Tunneling and Fluctuating Electron-Hole Cooper Pairs in Double Bilayer Graphene

Dmitry K. Efimkin,1, 2, ∗ G. William Burg,3 Emanuel Tutuc,3 and Allan H. MacDonald1

1The Center for Complex Quantum Systems, The University of Texas at Austin, Austin, Texas 78712-1192, USA2School of Physics and Astronomy and ARC Centre of Excellence in Future

Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia3Microelectronics Research Center, Department of Electrical and Computer Engineering,

The University of Texas at Austin, Austin, TX 78758, USA

A strong low-temperature enhancement of the tunneling conductance between graphene bilayershas been reported recently, and interpreted as a signature of equilibrium electron-hole pairing,first predicted in bilayers more than forty years ago but previously unobserved. Here we providea detailed theory of conductance enhanced by fluctuating electron-hole Cooper pairs, which are aprecursor to equilibrium pairing, that accounts for specific details of the multi-band double graphenebilayer system which supports several different pairing channels. Above the equilibrium condensationtemperature, pairs have finite temporal coherence and do not support dissipationless tunneling.Instead they strongly boost the tunneling conductivity via a fluctuational internal Josephson effect.Our theory makes predictions for the dependence of the zero bias peak in the differential tunnelingconductance on temperature and electron-hole density imbalance that capture important aspectsof the experimental observations. In our interpretation of the observations, cleaner samples withlonger disorder scattering times would condense at temperatures Tc up to ∼ 50K, compared to therecord Tc ∼ 1.5K achieved to date in experiment.

I. INTRODUCTION

The possibility of Cooper pairing in a system withspatially separated electrons and holes in semiconduc-tor quantum wells was first anticipated more than fortyyears ago [1, 2]. According to theory, strong Coulomb in-teractions allow pairing at elevated temperatures, whichwould provide a physical realization of dipolar superflu-idity that is potentially relevant for applications. Thepaired state is fragile however, and can be suppressed bydisorder [3, 4], or by Fermi-line mismatches due to the dif-ferences between electron and hole anisotropies [5, 6] thatare always present in conventional semiconductors [7]. Infact equilibrium pairing has until recently been observedonly in the presence of strong magnetic fields that quenchthe kinetic energies of electrons and holes and drive thesystem to the regime of strong correlations [8, 9].

Recent progress in fabricating single-atomic-layer two-dimensional materials has renewed interest in electron-hole pairing [11–25]. Graphene-based two-dimensionalelectron systems not only have high mobility and almostperfect electron-hole symmetry but they make it possibleto fabricate closely-spaced, and therefore strongly inter-acting, independently gated and contacted double layerstructures. Very recently low-temperature enhancementof the tunneling conductance between graphene bilayershas been observed at matched concentrations of electronsand holes [10]. A typical conductance trace is presentedin Fig. 1, where we see a tunneling conductance that ap-pears to diverge at T0 ≈ 1.5 K, signaling equilibrium paircondensation. This observation provides the first clearexperimental signature of equilibrium electron-hole paircondensation in the absence of magnetic field [26, 27].

Enhanced tunneling conductance has been observed

0 20 40 600

10

20

0 5 10 15 200.0

0.1

0.2

FIG. 1. (a) Temperature dependence of the tunneling con-ductance GT between graphene bilayers at zero voltage bias(V = 0). The red curve corresponds to the calculation thatincorporates the effect of fluctuating Cooper pairs (Eq. (27))and accurately fits the experimental data presented by bluedots (Fig. 3-b in Ref. [10]). The purple dashed curve corre-sponds to the model of non-interacting electrons and holes(Eq. (26)). The fitting details and parameters are presentedin Sec. V. Fluctuating Cooper pairs above the critical temper-ature T0 = 1.5 K strongly enhance GT and are responsible forits critical behavior GT ∼ (T − T0)−2. The latter is derivedand discussed in Sec. IV-E and reasonably matches with theexperimental data, as it is clearly seen in the inset (b).

previously in semiconductors bilayers in the strong fieldquantum Hall regime, [28] and has been interpreted asan internal Josephson effect [8]. The differential conduc-tance, does not diverge however, and instead has a sharppeak at zero bias. The property that the conductancepeak width is smaller than temperature, and smaller thanthe single-electron scattering rate (i.e. the Landau levelwidth) nevertheless points to a collective origin of thepeak. Bilayers in the quantum Hall regime are predicted

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to support dissipationless Josephson-like tunneling cur-rents in the presence of long-range electron-hole coher-ence [29, 30]. The development of a quantitative theory ofenhanced tunneling in quantum Hall systems [31–34] thatfully explains the peak width has been challenged by theimportance of inhomogeneity and disorder, and by stronginteractions in the presence of dispersionless Landau lev-els. Phase fluctuations that are inevitably present due tothe two-dimensional Berezinskii-Kosterlitz-Thouless na-ture of the phase transition [35–38] also play a role. Thetheory of enhanced tunneling is simpler at zero mag-netic field, at least in the weak coupling regime wherethe electron-hole pairing energy is small compared to theFermi energy, allowing experiments to be explained morefully as we demonstrate below.

The enhancement of the tunneling conductance in thedouble bilayer graphene system has been observed overa wide density range 4 · 1010∼ 1012 cm−2, where elec-tronic correlations vary from moderate to weak [39]. TheBardeen-Cooper-Schrieffer (BCS) theory of electron-holepairing has greater validity at weaker pairing. True in-ternal Josephson behavior in this case occurs only belowa critical temperature T0 and is preceded by enhance-ment of the tunneling conductance that diverges as T0is approached as illustrated in Fig. 1. This critical be-havior has been predicted by one of the authors [40] andhas been interpreted as a fluctuational internal Joseph-son effect. It originates from partly coherent fluctuat-ing electron-hole Cooper pairs [40–42] that are a precur-sor of equilibrium pairing and reminiscent of Aslamazov-Larkin and related effects in superconductors [43–45].Above T0 fluctuating Cooper pairs have a finite coher-ence time [19, 46–48] and cannot support a dissipation-less tunneling current. While the recent observations doqualitatively agree with earlier theory, the double bilayergraphene system has some important differences com-pared to the double parabolic electron gas models con-sidered previously. These are related to the system’s wellknown 2π momentum space Berry phases. We show herethat accounting properly for these differences provides abetter account of the low-temperature tunneling anoma-lies.

In the present work we have developed a theory ofthe fluctuational internal Josephson effect in a systemwith closely spaced graphene bilayers. As we show be-low, the presence of valley and sublattice degrees of free-dom provides three competing electron-hole channels forboth intra-valley and inter-valley Cooper pairs. We showthat three channels are nearly independent and have havedifferent condensation temperatures and different sub-lattice structure. The experimental enhancement of thetunneling conductance by fluctuating Cooper pairs canbe explained only by the presence of competing channelsthat dominate in different temperature ranges. In thevicinity of T0, the tunneling conductance at zero bias ispredicted to have a critical divergence GT ∼ (T − T0)−2,

1221

(a)top

(b)

bottom

top gate

bottom gate

𝐈𝐓

𝐕

FIG. 2. (a) Schematic of a double bilayer graphene systemwhich specifies our sublayer labeling. External gates inducean excess of electrons (holes) in the top (bottom) bilayer andopen a gap in the spectrum of isolated bilayers. (b) Elec-tronic structure of the system in the case of the matched con-centration of electrons and holes that most strongly favorselectron-hole Cooper pairing.

that matches the experimental data well as we see inFig. 1. The calculated dependence of the tunneling con-ductivity on inter-layer voltage bias and carrier-densityimbalance also match the experimental data [10] reason-ably. We conclude that the observed enhancement of thetunneling conductance in double bilayer graphene is wellexplained by our fluctuational internal Josephson effecttheory.

The paper is organized as follows. In Sec. II we in-troduce a model that describes the low-energy physics oftwo closely spaced graphene bilayers. Sec. III is devotedto a description of fluctuating Cooper pairs above thecritical temperature T0. In Sec. IV we use these resultsas a starting point for a theory of the tunneling conduc-tance. In Sec. V we compare our calculations with therecent experimental data. Finally in Sec. VI we discusslimitations of our theory and aspects of the experimentaldata that are still not well understood, and present ourconclusions.

II. MODEL

A. Weak and strong coupling regimes

The system of interest contains two graphene bilayersseparated by an insulator as sketched in Fig. 2-a. Anexternal electric field perpendicular to the bilayers in-duces an excess of electrons in the top bilayer (t) andtheir deficit in the bottom bilayer (b). Inevitably, it alsoresults in gaps 2|u| in isolated bilayer graphene spectra.The latter are in Fig. 2-b in the case of matched con-centrations for electrons and holes, the most favorableregime for the Cooper pairing. Double bilayer physics isvery rich and has been considered in a number of recentpapers [19–21, 23, 39] that aim to provide realistic predic-tions of the critical temperature T0 based on the micro-scopic model. Here we follow a different route and con-

3

sider a minimal phenomenological model that accountsfor disorder and describes the instability of the system to-wards electron-hole pairing in the weak coupling regimeand can explain the observed enhancement of the tunnel-ing conductance between graphene bilayers [10].

