Superconductivity in 2D electron gas induced by high energy optical phonon mode and large polarization of the STO substrate

1.1 MB·

by  Baruch Rosenstein

Superconductivity in 2D electron gas induced by high energy optical phonon mode
and large polarization of the STO substrate.
Baruch Rosenstein,1,2,∗ B.Ya. Shapiro,3,† I. Shapiro,3 and Dingping Li4,5,‡
1Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, R. O. C
2Physics Department, Ariel University, Ariel 40700, Israel
3Physics Department, Bar-Ilan University, 52900 Ramat-Gan, Israel
4School of Physics, Peking University, Beijing 100871, China
5Collaborative Innovation Center of Quantum Matter, Beijing, China
Pairinginoneatomiclayerthicktwodimensionalelectrongasbyasingleflatbandofhighenergy
6
longitudinal optical phonons is considered. The polar dielectric SrTiO (STO) exhibits such an
1 3
0 energetic phonon mode and the 2DEG is created both when one unit cell FeSe layer is grown
2 on its (100) surface and on the interface with another dielectric like LaAlO3 (LAO). We obtain
a quantitative description of both systems solving the gap equation for Tc for arbitrary Fermi
r
p energy (cid:15)F, electron-phonon coupling λ and the phonon frequency Ω, and direct (RPA) electron-
A electron repulsion strength α. The focus is on the intermediate region between the adiabatic,
(cid:15) >> Ω, and the nonadiabatic, (cid:15) << Ω, regimes. The high temperature superconductivity in
F F

7 1UCFeSe/STO is possible due to a combination of three factors: high LO phonon frequency, large
2 electron-phonon coupling λ ∼ 0.5 and huge dielectric constant of the substrate suppression the
Coulomb repulsion. It is shown that very low density electron gas in the interfaces is still capable

] of generating superconductivity of the order of 0.1K in LAO/STO.
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I. INTRODUCTION
Singlelayerofironselenide(FeSe)grownonastrongpolarinsulatorSrTiO (001)(STO)exhibitssuperconductivity1–6
3
at surprisingly high temperature 70−100K. This is an order of magnitude larger than the parent bulk material with
the superconducting transition temperature7 T of 8K. This suggests that the dominant mechanism of creation of
c
the superconductivity in the FeSe layer might be different from that of the bulk FeSe and is caused by influence of
the STO substrate. To strengthen this point of view the high-resolution angle-resolved photoemission spectroscopy
(ARPES) experiments5 and the ultrafast dynamics3 demonstrated the presence of high-energy phonons in STO. The
frequency of the oxygen longitudinal optical (LO) mode reaches Ω ≈ 100meV. In addition it turns out that the
phononscouplestronglytotheelectronsintheFeSelayer(thecouplingconstantwasestimatedtobe3 λ∼0.5,much
larger than in the parent material, λ = 0.19). The band is flat with only a small momentum transfer to electrons.
This identification is supported by the earlier ARPES on STO surface states, which shows a phonon-induced hump
at approximately 100meV away from the main band and through inelastic neutron scattering8. The role of substrate
in assisting superconductivity is not limited to generation of phonons. The polar STO has a huge dielectric constant
(estimated to be above (cid:15)=1000 on the surface) and hence suppresses Coulomb repulsion inside the FeSe layer.
The nature of electronic states within the FeSe layer is by now quite settled experimentally. The Fermi surface of
the single unit cell (1UC) consists of two electron-like pockets centred around the crystallographic M-point (Brillouin
zone corners) with a band bottom below the Fermi level5 (cid:15) = 60meV. This means that electrons form a two
F
dimensional electron gas (2DEG) with small chemical potential. The novelty of the superconducting system is that
the occupied states are close to the band edge, very far from the classic case. In both conventional (BCS) and
unconventional superconductors the chemical potential is the largest energy scale in the problem (even in quasi 2D
high T cuprates the chemical potential is order of magnitude higher). STM measurements in the superconducting
c
state demonstrates that there are no nodes6 (no sign change of the order parameter). It shows at 4K a fully peaked
gap (with double peaks at 10mev,15mev with minimum at 5eV) which is suppressed only by magnetic impurities,
similar to a conventional 2D s-wave superconductor. Absence of nesting indicates that there are no effects like charge
density waves.
