Nonconstant electronic density of states tunneling inversion for A15 superconductors: Nb3Sn

11 Pages·0.32 MB·

by  J. K. Freericks

Nonconstant electronic density of states tunneling inversion for A15 superconductors:
Nb Sn
3
J. K. Freericks, Amy Y. Liu, A. Quandt
Department of Physics, Georgetown University, Washington, DC 20057-0995, U.S.A.
2 J. Geerk
0
Forschungszentrum Karlsruhe, Institut fu¨r Festk¨orperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany
0
(February 1, 2008)
2
Were-examinethetunnelingdataon A15superconductorsbyperforming ageneralized McMillan-
n
a Rowell tunneling inversion that incorporates a nonconstant electronic density of states obtained
J from band-structure calculations. For Nb3Sn, we find that the fit to the experimental data can

4 be slightly improved by taking into account the sharp structure in the density of states, but it is
likelythatsuchananalysisaloneisnotenoughtocompletelyexplainthesuperconductingtunneling

] characteristics of this material. Nevertheless, the extracted Eliashberg function displays a number
n
of features expected to be present for thehighest quality Nb3Sn samples.
o
c
Primary: 74.70.Ad, 71.20.Be, 74.25.Jb, 74.25.Kc
-
r
p I. INTRODUCTION Alternatively, it has been suggested that the rapid
u overswinginthehigh-energyregimeinthetunnelingden-
s
. Twentyyearsago,theA15superconductorsintheA3B sity of states is related to the presence of sharp struc-
at structure, with A a transition metal and B an sp-metal, tures in the electronic density of states near the Fermi
m werethehighesttransitiontemperaturesuperconductors level.6 Band-structure calculations9 show that the elec-
- known. The first such superconductor found was V3Si, tronic density of states in these materials has peaks of
d whichwasdiscovered1inthe1950’stohaveaT ofabout width on the order of 100 meV near the Fermi energy.
c
n 17K.Intheensuingyears,otherequilibriumcompounds Such sharp structures require, at the very least, a refor-
co likeV3GaandNb3SnwerediscoveredtohaveTc’sinthe mulationoftheMigdal-Eliashberg10many-bodyanalysis
[ rNabn3gGeao,fN15b-31G8eK,.aBnd-elNemb3eSnit-wpeoroeraclosmo fpoouunndd,swliiktheNTbc’3sAals, tsotatinesc1lu1–d1e3ewffietchtisnoafnaennoenrcgoynrsatanngteeolencttrhoeniocrddeernsoiftythoef
1 high as 23 K. The structural, electronic, magnetic, elas- maximal phonon energies of the material.
v
tic, vibrational, and superconducting properties of these OthertheoriesofsuperconductivityinA15compounds
8
4 materials were widely studied.2,3 go beyond simply generalizing the standard Migdal-
0 The use of tunneling spectroscopy to probe supercon- Eliashberg theory to allow for energy dependence in the
1 ducting properties of the A15 materials was hindered by density of states. For example, Yu and Anderson14 ex-
0 a number of materials issues. The fabrication of high- amine what happens in an electron-phonon system that
2 quality tunnel junctions was difficult because the use is coupled strongly enough to have the single-electron
0 of native oxides for the tunneling barrier did not yield (polaronic) phase become unstable to bipolaronic (pre-
/
t reproducible results, and hence artificial barrier layers formedpair)phases. Suchsystemsdisplayquitedifferent
a
m had to be grown on top of the A15 superconductors.4,5 behavior, but these theories have not been developed to
As the quality of the tunneling data improved, it be- the point where direct comparison to materials-specific
d- came clear that these materials do not exhibit the sim- tunneling conductances can be made.