The physics of electron-hole Copper pairing dependson three dimensionless parameter: Wigner-Seitz interac-tion strength parameter rs = me

2/κ~2kF that scales theratio of interactions and kinetic energy in an individualbilayer [49], a parameter kFd that scales the inter-layerCoulomb interactions respect to the intra-layer one, anda parameter |u|/�F that determines the low-energy spec-trum of individual bilayers. Here kF and �F are Fermiwave vector and energy for electrons and holes, m istheir effective constant, κ and d are a dielectric con-stant for the spacer between bilayers and its thickness.If rs � 1, kFd � 1 and |u|/�F � 1 the system is in theweak coupling regime with pairing correlations only inthe vicinity of Fermi level for electrons and holes. Thisregime can be described by the BCS theory for electron-hole pairing. The nature of the strong coupling regime(rs � 1 and kFd ∼ 1) depends on the ratio |u|/�F. If|u| � �F the state is a Bose-Einstein condensate (BEC)of indirect excitons that represent a bound state of elec-tron and hole. It has been argued that the most favor-able conditions for observation of electron-hole conden-sation are reached near the mid-point of the BEC-BCScrossover [19–21, 39, 50]. In the opposite case the pairedstate is a multi-band BCS-like paired state [15–18, 51]where pairing correlations also span to remote bands (va-lence band in the layer with excess of electrons and con-duction band in the layer with excess of holes).

Enhanced conductance has been seen over a rangeof electron and hole densities that covers 4 · 1010 ∼1012 cm−2 and corresponds to rs = 3.1 ∼ 0.72 andkFd = 0.07 ∼ 0.35 [52]. This range suggests to the transi-tion from moderate to weak coupling regime occurs withincreasing of electron and hole concentrations. In theexperimental setup [10] the Fermi energy �F of chargecarriers and the gap 2|u| in the spectrum of individualbilayers are not controlled independently, but with thesame gate (that produces the electric field perpendicularto bilayers). While the density can be tuned within awide range, the ratio |u|/�F ≈ dBG/d ≈ 0.15 is approx-imately fixed. Here dBG is the thickness of individualbilayers. This favors the multi-band BCS-like state atstrong coupling and extends the region of the applicabil-ity for the phenomenological weak coupling BCS theorythat will be employed below.

B. Phenomenological model in the weak couplingregime

The spectrum sketched in Fig. 2-b has spin and valleydegeneracy. The spin degrees of freedom simply added a

factor of 2 in the tunneling conductance when the con-densed state preserves spin-invariance and do not needto be treated explicitly. The low energy states in bilayergraphene are concentrated around two inequivalent val-leys K (v = 1) and K ′ (v = −1) situated at the cornersof the first Brillouin zone, and are described by the two-band Hamiltonian [53]

H0 =∑p

[ψ̂+tp(ĥtp − µt)ψ̂tp + ψ̂+bp(ĥbp + µb)ψ̂bp

]. (1)

where ψ̂p = {ψl1p, ψl2p} is a spinor of annihilation oper-ators for electrons in both layers with sublattice indexesσ = 1, 2, that are numerated according to the sketch inFig. 2-a. We assume that there is a deficit of electrons inthe bottom layer, but it is instructive not to perform thetransformation to field operators of holes. µt = �F + hand µb = �F − h characterize the electric potentials inthe top and bottom bilayers. Here �F is the average ofthe electron and hole Fermi energies at neutrality, while2h � �F is their difference. The matrices ĥtp and ĥbpare

ĥtp =

(u

p2v̄t2m

p2vt2m −u

), ĥbp =

−u p2vb2mp2v̄b2m u

. (2)Here v = ±1 and v̄ = ∓1 are valley indexes, m is theelectron mass, and pv = px+ ivpy. The electric field per-pendicular to bilayers opens a gap 2|u| separating con-duction �cp = �p and valence �vp = −�p bands in eachbilayer with �p =

√u2 + (p2/2m)2. In the weak coupling

regime the pairing correlations appear in the vicinity ofFermi lines for electrons and holes, and the presence ofremote bands (valence band in the layer with excess ofelectrons and conduction band in the layer with excessof holes) can be neglected. In this regime only the con-duction band of the top bilayer and the valence band ofthe bottom bilayer are relevant, and the correspondingspinor wave functions are

|tcp〉 =(

cp e−ivtφp

sp eivtφp

), |bvp〉 =

(cpe

ivbφp

−spe−ivbφp

). (3)

Here φp is the polar angle; cp = cos(ϑp/2) and sp =sin(ϑp/2) with cos(ϑp) = u/�p. The spinors have val-ley dependent chirality ±vt(b)φp that defines a sublatticestructure of fluctuating electron-hole Cooper pairs as willbe shown below. We introduce disorder with the help ofphenomenological scattering rates γt(b). It is importantin this theory to observe that because the electron andhole components of the Cooper pair are spatially sepa-rated and have opposite charges, both short-range andlong-range Coulomb disorder lead to pair-breaking [3, 4].

In experiment [10] the relative angle θ betweengraphene bilayers can be adjusted. Since valleys K andK ′ reside at the corners of the first Brillouin zone, the

4

(a)

Γ = + Γ

(b) (c)

T T+ T ′ T+

(d)

T ′ = T + T ′

1

FIG. 3. (a) Bethe-Saltpeter equation in the electron-holechannel. Divergence of the many-body vertex Γ(ω = 0,Q0)signals the double-bilayer electron-hole pairing instabilitywith a momentum Q0. (b) The non-interacting tunneling re-sponse function χ0(ω,q) introduced in Eq. (18). (c) The tun-neling response function χ(ω,q) with the renormalized vertext′ that captures the effect of fluctuating Cooper pairs. (d)Renormalization of t′ that illustrates Eq. (21).

valley momenta in the two layers do not match in thepresence of a twist. For momentum conserving tunnel-ing, the current is maximized when the layers are aligned(θ = 0) or twisted by θ = nπ/3. For even n, valley K (K’)in one layer is aligned with valley K (K’) in the other layerwhereas for n odd valley K (K’) in one layer is alignedwith valley K’ (K) in the other layer. When states arelabeled by their momenta relative to the Brillouin-zonecorners, the tunneling Hamiltonian for θ close to nπ/3

Ht = T+ + T =

∑p

[ψ̂+bp t̂

+ ψ̂tp+Q + ψ̂+tp+Qt̂ ψ̂bp

]. (4)

Here Q = Qθ + QB is the momentum splitting betweenvalleys in the different layers. The twist contributionat small relative angle θ � 1 can be approximated asQθ = −[qK × ez]θ where qK is the momentum for theBrillouin-zone corner in bilayer graphene. It has oppo-site directions for valleys K and K ′, and its magnitudeis qK = 4π~/3a0 with a0 the corresponding Bravais lat-tice period. The contribution induced by an in-planemagnetic field B|| is the same for two valleys and isequal to QB = ed[B|| × ez]/~c. Because each bilayeris represented by a two-band model, the matrix t̂ hasfour matrix elements which we treat in a phenomeno-

logical way below, with the expectation that since t22corresponds to the tunneling between adjacent sublay-ers while t11 to the tunneling between remote sublayers,|t11| � |t12| ≈ |t21| � |t22|.

III.FLUCTUATING COOPER PAIRS

A. The Cooper instability

Due to Coulomb interactions between electrons andholes, the double-bilayer system is unstable towardsCooper pairing. Here we omit repulsive interactionswithin each graphene bilayer since its main effect in theconsidered weak coupling regime is a simple renormal-ization of the quasiparticle spectra. The inter-bilayer at-traction

Hint =∑pp′q

∑σtσb

Uqψ+t,p+q,σtψ

+b,p′−q,σbψbp′σbψtpσt . (5)

Here Uq is the screened Coulomb potential estimated inour previous work [14–16]. Below we employ a multi-pole decomposition of the interaction and set the mo-menta magnitudes to the Fermi momentum so that Uqreduces to a constant Ul for each orbital angular mo-mentum channel l. The set of Ul parameters are alsotreated as phenomenological parameters with the expec-tation that the s-wave moment Us ≡ U0 is largest. Sincevalleys are well separated in momentum space we neglectinter-valley scattering for electrons and holes.