An early theory9 focused on the screening due to the STO ferroelectric phonons on antiferromagnetic spin fluctu-
ations mediated Cooper pairing in parent material FeSe. It suggested that the phonons significantly enhance the
Cooper pairing and even change the pairing symmetry. Naively the spin fluctuation interaction by itself should lead
to nodeless d-wave pairing. For the electron-phonon coupling λ ∼ 1 the enhancement was large, although perhaps
notenoughtoexplaintheexperiment. Whentheinter-pocketelectron-phononscatteringisalsostrong, opposite-sign
pairing will give way to equal-sign pairing. Later5 it was suggested that the interfacial nature of the coupling assists
superconductivity in most channels, including those mediated by spin fluctuations.
Another idea10 is to use both the electron pockets at the Fermi surface band and the ”incipient” hole band below
it also found in ARPES, namely generalizing to the multiband model. The conclusion was that ”a weak bare phonon
interaction can be used to create a large T , even with a spin fluctuation interaction which may be weakened by the
c
incipient band.” The difficulty is that the forward scattering nature of the essential phonon processes then means
that LO phonons cannot contribute to the inter-band interaction. Gor’kov considered11 polarization on the surface,
screening and the STO surface LO phonon pairing. His conclusion is that the LO phonon mediated pairing alone
cannot account for superconductivity at such high T .
c
ThesmallchemicalpotentialistypicalfortheSTOsystems. Anotherrelatedsuperconducting(withmuchlowerT )
c
2DEG system with even much smaller chemical potential is the LaAlO (LAO)-STO interface observed earlier12. The
3
microscopic origin of the superconductivity in the LAO/STO system is already quite clear13. It is the BCS - like s-
wavepairingattributedtothesameLOphononmodesdiscussedaboveincontextofthe1UCFeSe/STOsystem. Spin
fluctuations seem not to play any role in the pairing leading to superconductivity. The phase diagram of LAO/STO
is qualitatively similar to the dome-shaped phase diagram of the cuprate superconductors: in the underdoped region
the critical temperature increases with charge carrier depletion.
The theoretical effort to understand the LAO/STO system14 resulted in realization that the Migdal-Eliashberg
theory of superconductivity, valid when the phonon frequencies are much smaller than the electron Fermi energy,
shouldbegeneralized. ThisisnotthecaseforpolarcrystalslikeSTOwithsufficientlyhighoptical-phononfrequencies,
and consequently the dielectric function approach proposed long ago by Kirzhnits15 and developed in ref16 proved to
be useful. It was shown that the plasma excitations are important at larger µ (reduce the electron-phonon coupling)
and enable to explain the non-monotonic behavior of T as function of bias that changes chemical potential.
c
Inthispaperwefurtherdevelopatheoryofsuperconductivityin1UCFeSe/STOandLAO/STObasedonphononic
mechanismincludingeffectsofthescreenedCoulombrepulsion. Inthefirststage asimplemodelof2DEGwithpairing
mediated by a dispersionless LO phonons is proposed with Coulomb repulsion assumed to be completely screened by
huge polarization of STO ((cid:15) ∼ 3000 in 1UCFeSe/STO). In this case the gap equations of the Frohlich model can
be reduced (without approximations) to an integral equation with one variable only and are solved numerically for
3
arbitrary Fermi energy (cid:15) , phonon frequency Ω and electron-phonon coupling λ<1. An expression for the adiabatic
F
andnonadiabaticlimitsarederivedandresultsforT comparewellwithexperimentson1UCFeSe/STO.Then,inthe
c
second stage weincludetheRPAscreenedCoulombrepulsion(forsomewhatsmallervaluesofdielectricconstantsare
estimated17 to be (cid:15)=186 on the STO side and (cid:15)=24 on the LAO side) and solve a more complicated gap equations
numerically (without making use of the Kirzhnits Ansatz) for various (cid:15) and Coulomb coupling constant. Both the
F
adiabatic,(cid:15) >>Ω,(conventionalBCS)andthenonadiabatic,(cid:15) <<Ω,casesareconsideredandcomparedwiththe
F F
localmodelstudiedearlierinthecontextofBECphysics18–21. TheCoulombrepulsionresultsinsignificantreduction
or even suppression of superconductivity. A phenomenological model for dependence of (cid:15) and λ on electric field for
F
the LAO/STO is proposed.