n ple behavior seenin conventionalstrong-couplings-wave Since the discovery of the high-Tc cuprates, work on
o superconductors like Pb, Hg, and Nb. In particular, the A15 compounds has virtually ceased - this despite
c the reduced tunneling density of states displays a rapid the many fundamental questions about these materials
v: overswing6 followed by a sharp return to zero at ener- that remain open. In this paper, we investigate whether
i gies near and above the maximum phonon energy. A high-quality tunneling data for the A15 materials can
X
revision of the McMillan-Rowell tunneling analysis7 by be understood within a conventional Migdal-Eliashberg
ar ArnoldandWolf8 allowedthis datato be fitto highpre- framework generalized to include an energy-dependent
cisionbyassumingthe presenceofanadditionalnormal- electronic density of states. We perform this analysis
metal layer (characterized by a width and a scattering and extract an experimentally fit Eliashberg function
strength) between the superconductor and the insulat- α2F(Ω), and Coulomb pseudopotential µ∗, for Nb3Sn,
ing barrier. However, to fit the most recent experimen- which has Tc = 18 K. Data from high-quality tunnel
tal data5 within this proximity-effectmodified tunneling junctions grown on this material are available.5
theory, one needs to assume that the width of the nor- InSectionII,wederivetheformalismneededtonumer-
mal region approaches zero while its scattering strength ically perform the tunneling inversion including a non-
becomes unusually large. constantelectronicdensityofstates. Thisderivationand
the computationalalgorithmaredifferentfromthosede-
1
velopedpreviouslyinthatanexactanalyticcontinuation large to perform calculations efficiently.
that properly treats the Coulomb pseudopotential and We adopt an alternative scheme here. We start by
allows the calculations to be performed relative to the first calculating the normal-state self-energy at T = 0.
normal state is employed. In Section III, we present our Since the Matsubara frequencies become a continuum at
numerical results for the tunneling inversion with both T =0,suchacalculationcanbeperformedsimplybyre-
constant and nonconstant electronic density of states. placingtheMatsubarasummationsbyintegralsalongthe
Conclusions are presented in Section IV. imaginary axis, which are computed using conventional
quadrature techniques. Next, we choose the energy cut-
offtobe thesameasthatusedinthe constantdensityof
II. FORMALISM statescalculations,namely6Ωmax. Wecalculatetheself-
energy in the normal state at T (relative to the normal
Our analysis begins with the calculation of the band state at T = 0) with the finite energy cutoff, and then
structure and electronic density of states (DOS). Cal- add the T = 0 normal-state self-energy (with no cutoff)
culations are carried out using the VASP package,15 a to the finite-temperature self-energy. Then we calculate
plane-wavebased density-functional code using ultrasoft the superconducting self-energy relative to the normal
pseudopotentials.16Theelectronicwavefunctionsareex- state atT using the same cutoff. Finally, weaddthe full
panded in plane waves up to a cutoff of 219 eV, and normal-stateself-energyatT to getthe superconducting
the electron-electron interaction is treated within the self-energy.
local density approximation (LDA) with the Ceperley- Our many-body formalism on the imaginary axis fol-
Alder exchange-correlation functional.17 The Brillouin lows most closely to Ref. 13. We use a Nambu-Gor’kov
zone is sampled on a Monkhorst-Packmesh18 of at least formalism and evaluate the self-consistent perturbation
20×20×20points. We find the optimized crystalstruc- theory (using dressed phonons) in the Hartree-Fock ap-
ture to be tetragonal, with a small distortion of the Nb proximation. (TheHartreetermprovidesjustaconstant
sublattice.19 The peaks in the density of states near the shift to the chemical potential and is ignored.) Pertur-
Fermilevelareverysensitivetothissublatticedistortion. bative calculations are performed for the normal state
In conventional Migdal-Eliashberg theory,10 the elec- and the superconducting state. We begin our analysis
tronic density of states is chosen to be a constant (with in the normal state, where the resulting self-consistent
an infinite “bandwidth”), and the energy cutoff of the equations are
theory is provided by the finite range of the Eliashberg
functionα2F(Ω),whichmeasurestheabilityofaphonon χ′m =T λm−n
of energy Ω to scatter electrons on the Fermi surface. n
Xρ(ǫ) µ −ǫ−χ′