The instability of double layer system towardselectron-hole Cooper pairing is signaled by diver-gence of the electron-hole channel scattering vertex

Γσ′tσtσ′bσb

(ω,p′,p,q) at zero frequency ω. Note that in

our approximation scattering conserves valley indices forelectrons (vt) and holes (vb). The Γ-vertex satisfies theBethe-Saltpeter equation presented in Fig. 3-a, and isalgebraic within the multipole approximation. It is in-structive to combine σt and σb into a single index |σtσb〉that varies between 1 and 4 as |1〉 = |11〉, |2〉 = |12〉,|3〉 = |21〉 and |4〉 = |22〉. With this definition the Bethe-Saltpeter equation can be written (See Appendix A fordetails) in a compact matrix form:

Γ̂l′l = Ul δl′l +∑l′′

Ul′ M̂l′−l′′ Π Γ̂l′′l. (6)

Here momentum and frequency dependence are sup-pressed, l (l′) is the orbital momentum for the relativemotion of two particles before (after) scattering, and Γ̂l′lis the corresponding scattering matrix. We have sepa-rated a factor of Π(ω,q) which also appears as the single-step pair propagator in the Cooper ladder sum of a bi-layer system without sub lattice degrees of freedom:

5

Π(ω,q) = NF

ln [ �c2πT ]− 12 ∑ζ=±

〈Ψ

(1

2+i[ω + ζ(h+ vFq cosφp)] + γ

4πT

)〉φp

. (7)

Here �c is an energy cutoff that is required formomentum-independent interactions. The average 〈· ··〉φp is calculated respect to a polar angle φp. Ψ(x) isthe logarithmic derivative of Γ-function (or the digammafunction), while ζ = ±1 is the summation index. γ =γt + γb is the pair breaking rate, which is the sum of thescattering rates for electrons γt and holes γb. The expres-

sion for Π(ω,q) in Eq. (7) is well known [43, 44] from pre-vious work on systems without layer degrees of freedom.The chiral nature of the bilayer graphene charge carriersis captured by the nontrivial matrix form-factor M̂l inthe Bethe-Salpheter Eq. (6). The matrix form-factor M̂lis defined as the multipole moment of the two-particlematrix element

M̂σ′t,σtσ′b,σb

(φp) = 〈σ′t|tcp +q

2〉〈tcp + q

2|σt〉〈σb|bvp−

q

2〉〈bvp− q

2|σ′b〉. (8)

Corrections to the form-factor M̂l due to finite Cooperpair momentum q, |∆M | = q2|u|/4p2F�F, are negligible in

the weak coupling regime since q � pF. As a result, thematrix (8) can be approximated as follows

M̂(φp) =

c4 −c3se−2ivbφp c3se−2ivtφp −c2s2e−2i(vt+vb)φp

−c3se2ivbφp c2s2 −c2s2e2i(vb−vt)φp cs3e−2ivtφpc3se2ivtφp −c2s2e2i(vt−vb)φp c2s2 −cs3e−2ivbφp

−c2s2e2i(vt+vb)φp cs3e2ivtφp −cs3e2ivbφp s4

. (9)

Here the coefficient c and s correspond to the coherencefactors cp and sp in (3) evaluated at the average Fermienergy �F for electrons and holes and are given by

c2 =1

2

(1 +

u

�F

), s2 =

1

2

(1− u

�F

). (10)

The matrix form-factor (9) shapes the sublattice struc-ture of fluctuating electron-hole Cooper pairs in double-bilayer graphene. Note that it couples scattering chan-nels Γ̂l′l with different orbital momenta. Importantly,M̂l has only even harmonics l = 0,±2,±4 that forbidscattering between states with even and odd orbital mo-menta. For isotropic Coulomb interactions, the s-wavemoment Us is expected to be largest. It is instructiveto start by neglecting all other moments. In that caseonly Γ̂00 is nonzero. The latter depends on the s-wavemoment of the form-factor M̂0 that has a different formfor intra-valley and inter-valley Cooper pairs. We discussthese two case separately below.

B. Intra-valley Cooper pairs

For intra-valley (vt = 1 and vb = 1) electron-holeCooper pairs the s-wave moment of the form-factor M̂0 =

〈M̂(φp)〉φp is given by

M̂0 =

c4 0 0 00 c2s2 −c2s2 00 −c2s2 c2s2 00 0 0 s4

. (11)The scattering problem decouples into the three channelsidentified in Ref. [21] and the corresponding scatteringvertex is given by

Γ̂00Us

=

1L11

0 0 0

0 1−c2s2UsΠL12-21

−c2s2UsΠL12-21

0

0 −c2s2UsΠL12-21

1−c2s2UsΠL12-21

0

0 0 0 1L22

. (12)Here Lα = 1−λsαΠ/NF is a dimensionless inverse Cooperpropagator for channel α and λsα is the correspondingcoupling constant specified in Tab. (I). Lα vanishes atthe critical temperature Tα for the electron-hole pairinginstability in channel α.

In the absence of disorder and electron-hole densityimbalances, the critical temperatures are given by T̄α =2eCΛ exp[−1/λsα]/π, where C = 0.577 is the Euler con-stant. Although the coupling constant values λα can be

6

0.0

0.4

0.8

0.0 0.2 0.4 0.60.0

0.4

0.8

FIG. 4. Dependence of coupling constants λα for differentpairing channels on the displacement field parameter u. Forintra-valley (a) Cooper pairs a competition between channelsis possible, but in the considered regime |u| � �F the mixedone 12-21 dominates. For inter-valley Cooper pairs (b) thehierarchy between channels does not depend on ratio between|u| and �F and the mixed channel 11-22 is the dominant one.

fine-tuned by the displacement field, as it is presented inFig. 4-a, their hierarchy is universal for the case |u| � �F.The coupling constant λs12-21 ≈ 1/2 is almost twice aslarge as the constants λs11(22) ≈ 1/4, ensuring domina-tion of the mixed channel 12-21. Physically the presenceof two sub lattice combinations (|12〉 and |21〉) doublesthe number of states that take part in the Cooper pairing.

C. Inter-valley Cooper pairs

For inter-valley (vt = 1 and vb = −1) electron-holeCooper pairs the s-wave moment of the form-factor M̂0 =〈M̂(φp)〉φp is given by

M̂0 =

c4 0 0 −c2s2

0 c2s2 0 0

0 0 c2s2 0

−c2s2 0 0 s4

. (13)

TABLE I. Coupling constants λα for intra-valley Cooper pairs

Channel, α s-wave, λs d-wave, λd

11 c4NFUs 2c2c2NFUd

22 s4NFUs 2c2c2NFUd

12-21 2s2c2NFUs (c4 + s4)NFUd

FIG. 5. Dependence of the critical temperature T0 (a) andthe instability momentum of Cooper pairs Q0 on the pair-breaking rate γ and electron-hole imbalance, parameterizedby the difference between electron and hole Fermi energies2h. When disorder is weak γ . T0 the imbalance can stabilizethe Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with thefinite Cooper pair momentum.

This case also decouples into three independent channels,with scattering vertex

Γ̂00Us

=

1−s4UsΠL11-22

0 0 −c2s2UsΠL11-22

0 1L12 0 0

0 0 1L21 0−c2s2UsΠL11-22

0 0 1−c4UsΠ

L11-22

. (14)Interestingly, the sublattice structure of the Cooper pairsand the corresponding coupling constants are different inintra-valley and inter-valley cases. The latter are pre-sented in the Tab. (II) and their dependence on displace-ment field parameter u is shown in Fig. 4-b. In that casethe hierarchy between coupling constants is universal anddoes not depend on the ratio between |u| and �F. Themixed channel 11-22 has the highest critical temperature.

The separation of scattering problem into three chan-nels is not an artifact of the s-wave truncation, but is

TABLE II. Coupling constants λα for inter-valley Cooperpairs

Channel, α s-wave, λs d-wave, λd

12 c2s2NFUs (c4 + s4)NFUd

21 c2s2NFUs (c4 + s4)NFUd

11-22 (c4 + s4)NFUs 2c2s2NFUd

7

maintained when higher multipole momenta of interac-tions Ul are taken into account. When the d-wave in-teraction Ud ≡ U±2 is nonzero (s- and p-wave momentaare decoupled and the latter is irrelevant) the scatteringmatrix Γl′l is nonzero for l = −2, 0, 2. We show be-low that the only effect of the d-wave momentum Ud onthe tunneling conductance between graphene bilayers is arenormalization of coupling constants λα = λ

sα+λ

dα. The

d-wave coupling constants λdα are summarized in Tabs. Iand II.

D. Disorder and density imbalances

Disorder and electron-hole density imbalances both re-duce the critical temperature T̄0 (index 0 corresponds

to the channel α that has the largest critical temper-ature of electron-hole pairing). The latter can stabi-lize the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pair-ing state [5, 6, 54–57] with finite Cooper pair momen-tum Q0 (Ref. [56, 57] also outlines a stabilization of theSarma-phases [58] within the BEC-BCS crossover. Thesephases are unstable in the weak-coupling regime consid-ered here). The critical temperature T0 and the insta-bility momentum Q0 for a channel with highest criticaltemperature satisfy the equation L0(0,Q0) = 0, whichcan be recast as follows:

ln

[T0T̄0

]+

1

2

∑ζ=±

〈Ψ

(1

2+iζ(h+ vFQ0 cosφp) + γ

4πT

)−Ψ

(1

2

)〉φp

= 0. (15)

The dependence of the critical temperature T0 on thepair-breaking rate γ and the electron-hole imbalance ex-pressed as a difference in Fermi energies, h, are illus-trated in Fig. 5-a. The electron-hole pair instability issuppressed when the pair-breaking rate exceeds a criticalvalue γ ≈ 1.78 T̄0. If the rate does not exceed γ ≈ T0the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state withfinite Cooper pair momentum Q0 is stabilized by finiteimbalance. The dependence of the instability momen-tum Q0 is presented in Fig.5-b. Its magnitude can beapproximated by vFQ0 ≈ h, corresponding to the differ-ence between the Fermi momenta for electrons and holes.