The paper is organized as follows. The basic 2DEG phonon superconductivity model is introduced in Section II.
ThegeneralGaussianapproximationforweakelectron-phononinteractionsandRPAscreeningisdescribedinSection
III. The superstrong screening case (neglecting Coulomb repulsion altogether) case is solved Section IV. The same
calculation is performed using the Kirzhnits approach in Section V. The general case including the RPA screened
Coulomb repulsion is investigated numerically in Section VI. The phenomenology of 1UCFeSe/STO and LAO/STO
and comparison with experiments are discussed in Section VII followed by Discussion and summary. Appendices A
andBcontainthederivationofGorkovequationsandthe2DRPAneutralizingbackgroundcontributionrespectively.
II. THE LO PHONON MODEL OF PAIRING IN 2DEG
As mentioned above various STO systems including 1UCFeSe/STO (medium to low density) and interface
LAO/STO the (very low density) electron gas appears localized in a plane of width of one unit cell (in FeSe layer or
on the STO side respectively). The Hamiltonian of the system contains three parts
H =H +H +H . (1)
e ph e−ph
A. Description of 2DEG
We use a continuum parabolic 2DEG model one ”flavours” (up and down spin projections and two valleys in
1UCFeSe/STO) with effective mass close to mass of electron14). The 2DEG Hamiltonian in terms of creation
operators ψ† (r,t), σ ={↑,1},...{↑,N},{↓,1},...{↓,N} electrons thus is
σ
(cid:90) (cid:18) (cid:126)2∇2 (cid:19) 1 (cid:90)
H = ψ† − −µ ψ + n(r)v(r−r(cid:48))n(r(cid:48)), (2)
e σ 2m σ 2
r r.r(cid:48)
where the charge density operator is
n(r)=ψσ†(r)ψσ(r), (3)
andµisthechemicalpotential(Fermienergy). Theelectron-electroninteractions,notrelatedtothecrystallinelattice,
are described by potential v(r). The electrostatics on the surface/interface is quite intricate17, and we approximate
it by the Coulomb repulsion:
e2
v(r)= , (4)
(cid:15)r
where (cid:15) is and effective 2D dielectric constant of the system. As mentioned in Introduction the effective dielectric
constant is huge in STO at low temperatures due to the ionic movements.
B. Phonons and electron-phonon interactions
Crystal vibrations in STO are highly energetic. The single phonon band8,13 near Ω = 100meV is most probably
associatedwithpairingattractiveelectron-electronforceistheferroelectricLOthatinvolvestherelativedisplacement
of the Ti and O atoms. The high energy STO oxygen LO phonon band mode is separated from all the other phonon
bands by a substantial energy gap8. The single branch of the optical phonons described by the bosonic field22
(cid:16) (cid:17)
φ(r)=(cid:80)k √12 b†ke−ikr+bkeikr . The phonon part of the Hamiltonian therefore is:
4
1(cid:90)
H = φ(r)v (r−r(cid:48))φ(r(cid:48)), (5)
ph 2 ph
r,r(cid:48)
where the phonon energy density v (r−r(cid:48)), for the nearly flat LO band is approximately local:
ph
v (r)=(cid:126)Ωδ(r). (6)
ph
Experiments demonstrated a substantial electron–phonon coupling g. In fact the collective mode energy is greater
or comparable to the width of the electron band. Importantly, the electron–phonon coupling allows only a small
momentum transfer to the electron.
