4.10);theparameter
+τπ |ω|−EZ∗ θ(E(cid:12)(cid:12)Z2 −ω2)(cid:12)(cid:12)−θ(E(cid:12)(cid:12)Z∗2−ω2)(cid:12)(cid:12) b′ is now: C
(cid:0)Fσ 2(cid:1)(cid:16) (cid:17) b′(L )= i ω L E∗ + 1 ;
0 πτ ω θ(E2 ω2) . z − − z Z τ
− 1+Fσ | | Z −
(cid:18) 0 (cid:19) (cid:21) (cid:0) (cid:1)
all the other quantities, namely b [Eq. (A4)] and
Next, Isubstitute thisresultintoEq.(A2)andtheninto
Eq.(A1);takingthetemperaturederivativeandrescaling ω2∂N Fσ
the frequency [ω →2πTω] I finally obtain: N˜ =vFτ T ∂ωP 1+0F0σ
δCV(H)−δCV(0)= are unchanged.
41DT I1 2EπZ∗T − 1+1FσI1 2EπZT toTboe aprerrifvoermatedt,haeloknegrnweilth∆Bth1e, rtehmeasiunmingoivnetregLrazlnoeveedr
(cid:20) (cid:18) (cid:19) 0 (cid:18) (cid:19)(cid:21)
1 T Fσ 2 E E∗ (A6) the magnitude of the momentum q. This integral was
+ (Tτ) 0 f Z +f Z performed in Ref. 6 with logarithmic accuracy; in the
πD 1+Fσ 3 2T 3 2T
(cid:20)(cid:18) 0 (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) present case however both the infrared divergence (due
f EZ EZ∗ f EZ∗ f EZ to the long-range part of the Coulomb interaction) and
− 3 2T − 2T 2 2T − 2 2T the ultravioletdivergenceareabsent–the latterbecause
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)(cid:21)
I consider here the kernel difference, the former because
with I1 defined in Eq. (4.16) and thetripletchannelinteractionisshort-range–andIpro-
z ωn ceed in a different manner. By rescaling all the dimen-
fn(z)= dω sinh2ω . (A7) sionfulquantitiesω,EZ∗,1/τ andvFqbythetemperature
Z0 T, I find that the contributions due to Eq. (6.38a) are
Thefunctionsf canbegivenintermsofpolylogarithms smallerthanthoseofEq.(6.38b)by1/Tτ andcanthere-
n
as: fore be neglected. Similarly, to find the leading term in
the (Laurent) expansion over 1/Tτ one can set 1/τ 0
f (z) = π2 +2zlog 1 e−2z Li e−2z in Eq. (6.38b). Following this procedure, I get: →
2 2
6 − −
+z2(1 cot(cid:0)hz) (cid:1) (cid:0) (cid:1) d2q dn dn
− 1 2n Re σ(L )δ N1
f3(z) = 3ζ(3)+3z2log 1 e−2z 3zLi2 e−2z Z (2π)2Z Ω2 1µ L z ν 12
2 − − τω Fσ ∞(cid:8) (cid:9) 1
−32Li3 e−2z +(cid:0)z3(1−co(cid:1)thz) (cid:0) ((cid:1)A8) ≃−δµν8πvF2 1+0F0σN˜Z0 dp"Re p−(1−LzEZ∗/ω)2
The second line(cid:0)in Eq(cid:1). (A6) is the correction to the Re 1 p
specificheatinthediffusivelimit,whilethelasttwolines × p−(1−LzEZ∗/ω)2−i1+F0Fσ0σ
becomedominantinthequasiballisticlimit. Inthislimit
athnedrfeosruwlteoafkRineft.er1a9c.tion |F0σ|≪1, Eq. (A6) reproduces ×(cid:18)1p− (1−LzpEZ∗/ω)2(cid:19)#,
10
where in terms of the original variables p = (vFq/ω)2, ∆ 11 e2 τ π s(ω L E ω L E∗)
and the approximate equality indicates that I am ne- S ≃ σ 2π2ω 2 − z Z| − z Z
glecting higher order terms in 1/Tτ. The integral over p D LzX=±1(cid:20)
Fσ
is logarithmically divergent, but the difference between (ω L E∗) ω 0 s(ω L E ω)
the above expression and the similar one at zero paral- × − z Z − 1+F0σ − z Z| (cid:21)
lel field is finite. The exact integration of this difference (C2b)
gives finally the expression for the kernel ∆ 1 given in
B
Eqs. (4.24).
e2 τ π
∆ 12 s(ω L E ω L E∗)
DEAPPEPNEDNEDNICXECO:FPATRHAELELLEELCFTIERLICDAL S ≃ σD2π2ω 2 LzX=±1(cid:20) − z Z| − z Z
2ωFσ
CONDUCTIVITY. (ω L E∗) 0
× − z Z ω(1+2Fσ) L E
0 − z Z
The aim of this appendix is to compare the present ω F0σ s(ω E ω) . (C2c)
approach to the one of Ref. 12 for the calculation of the − 1+Fσ − Z|
0 (cid:21)
parallel field magneto-conductivity ∆σ, which is given Eqs. (C1) and (C2) lead to the quasiballistic limit re-
by: sultofRef.12;inparticular,thesumofthefirsttermsin
square brackets gives the contribution denoted there by
∂ K , while the remaining terms give the K contribution.
∆σ =σDZ dωh∆E(ω)+∆Sel(ω)i∂ωhωNP(ω)i. (C1) tHhe2erequIapsoiibnatlloisutticthmaatginnetthoe-cloownd-fiuecltdanlicmeiitsE1gZiv,eEnZ∗by≪: 2T,
This formula follows from Eq. (6.8) of Ref. 6, and the
kernels are defined in the subsequent Eq. (6.9). e2 2Fσ 1 E 2
Inthediffusivelimitonly∆ isrelevant,sincetheker- ∆σ 0 Tτ Z f(Fσ), (C3)
nel el givescontributionssmaEllerbythe factorTτ 1, ≈ π 1+F0σ 3(cid:18)2T (cid:19) 0
S ≪
as discussedin Ref. 6; it is straightforwardto verify that
substituting Eq.(4.14a)intoEq.(C1),the diffusivelimit where
resultof Ref. 12is recovered. Viceversa,it canbe shown
that in the quasiballistic limit the kernel can be ne- z 1 1 1
glected since larger corrections are due to Eel – this can f(z)=1− 1+z 1 + 21+2z + (1+2z)2 (C4)
S (cid:20)
be done for example by rescaling all dimensionful quan- 2(1+z)ln2(1+z)
tities by the temperature, as described in Appendix B. .
− (1+2z)3
Proceeding as detailed there, i.e. dropping higher order (cid:21)
terms in 1/Tτ, and introducing the shorthand notation:
This expression corrects the wrong definition of f(z)
givenafterEq.(14)ofRef.12.28 While thedifferencebe-
s(ab) sgn(a) sgn(b),
| ≡ − tween the two definitions is numerically small (less than
I obtain for the kernel ∆Sel in the quasiballistic regime: 4th%is)pfaorra−m0et.5e7rr<∼angFe0σ,a<∼nd0u.s1e4,ofitEqg.ro(wCs4)rraaptihdelyrtohuatnsiidtes
counterpartofRef.12maybeimportantinacomparison
∆ el =∆ 11+∆ 12 (C2a) between theory and experiment.
S S S
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