E. Coherence time and length of fluctuating Cooperpairs

In the absence of electron-hole imbalance, the Cooperpropagator L−10 of the dominating channel at small fre-quencies and momenta simplifies to

L−10 (ω,q) =1

λ0

1

iωτ − �q, �q = �+

ξ2q2

~2. (16)

Here � = ln [T/T0] is the energy scale that is requiredto create a uniform fluctuating Cooper pair. It vanishesat the critical temperature T0 and is linear � ≈ ∆T/T0in its vicinity ∆T ≈ T0. Here ∆T = T − T0. τ and ξare characteristic time and spatial scales for Cooper pairsthat are given by

τ =~Ψ′

(12 +

γ4πT

)4πT

, ξ =~vF |Ψ′′

(12 +

γ4πT

)| 12

8πT. (17)

They are connected to the coherence time and length ofCooper pairs by τ∗ = τ/2� and ξ∗ = ξ/

√� that diverge

at the critical temperature for electron-hole condensa-tion T0. The Cooper propagator (16) has its only poleon the imaginary frequency axis at ωq = −i/2τ∗q withτ∗q = τ/2�q, reflecting the dissipative nature of Cooperpairs dynamics. Due to the presence of a finite temporalcoherence time τ∗, fluctuating Cooper pairs do not pro-vide a dissipationless Josephson current, but do stronglyenhance the tunneling conductance at zero voltage bias.

IV. TUNNELING CONDUCTIVITY

A. Linear response theory for the tunnelingconductance

When inter-layer tunneling amplitudes are treated inleading order of perturbation theory, the inter bilayertunneling conductance at finite voltage bias V is [59–61]

GT(V ) =8Ae2

~Im[χ(eV,Q)]

eV. (18)

Eq. (18) accounts for the fourfold degeneracy due to thepresence of valley and spin degrees of freedom and A isthe sample area. Here χ(Ω) is the retarded correlationfunction corresponding to the imaginary-time orderedcorrelation function constructed from tunneling opera-tors

χ(τ,q) = −〈TMT (τ,q)T+(0,q)〉. (19)

χ(ω,q) can be constructed from χ(τ,q) by the usualFourier transform and analytical continuation steps.

8

Without Coulomb interactions between electrons andholes the response function χ(ω,q) corresponds to thesingle electron-hole loop diagram depicted in Fig. 3-b andis given by

χ0(ω,q) = t̂+M̂0 Π t̂, (20)

Here t̂ = {t11, t12, t21, t22} is the vector of tunneling ma-trix elements in the compact representation, and Π ≡Π(ω,q) is a single step pair propagator in the Cooperladder defined in Eq. (7). In the presence of Coulombinteractions, the single-particle Green functions and thetunneling vertex need to be renormalized. The renormal-ization of Green functions results in a dip of the densityof states at the Fermi level [42], that does not produceany singularities in the tunneling conductivity and thusis unimportant and can be neglected. The renormalizedtunneling vertex t̂ → t̂′ ≡ t̂′(ω,q) diverges at the criti-cal temperature T0 and is responsible for the drastic en-hancement of the tunneling conductivity in its vicinity.The renormalization of tunneling vertex is presented inFig. 3-d and the corresponding equation for t̂′ can bewritten as

t̂′l = t δl0 +∑l′

Ul M̂l−l′ Π t̂′l′ . (21)

The matrix form-factor M̂ couples even orbital channelsand we neglect all multipole moments except for s- and d.(The p-wave multipole moment is decoupled from the s-wave one and is irrelevant.) Since the form-factors M̂ aredifferent for intra-valley and inter-valley Cooper pairs, weagain consider these two cases separately.

B. Intra-valley tunneling

Without Coulomb interactions between electrons andholes the response function χ(ω,q) is a sum of three non-interfering terms that correspond to three channels α in-troduced in Sec. III and identified in Ref. [21]:

χ0 = (c4|t11|2 + s4|t22|2 + |t12 − t21|2c2s2) Π (22)

The three channels are not coupled by Coulomb interac-tions and are renormalized independently as follows:

χ =

(c4|t11|2

L11+s4|t22|2

L22+|t12 − t21|2c2s2

L12-21

)Π. (23)

Here Lα is the inverse Cooper propagator for each chan-nel and λα = λ

sα +λ

dα is the corresponding coupling con-

stant. Importantly the only role of the d-wave interactionmoment is the renormalization the coupling constant λα.

C. Inter-valley tunneling

When opposite valleys are aligned by a twist angle be-tween bilayers close to θ = π/3, the response functionχ(ω,q) for noninteracting electrons and holes is

χ0 = (c2s2|t12|2 + c2s2|t21|2 + |c2t11 − s2t22|2)Π. (24)

It again is a sum of three non-interfering terms that cor-respond to three channels α. Coulomb interactions donot couple the channels but renormalize as follows

χ =

(c2s2|t12|2

L12+c2s2|t21|2

L21+|c2t11 − s2t22|2

L11-22

)Π.

(25)Each channels acquires its own Cooper propagator Lαwith the coupling constant λα = λ

sα + λ

dα.

D. Tunneling conductance

Due to the remarkable conservation of momentum forelectron tunneling, the conductance GT is observableonly if graphene bilayers are aligned with twist angleθ = 0 or θ = π/3. In the former case the tunnelingis intra-valley, while in the latter case it is inter-valley.Therefore we refer to the cases θ = 0 and θ = π/3 as tointra-valley and inter-valley alignments. As we explainbelow, there important differences in the enhancementof tunneling conductance by fluctuating Cooper pair inthese two cases.

Within the model of non-interacting electrons andholes the tunneling conductance is dominated by tunnel-ing between adjacent sublayers t22 and can be approx-imated (both for intra- and inter-valley alignments) asfollows

G0T =8Ae2

~s4|t22|2Im[Π(eV,Q)]

eV. (26)

In the presence of interactions, the enhancement of tun-neling conductance by fluctuating Cooper pairs worksvery differently for intra- and inter-valley alignments.The reason is a drastic difference in the sublattice struc-ture for the pairing correlations.

For intra-valley fluctuating Cooper pairs, the dominat-ing channel 12-21 does not involve pairing correlations atadjacent sublayers. As result, the corresponding contri-bution of the channel 12-21 is weaker than that of 22,since the latter involves tunneling between adjacent sub-layers, except in the vicinity of the critical temperatureT0. The tunneling conductance is governed by the com-petition of two channels and can be approximated as fol-lows

9

GT =8Ae2

~

(s4|t22|2

|L22|2+|t12 − t21|2c2s2

|L12-21|2

)Im[Π(eV,Q)]

eV. (27)

As we explain in the next section, the competition be-tween channels is essential for the explanation of theexperimental data [10], that in our interpretation isstrongly depend on mutual orientation of graphene bi-layers.

For the inter-valley alignment, the dominant channelfor fluctuating Cooper pairs involves pairing correlationsat adjacent sublayers. As a result, the tunneling conduc-tance can be well approximated by a single term corre-sponding to the channel 11-22 and is given by

GT =8Ae2

~s4|t22|2

|L11-22|2Im[Π(eV,Q)]

eV. (28)

The inter-valley alignment case has not been studied ex-perimentally yet, but according to our theory is mostfavorable for observations of the fluctuational internalJosephson effect.

E. Critical behavior of tunneling conductance

At matched concentrations of electrons and holes thevoltage dependence of tunneling conductance GT inthe vicinity of the critical temperature T0 acquires aLorentzian shape which is governed by the factor

F (eV,Q) =Im[Π]

eV |L0|2=

τ

(eV τ)2 + �2Q. (29)

Here �Q = � + ξ2Q2/~2 and � = ln(T/T0) can be in-

terpreted as an energy of fluctuating Cooper pairs. Inthe absence of a valley splitting Q = 0, the amplitudeof the peak has a long high temperature tail F (0, 0) =τ/ ln2[T/T0]. It diverges near the critical temperatureT0 as a function of ∆T = T − T0 in the critical man-ner as F (0, 0) ≈ τT 20 /∆T 2 with the index 2. Its widthat half maximum eVHM = 1/τ

∗ = 2∆T/T0τ is equal tothe inverse coherence time τ∗ of fluctuating Cooper pairsand vanishes linearly at T0. In the presence of a val-ley splitting that can be induced by in-plane magneticfield or relative twist, fluctuating Cooper pairs with fi-nite momentum Q are probed in tunneling experiments.Temperature dependence of the peak width is modifiedas eVHM =

(1 + (ξ∗Q)2

)/τ∗. It has a universal form

as a function of coherence time τ∗ = 2τ/� and lengthξ∗ = ξ/

√� for fluctuating Cooper pairs that allow to ex-

tract them form the experimental data in the presence ofin-plane magnetic field.

For the case of the inter-valley alignment (θ = π/3),the temperature dependence of the tunneling conduc-tance (28) is well approximated by Eq. (29) in a widetemperature range. For the case of intra-valley align-ment (θ = 0) channels 12-21 and 22 compete with each

other. As a result, tunneling conductance Eq. (27) cannot be approximated by a simple analytical expressionover as wide temperature range. It is well approximatedby the critical behavior given by Eq. (29) only in thenarrow temperature range where the contribution of thechannel 21-21 dominates.