(cid:90)
H =g n(r)φ(r). (7)
e−ph
r
Despite the simplifications, the model is far from being solvable and standard approximations are applied in the
following section to obtain the critical temperature of the superconductor. Various ”bare” parameters like effec-
tive masses, Ω, the electron-electron and electron-phonon couplings are renormalized as the interaction effects are
accounted for.
III. THE PAIRING EQUATIONS
A. Matsubara Action
We use the Matsubara time τ (0 < τ < (cid:126)/T) formalism22 with action corresponding to the Hamiltonian
Eq.(1),(setting (cid:126)=1), A[ψ,φ]=A [ψ]+A [φ]+A [ψ,φ],with
e ph e−ph
(cid:90) 1(cid:90)
A = ψ∗(r,τ)D−1ψ (r,τ)+ n(r,τ)v(r−r(cid:48))n(r(cid:48),τ) (8)
e σ σ 2
r,τ r,r(cid:48),τ
1(cid:90)
A = φ(r,τ)d−1φ(r(cid:48),τ);
ph 2
r,r(cid:48),τ
(cid:90)
A =g n(r,t)φ(r,t).
e−ph
r,τ
Here the electron Green’s function is,
∇2
D−1 =∂ − −µ, (9)
τ 2m
while that of the phonon field is
d−1 =(cid:0)−∂2+Ω2(cid:1)δ(r−r(cid:48)). (10)
τ
In Fourier space the action reads
(cid:88) 1 (cid:88)
A = ψσ∗D−1ψσ + v ψσ∗ ψσ ψρ∗ ψρ ; (11)
e pω pω pω 2 p p1ω1 p1−p,ω1−ω p2ω2 p2+p,ω2+ω
pω pωp1p2ω1ω2
1(cid:88) (cid:88)
A = φ∗ d−1φ ;A =g ψσ∗ ψσ φ
ph 2 kω ω kω e−ph p1ω1 p1−p,ω1−ω pω
kω pp1ωω1
with electronic,
D−1 =iω+ε ; ε =p2/2m−µ, (12)
p,ω p p
and optical phonon
ω2+Ω2
d−1 = , (13)
ω Ω2
5
propagators respectively. The fermionic Matsubara frequencies are ω = πT (2n+1), while for bosons ω = 2πTn
n n
with n being an integer. In 2D
2πe2
v = . (14)
p (cid:15)p
The action can be treated with the standard gaussian approximation.
B. The pairing equations
The electronic action is obtained by integration of the partition function over the phonon field,
(cid:90)
Ze[ψ]= e−A[ψ,φ] =e−Aeeff[ψ]. (15)
φ
The gaussian integral is,
(cid:88)
Aeff[ψ]= ψσ∗D−1ψσ + (16)
e pω pω pω
ωp
1 (cid:88)
+ V ψσ∗ ψσ ψρ∗ ψρ ,
2 pω p1−p,ω1−ω p1ω1 p2ω2 p2−p.ω2−ω
ωω1ω2pp1p2
where V =VRPA+Vph. The part of the effective electron-electron attraction due to phonons is:
pω pω ω
Ω2
Vph =−g2 . (17)
ω ω2+Ω2
To take into account screening, we made the replacement v →VRPA (the random phase approximation) in 2D,
p pω
VRPA =v (cid:18)1+ Nmvp (cid:16)1−x/(cid:112)x2+1(cid:17)(cid:19)−1, (18)
pω p π
where x=|ω|/(v p) with v2 =2µ/m.