When performing the many-body theory calculations, dǫ N n , (1)
one begins on the imaginary axis, where the Coulomb ρ(EF)ωn2Zn′2+(µN −ǫ−χ′n)2
pseudopotential has a sharp cutoff,20 and then performs R
an exact numerical analytic continuation21 to calculate
T
real-axis properties. This technique allows for a proper Z′ =1+ λ
m ω m−n
treatmentofthe“soft”cutofffortheCoulombpseudopo- m
n
tential on the real axis. In addition, because the su- dǫ ρ(ǫ)X ωnZn′ . (2)
perconducting and normal states appear more and more ρ(E )ω2Z′2+(µ −ǫ−χ′ )2
F n n N n
similar at high energies, and because of the finite fre- R
quency cutoff (chosen to be 6Ωmax here), the self-energy A prime indicates the normal-state perturbation theory,
begins to deviate from the exact result in the normal iωn = iπT(2n + 1) is the Fermionic Matsubara fre-
state as the energy increases. One can correct for this quency, χ′m = χ′(iωm) = ReΣ′(iωm) is the real part
by performingthe perturbative expansionrelativeto the of the normal-state self-energy, and Zm′ = Z′(iωm) =
normal state. In this case, one adds the exact normal- 1−ImΣ′(iωm)/ωm is the so-calledrenormalizationfunc-
state self-energy minus the normal-state self-energy cal- tion determined from the imaginary part of the normal-
culatedwith the energycutoffused inthe superconduct- state self-energy. The symbols λm−n are the electron-
ing theory.22 Such a scheme was used when examining phonon coupling strengths
effects of vertex corrections23 in Pb. When an energy-
dependentelectronicdensityofstatesisused,suchacom- λm−n =λ(iωm−iωn)
∞ 2Ω
putational scheme becomes problematic because there is = dΩα2F(Ω) , (3)
no longer an exact analytic expression for the normal- 0 Ω2+(ωm−ωn)2
Z
state self-energy. Furthermore, a choice must be made
for the energy cutoff, because, assuming a bandwidth on andλ=λ0. Thefunctionρ(ǫ)istheelectronicdensityof
states as determined by the band-structure calculation,
the order of a few eV and a temperature on the order of
and ρ(E ) is the density of states at the T =0 chemical
0.1 meV, the number of Matsubara frequencies required F
potential of the band-structure calculation (E ). The
for an energy cutoff set by the bandwidth would be too F
chemical potential in the normal state is µ . In the
N
2
limit as T → 0, the number of Matsubara frequencies ω Z′
n n
becomes an infinite continuum, and the summations can ω2Z′2+(µ −ǫ−χ′ )2
" n n N n
be replaced by integrals. We find
ω Z′(iω )
χ′0(iω)= 1 dω′λ(iω−iω′) −ωn2Z0′2(iωn)n+0(µN0n−ǫ−χ′0)2(iωn)#
2π
dǫ Zρ(ǫ) µN0−ǫ−χ′0(iω′) , (4) +[Z0′(iωm)−1]. (7)
ρ(EF)ω′2Z0′2(iω′)+[µN0−ǫ−χ′0(iω′)]2 These equations would be exact if the summations over
R
Matsubara frequencies for the T = 0 quantities were re-
Z′(iω)=1+ 1 dω′λ(iω−iω′) placed by integrals. Since the normal-state self-energy
0 2πω does not depend too strongly on T for low temperature,
dǫ ρ(ǫ) Z ω′Z0′(iω′) , (5) this approximation is accurate for low T. Note that the
ρ(EF)ω′2Z0′2(iω′)+[µN0−ǫ−χ′0(iω′)]2 chemical potential µN typically changes by about one
R meV (at T = 1.2 K) from the zero-temperature value
with the subscript zero denoting the results are at T =
µN0 (in the nonconstant density of states case).
0. Evaluatingthese integralswith quadratureroutines is
The final set of equations we need are for the super-
muchmoreefficientthancalculatingtheMatsubarasums
conducting phase. The self-consistent equations are cal-
with a large energy cutoff at finite temperature.
culated “relative to the normal state” at T:
Ourstrategyfordetermining the functionsχ′(iω)and
0
Z′(iω) is to create a nonuniformly spaced grid on the ρ(ǫ)
0
χ =T λ dǫ
imaginaryaxis(we use 336points), with anupper cutoff m m−n ρ(E )
F
many times the electronic bandwidth. The grid is con- |ωnX|≤ωc Z
structed in the following fashion. The first grid point µ −ǫ−χ
S n
is chosen to lie at iω0 = 0. Subsequent grid points are "ωn2Zn2+(µS −ǫ−χn)2+∆2nZn2
chosen by adding the new step size to the old grid point
µ −ǫ−χ′
iωj+1 = iωj +iδj, with δj increasing by a factor of 1.1 − N n +χ′ , (8)
witheachstepfromitsinitialvalueδ0 =0.09meV[hence ωn2Zn′2+[µN −ǫ−χ′n]2# m
δ =0.09×(1.1)j]. Werestrictδ <400meV—ifδ would
j j j
be largerthan400meV,wesetitequalto400meV.The
grid is reflected about iω = 0 to construct the negative T ρ(ǫ)
Z =1+ λ dǫ
m m−n
frequency axis. The integrals are then evaluated using ω ρ(E )
m F
a simple Riemann sum over the nonuniform grid. Once |ωnX|≤ωc Z
the functions χ′ and Z′ are known on the grid points, ωnZn
0 0
we linearly interpolate to evaluate them at any point on "ωn2Zn2+(µS −ǫ−χn)2+∆2nZn2
the imaginary axis.