V. COMPARISON WITH EXPERIMENT

In Ref. [10] the twist angle between graphene layers canin principle be tuned to access the intra-valley (θ = 0)and inter-valley (θ = π/3) tunneling cases, although ex-perimental results for tunneling conductance GT have sofar been reported only for one alignment. A preliminaryanalysis of the data suggests that a divergent zero biaspeak appears on the top of a background with a weakertemperature dependence. This behavior can be explainedby the competition between channels 12-21 and 22. Thisis not surprising since top and bottom graphene bilay-ers originate from the same flake, and are aligned withthe twist angle θ = 0. Calculations for the intra-valleyalignment are presented below and compared with theexperimental data, while ones for the inter-valley one arepresented in Appendix B.

To fit the experimental data, the tunneling conduc-tance GT has been calculated with the help of Eq. (27).The model has a large number of fitting parameters. Toadjust their values and compare our results with the ex-perimental data we use the following strategy that in-volves six steps.

1) The experimental data has already been carefullyanalyzed [10] within a model of noninteracting chargecarriers. In the wide range of concentrations the non-interacting model explains the tunneling data very wellexcept in the case of opposite polarity charge carrierswith nearly equal electron and hole densities. Based onthe fits to experimental data away from matched electronand hole densities we can confidently assign values for the

10

adjacent layer tunneling amplitude, |t22| = 30 µeV, andthe disorder-broadening energies γt(b) = 4 meV. Notethat the measured disorder broadening parameter γt(b)is much larger than T0, where T0 is the temperature atwhich the tunneling conductance appears to diverge ex-perimentally. It is immediately clear therefore, even be-fore performing a detailed analysis, that the condensationtemperature must be strongly suppressed by disorder.

2) The colossal enhancement of the tunneling con-ductance has been observed in the wide density range4 · 1010 ∼ 1012cm−2. The strength of the interactionsin the system decreases with doping and we have chosenthe elevated doping level n = 7.4 · 1011 cm−2 because ofthe detailed experimental data is available in this case.This doping level corresponds to the average concentra-tion of electrons and holes n = (ne + nh)/2, while theelectron-hole imbalance ∆n� n can be present.

3) The transport experiments with similar double bi-layer graphene samples and with similar gating geome-try have already been performed [62]. The doping levelof charge carriers �F and the gap 2|u| in the electronicspectrum of bilayer graphene are independent and arecontrolled by the same gate. At the considered aver-age density n = 7.4 · 1011 cm−2, the effective massparameter of bilayer graphene can be approximated asm ≈ 0.04 m0, while the gap is equal to 2|u| ≈ 6.6 meVand is much smaller than the corresponding Fermi energy�F ≈ 20 meV. It follows that |u|/�F ≈ 0.16, which im-plies sublayer polarization within the bilayers c2 ≈ 0.58and s2 ≈ 0.42 to be modest.

4) The bare critical temperature without disorder T̄0 ≈50 K can be recalculated from the actual critical temper-ature at the considered doping level T0 = 1.5 K and theCooper pair-breaking rate γ = γt + γb = 8 meV that hasbeen chosen above with help of Eq. (15). The system isin the regime of strong pair breaking T0 � T̄0 ∼ γ andthe value γ/T0 ≈ 1.74 is very close to the critical value1.78. Experimentally, singular behavior of the tunnelingconductance is observed only in the cleanest samples.

5) The bare critical temperature T0 of Cooper pair-ing in the weak-coupling regime is given by T̄0 =2eCΛ exp[−1/λ0]/π, where C = 0.577 is the Euler con-stant. Here λ0 = λ

s0 + λ

d0 corresponds to the mixed-

channel 12-21 that dominates in the considered regime|u| � �F. Approximating the high energy cutoff as�c ≈ 2�F and employing the values of T̄0 and �F chosenabove we get λ0 ≈ 0.44. The applicability condition forthe weak-coupling approach λ0 � 1 is not well satisfied,but this value for λ0 corresponds to moderate couplingregime λ0 . 1 that justifies the approximations used inour theory.

6) The relative contribution to the tunneling conduc-tance GT of channels 12-21 and 22 can be characterizedby a dimensionless parameter r = |t12 − t21|2/|t22|2 thatwe treat in the phenomenological way. Along with thesample area A, r and A are only free parameters of the

-1.0 -0.5 0.0 0.5 1.00

25

50

75

100

125

FIG. 6. Dependence of the tunneling conductance GT on biasvoltage V at different temperatures T . The concentrations ofelectrons and holes do match. The solid curves correspondto calculations, while the dotted lines are experimental data(Fig. S2-b in Ref. [10]). For clarity the curves at adjacenttemperatures are offset by ∆GT = 12 mS. The theory thatincorporates the effect of fluctuating Cooper pairs (Eq. (27))reasonably fits temperature dependence of the peak height,but overestimates its width and does not capture its asym-metry. Possible origins of the asymmetry are discussed inSec. VI.

model that have not yet been assigned. We adjust themby fitting the measured temperature dependence for tun-neling conductance at zero voltage bias V = 0 (and alsoat matched concentrations of electrons and holes andzero in-plane magnetic field), that is presented in Fig. 1,with the theory that incorporates the effect of fluctuatingelectron-hole Cooper pairs (Eq. 27). The area A can bebe obtained by matching the high temperature behaviorof GT since the contribution of 12-21 in this case is neg-ligible small. It results in A ≈ 397 µm2. The value ofr ≈ 7.6 · 10−5 is obtained by fitting the singular behaviorof the tunneling conductance in the vicinity of the criticaltemperature. The corresponding theoretical curve is alsopresented in Fig. 1 and matches with the experimentaldata reasonably well over a wide temperature range.

In Fig. 1 we also present calculations within the modelof noninteracting electrons and holes (Eq. (26)). Thismodel severely underestimates the tunneling conduc-tance GT in the case of matched electron and hole con-centrations, where interactions are crucial to explain en-hanced tunneling conductance at low temperature andsingular behavior in the vicinity of the critical tempera-ture T0 for Cooper pair condensation. We will keep allparameters chosen above in further calculations and in-vestigate an impact of finite voltage bias between bilayersV , electron-hole imbalance ∆n, and in-plane magneticfield B at tunneling conductance. We will also presentonly results of the theory that incorporates the effect offluctuating Cooper pairs.

11

FIG. 7. Dependence of the tunneling conductance GT onvoltage bias V and electron-hole density imbalance ∆n. Theiraverage density n = (ne +nh)/2 = 7.4 · 1011cm−2 is fixed andthe temperature is 3.5 K. The top subplot (a) correspondsto experiment(Fig. 5-c in Ref. [10]), and the bottom one (b)to theory. The cut of this plot at zero voltage bias V = 0 ispresented as Fig. 8.

A comparison between theory and experiment for thevoltage dependence of the tunneling conductance be-tween graphene bilayers GT is presented in Fig. 6. Thezero bias peaks emerge with a decreasing temperature ona top of a smooth background that corresponds to thechannel 22. The width of the background eVHM is gov-erned by the single-particle energy scales 2πT and γ andis approximately equal to the largest of them. The widthof the zero bias peak in the vicinity of T0 is much smallerthan the single-particle disorder scale, that demonstratesits collective origin. While the temperature dependenceof peak height is well fit by the theory, the width depen-dence is captured only qualitatively and is overestimatesby the factor of 2. The experimental data also exhibit avoltage asymmetry that becomes more prominent at lowtemperatures. Within our phenomenological model anasymmetrical voltage dependence of the tunneling con-ductance can be obtained if the scattering rates of elec-trons and holes γt(b) that define Cooper pair scatteringrate as γ = γt + γb are energy-dependent. The simplelinear dependence γ = γ + γ′ω with a phenomenologicalparameter γ′ does not capture the observed asymmetry

-2 -1 0 1 20

5

10

15

FIG. 8. The dependence of tunneling conductance GT at zerovoltage bias V = 0 on the electron-hole density imbalance ∆n.Their average e density n = (ne + nh)/2 = 7.4 · 1011cm−2 isfixed and the temperature is 3.5 K. The solid curve is theoryand the dotted curve is experimental data extracted from theV = 0 line from Fig. 5-c in Ref. [10]. The theory capturesthe curve profile reasonably well, but does not capture theasymmetry. Possible origins of the asymmetry are discussedin Sec. VI.

however. A quantitative understanding of the asymmetryrequires a microscopic understanding of disorder mecha-nisms that is outside the scope of the present work.