F F
Performing the standard gaussian approximation averaging, see appendix A, one arrives at the Gor’kov equations
(cid:68) (cid:69) (cid:68) (cid:69)
for the normal , ψ↑I†ψ↓J = δ δ δIJG (I,J = 1,...,N are flavours), and the anomalous, ψ↑Iψ↓J =
kω qν ω−ν k−q kω kω qν
δ δ δIJF , Greens functions. The result is
ω+ν k+q kω
−∆∗ F +D∗−1G =1, (19)
kω kω kω kω
and
∆ G =−D−1F , (20)
kω kω kω kω
where the gap function is defined by
(cid:88)
∆ = V F . (21)
kω p1ω1 p1−k,ω1−ω p1ω1
Near the critical point one can neglect higher orders in ∆ in Eq.(19), resulting in G = D∗. Substituting this into
Eq.(20), one gets:
(cid:88) |D |2V ∆ =−∆ . (22)
pν p−k,ν−ω pν kω
pν
Using the explicit form of the propagator D, Eq.(12), the equation takes a final form:
(cid:88) 2NT
V ∆ =−∆ . (23)
pm ω2 +ε2 p−k,m−n pm kn
m p
6
C. Simplification of the integral equations for critical temperature for the s-wave pairing.
Transforming to polar coordinates and using rotation invariance, ∆ = ∆ , p = |p|, and then changing the
pν pν
variables to ε =p2/2m−µ, the electronic part of the kernel of Eq.(23) is
p
(cid:90) Λ−µ mNT (cid:88) 1
P ∆ =−∆ . (24)
ε2=−µ π n2 ωn22 +ε22 ε1ε2;n1−n2 ε2n2 ε1n1
Here Λ is an ultraviolet cutoff of the order of atomic energy scale (cid:126)2/2ma2 with lattice spacing a. The phonon part
of the kernel, P =PRPA +Pph is
ε1,ε2,n ε1,ε2,n n
g2Ω2
Pph =− , (25)
n ω2 +Ω2
n
while in the screened Coulomb part is
(cid:40) (cid:112) (cid:41)−1
e2 (cid:90) 2π 2(s−rcosφ)+
PεR1P,εA2,n = (cid:15) φ=0 +2e(cid:15)2 (cid:16)1−|ωn|/(cid:112)ωn2 +4µ(s−rcosφ)(cid:17) . (26)
This formula along with the treatment of the neutralizing background is derived in Appendix B. Here we have used
abbreviations
s=ε +ε +2µ; (27)
1 2
(cid:112)
r =2 (ε +µ)(ε +µ).
1 2
To symmetrize the kernel viewed as a matrix, one makes rescaling of the gap function
1
η = ∆ , (28)
εn (cid:112) εn
ω2 +ε2
n
leading to eigenvalue equation
(cid:90) Λ−µ
(cid:88)
K η =η , (29)
ε2=−µ n2 ε1n1;ε2n2 ε2n2 ε1n1
where the symmetric matrix is
mNT 1
K =− P . (30)
ε1n1;ε2n2 π (cid:113) (cid:113) ε1ε2,n1−n2
ω2 +ε2 ω2 +ε2
n1 1 n2 2
CriticaltemperatureisobtainedwhenthelargesteigenvalueofthematrixK isunit. Thiswasdonenumericallyby
discretizing variable ε. The numerical results for the full model are presented in section IV, however since screening
of the STO is very strong we first neglect the Coulomb repulsion altogether. This allows a significant simplification.
IV. SUPERCONDUCTIVITY IN THE LO PHONON MODEL
In this case the theory Eqs.(2,5) has three parameters (in addition to temperature), the optical phonon frequency
Ω, the electron-phonon coupling g and chemical potential µ. We first relate the bare coupling g to the ”binding
energy E ” conventionally determined in the BCS-BEC crossover studies18,20,21. Then, since this simplified model
c
will be applied to the 1UC FeSe on STO, one prefers to parametrize the electron gas via carrier density n related
to the Fermi energy by (cid:15) = π(cid:126)2n/m instead of chemical potential µ. Following the standard practice, T is found
F c
by solving the second Gorkov equation Eq(22). This is compared with a simpler Kirzhnits approach applied to the
present case in the next section. To simplify the presentation and without too much loss of generality we take the
number of flavors N =1.