The comparison between theory and experiment forthe dependence of the tunneling conductance GT on theelectron-hole imbalance ∆n and the voltage bias V aresummarized in Fig. 7 and a zero-bias-voltage cut of thecomparison made in this color plot at zero voltage biasis presented in Fig. 8. The theoretical curves again agreereasonably well with the data. A density imbalance splitsthe Fermi lines of the electrons and holes and disfavorstheir Cooper pairing. As seen in Fig. 5, the FFLO statewith a finite Cooper pair momentum can not be stabilizedin the strong pair breaking regime realized in the experi-ment. The critical temperature T0 is maximal for Cooperpairs with zero Cooper pair momentum and decreasesmonotonically in the presence of imbalance and vanishesif the latter exceeds the critical temperature. As a result,the dependence of tunneling conductance on imbalance∆n and in-plane magnetic filed B that is presented inFig. 9 is smooth and featureless. Fig. 9 presents theoret-ical curves since the experimental data for magnetic fieldand density-balance dependence at this temperature isnot yet available. The theory qualitatively explains thedecrease of peak height with magnetic field studied exper-imentally at lower temperatures T0 ≈ 1.5 K, but consid-erably overestimates its effect. In this case the system isin the paired state whose behavior lies outside the rangeof validity of the present theory. Other less fundamen-tal limitations might also explain this discrepancy as wediscuss in more details in the next section. We conclude,

12

0

5

10

15

FIG. 9. Theoretical dependence of tunneling conductance GTon in-plane magnetic B and electron-hole density imbalance∆n. Their average density n = (ne + nh)/2 = 7.4 · 1011cm−2is fixed and the temperature is 3.5 K. The dependence ofGT is smooth and featureless that demonstrates for the set ofparameters used for fitting (the regime of strong pair breakingγ & T0) the FFLO state with finite Cooper pair momentumis not stabilized by the density imbalance ∆n.

that our theory of the fluctuational internal Josephsoneffect, combined with specific features of the multiple-channel structure of pairing in double bilayer grapheneprovides a reasonable overall description of experiment.

VI. DISCUSSIONS

Our theory of the internal fluctuational Josephson ef-fect does not account for interactions between fluctuatingCooper pairs. The Gaussian nature of the theory we em-ploy is more clearly seen within an alternate derivationof the tunneling conductance. The latter employs theauxiliary field approach and is presented in Appendix C.Interactions between fluctuating pairs can be safely omit-ted in the wide range of temperatures ∆T ∼ T0, and areimportant only in the critical region ∆TGi . GiT0 wherefluctuations are large and strongly interfere with eachother. Here Gi = T0/EF is the Ginzburg number calcu-lated in Appendix D. The double bilayer graphene systemstudied experimentally [10] is close to the weak couplingregime, and the critical region ∆TGi = T

20 /EF ≈ 10 mK

is much smaller than the temperature range ∆T ≈ 4 Kwhere the tunneling conductance is strongly enhanced.The picture of noninteracting fluctuating Cooper pairs istherefore well justified to address the basic phenomenonidentified in experiment.

Fluctuating Cooper pairs in conventional superconduc-tors alter the thermodynamics of the normal state only inthe critical region ∆TGi. The conductivity and the mag-netic susceptibility [43] are however not only singular atthe critical temperature, but have long high temperaturetails [45]. The high temperature tail for the diamag-

netic susceptibility χ ∼ χL/ ln2[T/T0], where χL is theLandau diamagnetic susceptibility in the normal state,was predicted [63–65] theoretically and observed in ex-periments [45]. The reason is the paired state is super-conducting and mediates the perfect diamagnetism thatmakes even a small number of fluctuating Cooper pairsimportant. Similarly the equilibrium paired state of spa-tially separated electrons and holes provides in fluctua-tional internal Josephson effect that colossally enhancesthe inter-layer tunneling. That is why even a small den-sity of fluctuating Cooper pairs can make a strong impacton the tunneling conductance above the critical temper-ature T0 which also has the high-temperature tail pro-portional to 1/ ln2[T/T0] that is clearly seen in Fig. 1.

In the vicinity of the critical temperature T0 zerobias peak shape is universal and governed by the fac-tor F (eV,Q) given by Eq. (29). It can be rewritten asF (eV,Q) = Im[L−10 (eV,Q)]/eV . Thus an imaginary partof the Cooper propagator L−10 (eV,Q), that can be in-terpreted as Cooper pair susceptibility [66], is directlyprobed in tunneling experiments [67]. It should be notedthat Cooper pair-susceptibility of a superconductor canalso be probed in tunneling Josephson junction in whichone side is near its critical temperature while the otheris well below its critical temperature [68, 69]. The junc-tion does not support dissipationless Josephson tunnel-ing current, but the tunneling current at finite voltage isstrongly enhanced by fluctuating Cooper pairs that growin the vicinity of T0. The latter has been observed ex-perimentally [70, 71].

The enhancement of intra-valley tunneling that corre-sponds to the alignment (θ = 0) has only been reportedso far. In this case the instability happens in one chan-nel, while the main contribution to the tunneling con-ductance comes form the different one. As a result thedivergent contribution to the conductance due to fluctu-ating Cooper pairs appears on the top of background withweak temperature dependence that dominates at highertemperatures. We predict an enhancement of inter-valleytunneling (θ = π/3) to be much more profound becauseCooper pairs in the dominating pairing channel involveelectrons and holes settled at adjacent sublayers. Wepresent calculations for this case in Appendix B.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) statewith finite Cooper pair momentum has been predictedin conventional superconductors more than sixty yearsago [54, 55]. It requires the splitting of Fermi sur-faces/lines for pairing electrons with opposite spins. Thesplitting can be induced by a magnetic field provided thatits paramagnetic effect is larger than its diamagnetic one(Chandrasekhar-Clogston limit [72, 73]). This conditionis rarely satisfied even in layered conventional supercon-ductors subjected to in-plane magnetic field. There arefew observations in heavy-fermion and organic supercon-ductors where FFLO state signatures have been claimedbut are still debated (See Ref. [74] and Refs. [75, 76] for

13

reviews of progress in solid state and cold atoms sys-tems). So far the FFLO state has not been unambigu-ously identified. In double bilayer graphene the densitiesof electrons and holes can be controlled separately in away that opens the FFLO state up for experimental studyin a condensed matter system [5, 6]. The FFLO can beunambiguously identified if it appears from the depen-dence of the zero-bias peak on imbalance and in-planemagnetic field, since the latter makes it possible to probeCooper pairs with finite momentum. In the vicinity ofthe instability to the uniform paired state, the tunnelingconductance monotonically decreases with in-plane mag-netic field (as presented in Fig. 9). In the vicinity of aninstability to the FFLO state the tunneling conductanceachieves a maximum at finite field-induced momentumshift QB = Q0 where Q0 is the corresponding momen-tum of Cooper pairs. We discuss how distinguish thesestates in more details in Appendix B, where calculationsfor the inter-valley tunneling are presented.

The sensitivity of Cooper pairing to a disorder opensa possibility of a granular electron-hole state in the pres-ence of its strong long-range variations. In this statethe pairing happens in disconnected or weakly coupledregions with minimal amount of disorder and does notsupport the spatial coherence. It makes the transportproperties of the system including Coulomb drag effect tobe different from ones in the uniform paired state. A tun-neling conductance in the granular state is still colossallyenhanced since the latter requires temporal coherence ofCooper pairs but not the spatial one.

The interpretation of experiment provided by our the-ory suggested that the pairing critical temperature wouldbe substantial if samples with weaker disorder could befabricated. This finding is perhaps a bit surprising sincethe experiments are for the most part conducted in theweak to moderate coupling regime (rs = 0.72 ∼ 3.3)where some researchers have argued that critical temper-atures should be strongly suppressed by screening, espe-cially accounting for spin and valley degeneracy [17, 77](See also arguments that this approach considerably un-derestimates the critical temperature [15, 16, 19, 20]).Our theory also suggests that high pairing temperaturesshould be achievable in double single-layer graphene sys-tems, since there is nothing in its structure that putssingle layers at a disadvantage relative to bilayer. More-over, for the linear spectrum in monolayer graphenethe Wigner-Seitz radius is defined in a different wayrs = 2.19/κ and could achieve even larger value rs = 1.1if hBN is used as a spacer between graphene sheets. Hereκ is the corresponding dielectric constant. Future ex-perimental work which seeks to weaken pair-breaking bydisorder and which explores double single-layer graphenesystems as well, is therefore important.

In summary, the theory of the fluctuational internalJosephson effect developed here explains the anomaliesin the tunneling conductance between graphene bilay-

ers observed experimentally at equal electron and holedensities, including their dependence on temperature,bias voltage bias and electron-hole imbalance. Some as-pects of the observations are nevertheless not understood.First of all the observed dependence of the tunneling con-ductance on bias voltage has an asymmetry between posi-tive and negative bias that becomes more prominent withdecreasing temperature. At first glance the asymmetry isunexpected and surprising since the electronic spectrumof two graphene bilayers with matched concentrations ofelectrons and holes is symmetric, as it is clearly seen inFig. 2. The symmetry can be broken by Coulomb im-purities if most of them are of the same charge. Forexample positive charges (ionized donors) provide repul-sive scattering for holes and attractive scattering for elec-trons. Our model takes the scattering rates for electronsand holes γt(b) to be momentum and energy independentand ignores these common complications. An asymmetrycan be introduced can be introduced in phenomenolog-ical way by making the assumption that Cooper pair-breaking time is energy dependent γ(ω). Approximatingit by a linear function does predict an asymmetry of thetunneling conductance that grows with decreasing tem-perature, but the shape of the experimental curves is notcaptured by this simple ansatz. To clarify whether or notthe observed asymmetry can be explained by the pres-ence of charge impurities, a more microscopic descrip-tion of their scattering characteristics is needed and thisis outside of the scope of the present work. Secondlyit is not clear whether or not our theory can capturethe dependence of tunneling conductance on magneticfield since more experimental data is needed. Compari-son with data obtained at T ≈ 1.5 K suggests that thetheory considerably overestimates the effect of magneticfield. Nevertheless, at such temperatures the system is inthe paired state or in the critical regime that is outsideof the applicability range of the theory of Gaussian fluc-tuations. This discrepancy can be due to other reasons.Bilayer graphene as other two-dimensional systems havelong-range density variations and if the correspondinglength is smaller than ~/QB the effect of the magneticcan not be reduced just to the relative shift of disper-sion for electrons from different layers. To better under-stand capabilities of the theory more experimental datais needed.