7
A. Binding energy
It is customary18,21 to relate the electron - phonon coupling g to the energy of the bound state E ≡ 2E created
b c
by this force in quantum mechanics in vacuum (the two - particle sector of the multiparticle Hilbert space). We use
the binding energy to estimate the parameter range in which chemical potential µ approaches the Fermi energy (cid:15)
F
definedabove. In2DthethresholdscatteringmatrixelementfortotalenergyE atzeromomentumobeystheintegral
Lippmann-Schwinger equation for scattering amplitude:
1 (cid:90)
Γ(ω,ν,2E)=−Vph − Vph f(ρ,E)Γ(ρ,ν,2E), (31)
ω−ν 2π ω−ρ
ρ
where
1 (cid:90) 1 1
f(ρ,E)= (32)
(2π)2 p2/2m+E+iρp2/2m+E−iρ
p
m (cid:90) Λ 1 m (cid:18) 2 E (cid:19)
= = 1− arctan .
2π ε2+ρ2 4|ρ| π |ρ|
ε=E
The equation Eq.(31) coincides with the sum of ”chain diagrams” at zero chemical potential in the many - body
theory with Γ being the ”renormalized coupling”23. The bound state (there is only one such bound state in 2D) with
binding energy 2E is found as a singularity of Γ(ω,ν,2E). It occurs at energy for which the matrix of the linear
c
equation Eq.(31) has zero eigenvalue, so that the eigenvector ψ(ρ) obeys
(cid:90) (cid:16) (cid:17)
2πδ(ω−ρ)+Vph f(ρ,E ) ψ(ρ)=0. (33)
ω−ρ c
ρ
Changing the variables, ψ(ρ)=f(ρ,E)−1/2η(ρ), this equation can be presented as the unit eigenvalue problem
mg2 (cid:90)
K(ω,ρ)η(ρ)=η(ω), (34)
2π
ρ
with a symmetric kernel
(cid:115)
1 1 (cid:18) 2 E (cid:19) 1 (cid:18) 2 E (cid:19) Ω2
K(ω,ρ)= 1− arctan c 1− arctan c . (35)
4 |ω| π |ω| |ρ| π |ρ| (ω−ρ)2+Ω2
Itturnsoutthattheuniteigenvalueisthemaximaleigenvalueofthispositivedefinitematrix. Thediscretizedversion
of Eq.(34) was diagonalized numerically. The results are presented in Fig. 1.
Solution found numerically is well fitted by
(cid:20) (cid:21)
2π 1 1 Ω(Ω+πE )
= ≈ sinh−1 c , (36)
mg2 λ 2 πE (Ω+E )
c c
where the 2D dimensionless electron - phonon coupling (per spin) is defined as λ= mg2. As will be demonstrated in
2π(cid:126)2
the following subsections, the interesting range of couplings will obey (cid:15) >> E and thus21 we always replace µ by
F c
(cid:15) .
F
It has the correct asymptotics at both weak and strong coupling, so that
 (cid:115) 
E 1  (cid:20)2(cid:21) (cid:18) (cid:20)2(cid:21)(cid:19)2 4 (cid:20)2(cid:21)
c = 1−sinh + 1−sinh + sinh . (37)
Ω 2sinh(cid:2)2(cid:3) λ λ π λ
λ  
At weak coupling
2
E /Ω= e−2/λ <<1 (38)
c π
8
0.4 (cid:230)
0.3 (cid:230)
(cid:87)
c 0.2 (cid:144) (cid:230)
E
0.1 (cid:230)
(cid:230)
(cid:230)
(cid:230)
(cid:230)
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Λ
FIG.1. The2DbindingenergyperelectronoftwoelectronsintheboundstatecreatedbytheattractionduetoLOdispersionless
phononbranchwithfrequencyΩ. The(bare)couplingstrengthλisinawiderangeλ∼0−3.5. Theessentialexactdependence
found numerically (dots) is compared with weak coupling (the solid line) and results obtained using the local model (dashed
line).
and hence one can use a local ”instantaneous” electron-phonon interaction model, with Eq.(25) approximated by
g2Ω2
Pph =− ≈−g2θ(|ω |−Ω), (39)
n ω2 +Ω2 n
n
to describe this limit. In the instantaneous model the electron - phonon interaction is assumed to vanish on the scale
of Ω. Therefore in this approximation for (cid:15) << Ω all the integrations can be cut off at this scale intercepting the
F
larger cutoff Λ. The results for E are consistent with BEC literature21, see dashed line in Fig. 1. Note that the
c
dimensionless pre-exponential factor in Eq.(38) is determined to be 2.
π
B. The energy independence of the gap function
The equation Eq.(24) in the limit e2 →0 is:
g2mT (cid:88) (cid:90) Λ−(cid:15)F 1 Ω2
∆ =∆ . (40)
2π n2 ε2=−(cid:15)F ωn22 +ε22(ωn1 −ωn2)2+Ω2 ε2n2 ε1n1
Since the left hand side of the equation is independent of ε , the gap function is independent of energy: ∆ = ∆ .
2 εn n
Substituting this, one gets a one dimensional integral equation
(cid:88) Ω2 (cid:90) Λ−(cid:15)F 1
λT ∆ (41)
n2 (ωn1 −ωn2)2+Ω2 n2 ε2=−(cid:15)F ωn22 +ε22
=λ(cid:88) Ω2f(ωn2) ∆ =∆ ,
n2 (ωn1 −ωn2)2+Ω2 n2 n1
where the integral is
(cid:18) (cid:20) (cid:21)(cid:19)
T Λ−(cid:15) (cid:15)
f(ω)= arctan F +arctan F . (42)
|ω| |ω| |ω|
(cid:112)
Changing of variables η = f(ω )∆ , makes the kernel matrix of the integral equation,
n n n
(cid:88)
K (T)η =η , (43)
n2 n1n2 n2 n1
9
0.10 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
(cid:230)
(cid:230)
(cid:230)
(cid:230)
0.08 (cid:230)
(cid:230)
(cid:230)
(cid:230)
(cid:230)
0.06
(cid:87) (cid:230)
(cid:230)
(cid:144)
Tc (cid:230)
(cid:230)
0.04 (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
(cid:230)(cid:230)
(cid:230)
(cid:230) (cid:230)
0.02 (cid:230)(cid:230)(cid:230)(cid:230)
(cid:230)
(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
0.00
0 2 4 6 8
Ε (cid:87)
F
FIG. 2. The critical temperature of a 2DEG - LO phonon superconductor (the Coulomb repulsion is assumed to screened out
bythesubstrate). T inunitsofthephononfrequencyΩisgivenasafunctionoftheFermienergyinwhalerangeof(cid:15) /Ω,for
c F
the dimensionless electron-phonon coupling (from top to bottom): λ = 0.5,0.34,0.25. The adiabatic (BCS) limit is a dashed
(cid:144)
line. Solid line is result of local theory.
symmetric,
(cid:112)
f(ω )f(ω )Ω2
K (T)=λ n1 n2 . (44)
n1n2 (ω −ω )2+Ω2
n1 n2
C. Numerical procedure and results
The eigenvalue equation Eq(43) is solved numerically by diagonalizing sufficiently large matrix K (T). The
n1n2
index −N /2<n<N /2 with the value N =256 used. At this value of N the results are already independent of
ω ω ω ω
the UV cutoff Λ. The critical temperature for given λ, (cid:15) and Ω is determined from the requirement that the largest
F
eigenvalueofK(T)is1. Theresultspresentedasfunctionsof(cid:15) inFig. 2inwholerangeof(cid:15) andFig. 3for(cid:15) <Ω.