To conclude, we have developed a theory of the fluc-tuational internal Josephson effect in the system of twoclosely spaced graphene bilayers. The presence of valleyand sublattice degrees of freedom provides three com-peting electron-hole channels for both intra-valley andinter-valley Cooper pairs. We show that three channelsare nearly independent and have different critical temper-atures of the condensation and sublattice structures. Theobserved enhancement of the tunneling conductance canbe explained only by the presence of competing channelsthat dominate in different temperature ranges. The the-

14

ory reasonably captures the dependence of the conduc-tance on temperature, voltage bias between bilayers andelectron-hole imbalance. We also argue that the enhance-ment is much stronger for inter-valley tunneling than forthe intra-valley one that has been reported recently. Wealso discuss how to distinguish the uniform state and theFFLO state with finite Cooper pair momentum that canbe stabilized in the system by an electron-hole imbalance.

This work was supported by the Army Research Of-fice under Award W911NF-17-1-0312, by ARO MURI3004628717, and by the Welch Foundation under grantF-1473. DKE acknowledges support from the AustralianResearch Council Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET).

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15

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APPENDIXES

A. Bethe-Salpether equation

Here we present a detailed derivation of Eq. (6) fromthe paper. The Bethe-Saltpeter equation that is illus-trated in Fig. 3 can be written as follows

Γσ′tσtσ′bσb

(ipn,p′,p) = Up′−pδ

σ′tσtσ′bσb

+ T∑iωnp′′

Up′−p′′Gtσ′tσ′′t

(ipn + iωn,p′′+)G

bσ′′b σ

′b(iωn,p

′′−)Γ

σ′′t σtσ′′b σb

(ipn,p′′,p). (30)

Here p± = p± q/2; ωn = (2n+ 1)π/T and pn = 2nπ/Tare fermionic and bosonic Matsubara frequencies. The

electron Green function Ĝt(b)(ıωn,p) in the sublatticespace can be presented as follows

http://dx.doi.org/ https://doi.org/10.1016/0039-6028(96)00355-Xhttp://dx.doi.org/ https://doi.org/10.1016/0039-6028(96)00355-Xhttp://dx.doi.org/10.1103/PhysRevB.99.144517http://dx.doi.org/10.1103/PhysRevB.99.144517http://dx.doi.org/10.1088/0034-4885/76/5/056503http://dx.doi.org/10.1088/0034-4885/76/5/056503http://dx.doi.org/10.1103/PhysRev.135.A550http://dx.doi.org/10.1103/PhysRevB.75.113301http://dx.doi.org/10.1103/PhysRevB.75.113301http://dx.doi.org/10.1103/PhysRevB.81.075436http://dx.doi.org/https://doi.org/10.1016/0022-3697(63)90007-6http://dx.doi.org/https://doi.org/10.1016/0022-3697(63)90007-6http://dx.doi.org/10.1103/PhysRevB.47.10619http://dx.doi.org/10.1103/PhysRevB.47.10619http://dx.doi.org/10.1103/PhysRevB.53.7403http://dx.doi.org/10.1103/PhysRevB.53.7403http://dx.doi.org/ 10.1088/2053-1583/aa7bcfhttp://dx.doi.org/ 10.1088/2053-1583/aa7bcfhttp://dx.doi.org/10.1103/PhysRevLett.30.648http://dx.doi.org/10.1007/BF00628207http://dx.doi.org/10.1007/BF00628207http://dx.doi.org/10.1103/PhysRevLett.24.1052http://dx.doi.org/10.1007/BF00655546http://dx.doi.org/10.1007/BF00115616http://dx.doi.org/10.1007/BF00115616http://dx.doi.org/10.1103/PhysRevLett.9.266http://dx.doi.org/10.1143/JPSJ.76.051005http://dx.doi.org/10.1143/JPSJ.76.051005http://dx.doi.org/10.1088/1361-6633/aaa4adhttp://dx.doi.org/10.1088/1361-6633/aaa4adhttp://dx.doi.org/https://doi.org/10.1016/j.physrep.2012.11.004http://dx.doi.org/https://doi.org/10.1016/j.physrep.2012.11.004http://dx.doi.org/10.1103/PhysRevB.78.241401http://dx.doi.org/10.1103/PhysRevB.78.241401http://dx.doi.org/10.1103/PhysRevLett.102.026804http://dx.doi.org/10.1103/PhysRevB.79.041305http://dx.doi.org/ 10.1103/PhysRevLett.101.246801http://dx.doi.org/10.1103/PhysRevB.80.125323http://dx.doi.org/10.1103/PhysRevB.80.125323http://dx.doi.org/ 10.1038/ncomms6824http://dx.doi.org/ 10.1038/ncomms6824https://doi.org/10.1038/nphys3143https://doi.org/10.1038/nphys3143http://dx.doi.org/10.1103/RevModPhys.85.299http://dx.doi.org/10.1103/RevModPhys.85.299http://dx.doi.org/10.1088/1361-6633/aa50e3http://dx.doi.org/10.1088/1361-6633/aa50e3

16

Gtσ′σ(iω̄tn,p) =

〈σ′|tcp〉〈tcp|σ〉iωn − �cp + iγtsgn[ωn]

, Gbσ′σ(iωn,p) =〈σ′|bvp〉〈bvp|σ〉

iωn − �vp + iγbsgn[ωn]. (31)

Here γt(b) are scattering rates for electrons (holes). Wehave also neglected the presence of the valence band inthe layer with excess of electrons and the presence of the

conduction band in the layer with the excess of holes.The product of Green functions that appears in (30) canbe written with as follows

Gtσ′tσ′′t (ipn + iωn,p+)Gbσ′′b σ

′b(iωn,p−) = M

σ′tσ′′t

σ′bσ′′b

(p′′)C(ipn,q, iωn,p).

Here the matrix form-factor

Mσ′tσ′′t

σ′bσ′′b

(p) = 〈σ′t|tcp+〉〈tcp+|σ′′t 〉〈σ′′b |tvp−〉〈bvp−|σ′b〉.

reflects the chiral nature of charge carriers in bilayergraphene, while C(ipn,q, iωn,p) contains informationonly about energy spectrum for electrons and holes andis given by

C(ipn,q, iωn,p) =1

(iωn + ipn − �cp+ + iγtsgn[ωn + pn])(iωn − �vp− + iγbsgn[ωn])(32)

In the weak coupling regime that we consider in thepaper pairing correlations do appear in the vicinity ofFermi lines for electrons and holes. As a result the ver-tex Γ̂(ipn, φp′ , φp), the Fourier transform of interactions

U(φp′−φp), and the form-factor M̂(φp) can be safely ap-proximated by their values at the Fermi level (|p| = pFand |p′| = pF) and depend only on the corresponding po-lar angles (φp and φp′). After decomposition over multi-

pole momenta Γ̂l′,l, Ul and M̂l the equation (30) becomesalgebraic and can be presented in a compact form

Γ̂l′l = Ul′δl′l1̂ +∑l1l2

Ul′Ml′−l1−l2Πl2(ipn,q)Γ̂l2,l. (33)

Here Πl(ipn,q) is given by

Πl(ipn,q) = T∑iωn,p

e−ilφpC(ipn,q, iωn,p)

Its s-wave component Π0(ipn,q) ≡ Π(ipn,q) has the ul-traviolet logarithmic divergence that is usually present inthe weak coupling pairing theories and coincides with thesingle-step pair propagator in the Cooper ladder sum ofa bilayer system without sublattice degrees of freedom.Its detailed derivation can be found in textbooks [43, 44]and its explicit expression is presented in the main partas Eq. (7). Therefor all information about the chiral na-ture of the bilayer graphene charge carriers is hidden inthe nontrivial matrix form-factor M̂l. At finite l the valueΠl(ipn,q) is nonzero only at finite Cooper pair momen-tum and are much smaller than Π0(ipn,q) and can be

neglected. As a result, the Eq. (33) reduces to Eq. (6)from the main text of the paper.

0 20 40 60 800

250

500

750

1000

FIG. 10. The temperature dependence of the inter-valleytunneling conductance GT between graphene bilayers at zerovoltage bias (V = 0). Three curves correspond to pair break-ing rates γ = 2, 4, and 8 meV. The pair breaking rate in-duced by scattering at impurities reduces the critical temper-ature T0 of electron-hole condensation, but weakly effect thecritical behavior above T0.