F F F
D. Adiabatic and nonadiabatic (local interaction model) limits
In the strongly adiabatic situation, (cid:15) >> Ω, one can take the (cid:15) → ∞ limit in which the matrix simplifies,
F F
f(ω)≈ πT,
|ω|
λ
KBCS(T)= . (45)
n1n2 (cid:112)|n +1/2||n +1/2|(cid:16)(cid:0)2πT (n −n )(cid:1)2+1(cid:17)
1 2 Ω 1 2
This can be fitted by the phenomenological McMillan like formula (dashed lines in Fig.2),
(cid:20) (cid:21)
1
Tadiab(λ)≈0.75 Ωexp − . (46)
c λ
In the opposite strongly non-adiabatic limit, E << (cid:15) << Ω, the local model defined in subsection A can be used.
c F
The gap equation Eq.(41) for frequency independent ∆ =∆ simplifies into
n
λ(cid:88)Ω/(2πTc) f(ω )∆=∆. (47)
n2=−Ω/(2πTc) n2
10
0.10
0.08 (cid:230) (cid:230)
(cid:230)
(cid:230)
(cid:230)
(cid:230)
(cid:230)
0.06
(cid:230)
(cid:87) (cid:230)
Tc (cid:144) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)
(cid:230)(cid:230)
0.04 (cid:230)(cid:230)
(cid:230)(cid:230)
(cid:230)(cid:230)
(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
(cid:230)(cid:230) (cid:230)
0.02 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
0.00
0.0 0.2 0.4 0.6 0.8 1.0
Ε (cid:87)
F
FIG. 3. The critical temperature of a 2DEG - LO phonon superconductor in the low temperatures range in units of the
phonon frequency Ω for λ=0.5,0.34,0.25. Solid line is the result of the local theory.
(cid:144)
The solution exists for
(cid:18) (cid:20) (cid:21)(cid:19)
λT (cid:88)Ω/(2πTc) 1 π +arctan (cid:15)F =1 (48)
c n=−Ω/(2πTc) |ωn| 2 |ωn|
At low temperatures the sum can be approximated by an integral
λ(cid:90) Ω 1 (cid:16)π (cid:104)(cid:15) (cid:105)(cid:17)
+arctan F =1, (49)
π ω 2 ω
ω=πTc
one gets the formula
(cid:114) (cid:20) (cid:21)
Tlocal(λ)=(cid:112)E (λ)(cid:15) = 2Ω(cid:15)F exp −1 . (50)
c c F π λ
ThecurvesaregiveninFig.3(dashedlines)andcompareswellwiththesimulatedresult(circles)forλ=0.5,0.34,0.25
(from top to bottom).
Thereexistsanalternativeapproachtosuchcalculations(beyondtheGaussianapproximationadoptedhere),see19
in which the correlator at zero chemical potential is subtracted. We don’t use it, but very recently Chubukov et al
found21 that for the local instantaneous model results are identical. It is instructive to compare the direct numerical
simulation with a simpler approximate semi - analytic Kirzhnits method that is applied to the model in the following
Section.
V. COMPARISON WITH THE KIRZHNITS ANSATZ
A. Application of the Kirzhnits method to LO phonon model
Integral equations in general case Eqs.(43) are very complicated and typically approximated by simpler one dimen-
sional integral equations. It was first proposed long time ago by Kirzhnits15,16 and later developed for the dielectric
function approach to novel superconductors14. In this section the units of (cid:126) = m = Ω = 1 and physical frequency
(notMatsubara)isused. Spectralrepresentationofthedispersionlessopticalphononcontributiontoinversedielectric
constant is:
σ(k,E)= (cid:15) λkδ(cid:0)1−E2(cid:1). (51)
e2