17

FIG. 11. The dependence of tunneling conductance GT onin-plane magnetic B and the electron-hole density imbalance∆n. Three subplots correspond to T = 45 K (a), 25 K (b)and 5 K (c). The dashed line corresponds to the phase bound-ary of equilibrium electron-hole paired state. In (b) and (c)the conductance achieves maximum at finite value of mag-netic field B that demonstrates that fluctuating Cooper pairswith finite momentum Q ≈ QB are the most intensive andthe system is in the vicinity of the instability to the FFLOstate. The presence of a kink in the phase boundary in (c)clearly demonstrates that the system is unstable towards theequilibrium FFLO state.

B. Inter-valley alignment and identification of theFFLO state

In the main text of the paper calculations for intra-valley tunneling (θ = 0) are discussed, while ones forthe inter-valley alignment (θ = π/3) are presented here.The tunneling conductance is dominated by the channel11-22 because of fluctuating Cooper pairs in the dom-inating channel involve correlations at adjacent sublay-ers. The tunneling conductance can be approximatedas Eq. (28). We use the same set of parameters that

has been used above to fit the experimental data ex-cept for pair-breaking rate γ. For the latter we useγ = 2, 4, and 8 meV. The first two values correspond tocleaner samples compared to ones that have been studiedexperimentally [10]. The temperature dependence of tun-neling conductance is presented in Fig. 10. Its enhance-ment of the conductance is considerably stronger thanthat for the inter-valley tunneling. The pair-breakingrate γ determines the critical temperature of pair con-densation, but weakly influences the temperature depen-dence of tunneling conductance above it.

For the pair-breaking rate γ = 2 meV the critical tem-perature is T0 ≈ 42 K. According to the phase diagramFig. 5 the ratio γ/T0 ≈ 0.52 is small enough to stabi-lize the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) stateby the electron-hole imbalance. In-plane magnetic filedresults in a relative shift of electronic dispersions betweenlayers and makes it possible to probe fluctuating Cooperpairs with finite momentum QB = edB||/~c. The depen-dence of tunneling conductance at zero voltage bias onthe electron-hole imbalance and magnetic field at temper-ature T = 55 K is presented in Fig. 11-a. The tunnelingconductance has a monotonic dependence demonstratingthat the system is far away from the instability to theFFLO state and Cooper pair fluctuations with zero mo-mentum Q = 0 are the most intensive. The dependenceat T = 25 K is presented in Fig. 11-b with a dashed linethat denotes a phase boundary of a paired state. Sincethe critical imbalance required to suppress the pairinginstability decreases with magnetic field the paired stateis the uniform BCS one. Nevertheless, the tunneling con-ductance achieves maximum at finite value of magneticfield that shows that the systems is close to the FFLOstate and fluctuations with finite Cooper momentum arethe most intensive. The dependence at T = 5 K is pre-sented in Fig. 11-c. The dashed line that denotes thephase boundary is non-monotonous and has a kink at fi-nite value of magnetic field. It clearly demonstrates thatthe equilibrium FFLO state is stabilized by the electron-hole imbalance.

C. Bosonic picture of the fluctuational internalJosephson effect

Here we present a bosonic picture of the fluctuationalinternal Josephson effect that can be employed with ahelp of the field integral formalism. We consider the con-tact interactions U between electrons and holes that cor-respond to the truncation of all multipole momenta of in-teractions except the s-wave one. In the main text of thepaper we have demonstrated that they are unimportant.The correlation function of tunneling operators χ(ω,q)that defines the tunneling conductance GT according toEq. (18) can be extracted from the corresponding imag-

18

inary time correlation function χ(τ, r) given by

χ(τ, r) =1

Z

δ2Z

δΛτ,rδΛ+0,0

. (34)

Here Z[Λ+τ,r,Λτ,r] is the statistical sum with an auxiliarybosonic field Λτ,r introduced to the action S of the systemas follows

S =

∫ β0

dτ

∫dr[ψ̂+t (∂τ + ĥtr − µt)ψ̂t + ψ̂+b (∂τ + ĥbr + µb)ψ̂b + ψ̂

+b Λ

+t̂+ ψ̂t + ψ̂+t t̂Λ ψ̂b + Uψ

+tstψ

+bsbψbsbψtst

].

Here ψ̂t ≡ ψ̂tτr and ψ̂b ≡ ψ̂bτr are spinor fermionic fieldsfor electrons from top (t) and bottom (b) layers with la-beling described in Sec. II. If they are integrated out andthe corresponding action is expanded in the lowest orderin tunneling matrix elements t̂, the Fourier transform χqdo appears in the action as follows S = −

∑q χq|Λq|2.

Here q = {qn,q} with bosonic Matsubara frequencyqn = 2πTn. For noninteracting electrons and holes cal-

culations are straightforward and result in

χ0q = t̂+M̂0NFΠq t̂. (35)

After analytical continuation we get the Eq. (20) from themain text. In the case of interacting electrons and holes itis instructive start with the Hubbard-Stratonovich trans-formation. It eliminates interactions but introduces thebosonic field ∆̂ ≡ ∆̂τr corresponding to electron-holeCooper pairs. The action S is modified as follows

S =

∫ β0

dτ

∫dr

[ψ̂+t (∂τ + ĥtr − µt)ψ̂t + ψ̂+b (∂τ + ĥbr + µb)ψ̂b + ψ̂

+b ∆̂

′+ ψ̂t + ψ̂+t ∆̂′ ψ̂b +

1

Utr[∆̂+∆̂

]]. (36)

where ∆̂′τr = ∆̂τr + t̂Λτr. Above the critical tempera-ture T0 of the electron-hole pairing the saddle point ofthe action is trivial 〈∆̂〉 = 0 and the field ∆̂ correspondsto Cooper pair fluctuations [86]. In the wide tempera-ture range ∆TGi . ∆T ∼ T0 outside the critical regime

∆T . ∆TGi fluctuations can be approximated by thenoniteracting Gaussian theory. Here ∆TGi = Gi T0 withGinzburg number Gi = T0/EF calculated in AppendixD. Integrating out fermions and expanding the action upto the second order in the bosonic field ∆̂′ results in

S =∑q

[∆̂+q ∆̂q

U− ∆̂′+q M̂0Πq∆̂′q

]=∑q

[∆̂+q Γ̂

−1q ∆̂q − ∆̂+q M̂0ΠqtΛq − Λ+q t̂+M̂0Πq∆̂q − Λ+q χ0qΛq

]. (37)

Here all matrices are in the compact representation,and Γ̂q is the scattering vertex calculated in the Sec. IIIof the paper. Some of its components vanish at the crit-ical temperature T0 of the electron-hole Cooper pairing.The last term corresponds to the response χ0q functionfor noninteracting electrons and holes that is given byEq.( 35). The action represents the bosonic picture ofthe Josephson effect and is valid outside the weak cou-pling regime. The action (37) is quadratic in the bosonicfield ∆q and after its integration we get the tunnelingresponse function χq to be given by

χq = χ0q + t̂

+M̂0ΠqΓ̂−1q M̂0Πq t̂ (38)

With a help of Eqs. (12) and (11) we get the tunneling

response function for the intra-valley tunneling (23). Inthe same way with a help of Eqs. (14) and (13) we getthe response function for the inter-valley one (25).

D. Ginzburg criterion

The developed theory of the fluctuational internalJosephson effect implies that Cooper pair fluctuations areGaussian and do not interact with each other. The inter-actions can be safely omitted in the wide range of temper-atures ∆T ∼ T0 except the critical region ∆T . ∆TGiwhere fluctuations are overgrown are strongly interferewith each other. The range ∆TGi can be estimated

19

from the Ginzburg criterion [87] that compares the con-tribution of Gaussian fluctuations to the heat capacityCFL = T0/4πξ

2∆T with predictions of the mean-fieldtheory below the critical temperature CMF = NF/gT0 =�F/4πξ

2T0. Here g = 4ξ2/(~vF)2 is the strength of con-

tact interactions between fluctuating Cooper pairs ne-glected so far. The contribution of fluctuations CMF

grows with decreasing of temperatures and dominatesin the temperature range ∆TGi = GiT0 with Ginzburgnumber Gi = T0/EF. It does not depend explicitly onthe pair-breaking rate γ but only on the critical temper-ature T0. It should be noted that the Ginzburg criterioncan be derived microscopically in a more strict way byexplicit analysis of the role of interactions between fluc-tuating Cooper pairs [88].

Tunneling and Fluctuating Electron-Hole Cooper Pairs in Double Bilayer GrapheneAbstract I. Introduction II. Model A. Weak and strong coupling regimes B. Phenomenological model in the weak coupling regime

III.Fluctuating Cooper pairs A. The Cooper instability B. Intra-valley Cooper pairs C. Inter-valley Cooper pairs D. Disorder and density imbalances E. Coherence time and length of fluctuating Cooper pairs

IV. Tunneling conductivity A. Linear response theory for the tunneling conductance B. Intra-valley tunneling C. Inter-valley tunneling D. Tunneling conductance E. Critical behavior of tunneling conductance

V. Comparison with experiment VI. Discussions Acknowledgments References Appendixes A. Bethe-Salpether equation B. Inter-valley alignment and identification of the FFLO state C. Bosonic picture of the fluctuational internal Josephson effect D. Ginzburg criterion