ω Z′
We employ the T =0 normal-statesolutions as anap- − n n +[Z′ −1], (9)
ω2Z′2+[µ −ǫ−χ′ ]2 n
proximate solution for high frequency. This is included n n N n #
by first calculating the normal-state perturbation the-
and
ory at finite temperature, using the same Matsubara
frequency cutoff as used in the superconducting state
ρ(ǫ)
ωc = 6Ωmax, and then adding the difference between ∆mZm =T (λm−n−µ∗) dǫρ(E )
the T = 0 normal state solution and the finite-T solu- |ωnX|≤ωc Z F
tion to the superconducting solution, as shown below. ∆nZn
. (10)
We solve the following self-consistent equations for the ω2Z2+(µ −ǫ−χ )2+∆2Z2
n n S n n n
normal-state self-energy at temperature T:
Here ∆m = ∆(iωm) = Σ12(iωm)/Zm is the supercon-
ρ(ǫ)
χ′ =T λ dǫ ductinggapdeterminedfromtheoff-diagonalself-energy,
m |ωnX|<ωc m−nZ ρ(EF) µca∗lipsottheentCiaoluilnomthbepssuepuedrocpoontdeuncttiainl,gasntdatµe.SNisottheethchaetmwie-
µ −ǫ−χ′
N n add and subtract the normal-state results at finite tem-
"ωn2Zn′2+(µN −ǫ−χ′n)2 peratureinordertoensurethattheMatsubarafrequency
− µN0−ǫ−χ′0(iωn) sfruemqumeantciyo;nsthceonhvigehrg-fererqaupeidnlcyy(tωacils=a6reΩamlarxeaidsythineccluudtoeffd
ωn2Z0′2(iωn)+[µN0−ǫ−χ′0(iωn)]2# in the normal-state self-energy). This procedure allows
+χ′(iω ), (6)
0 m for a rapid computationof the many-body Green’s func-
tions when there is an energy-dependent electronic den-
T ρ(ǫ) sityofstates. We onlycalculatethe Green’sfunctions at
Z′ =1+ λ dǫ
m ω m−n ρ(E ) the Matsubara frequencies here.
m F
|ωnX|<ωc Z
3
The chemical potentials in the normal and supercon- Z′(ω)=1− Σ′0(ω+iη)−Σ′0∗(−ω+iη), (15)
ducting states are determined by the requirement that 0 2ω
the electron density be equal to the equilibrium density
and
of electrons for the given band. Our first step is to find
the T = 0 Fermi level for the band-structure density of Σ′(ω+iη)+Σ′∗(−ω+iη)
χ′(ω)= 0 0 , (16)
states, which satisfies 0 2
EF where the ∗ denotes complex conjugation. The term
ρe =2 dǫρ(ǫ), (11) λ(ω −iω′) is found from the spectral formula and the
Z−∞ given Eliashberg function α2F(Ω):
with the factor of 2 coming from spin. In the normal
state at T = 0, we use the T → 0 limit of the identity ∞ α2F(Ω)
λ(ω−iω′)=− dΩ . (17)
ρ = 2T G(iω ) and Eq. (11) to produce the self- ω−iω′−Ω
e n n Z−∞
consistent equation for µN0:
P Now we calculate the finite-T normal-stateresults rel-
EF 1 ative to the T =0 calculation:
2 dǫρ(ǫ)= dǫρ(ǫ) dω
π
ZµN0 Z Z Σ′(ω+iη)= T λ(ω−iω )
1 ρ(E ) n
F
"iωZ0′(iω)+µN0−ǫ+χ′0(iω) |ωnX|<ωc
ρ(ǫ)
1 × dǫ
−iω+µN0−ǫ#. (12) Z "iωnZn′ +µN −ǫ−χ′n
ρ(ǫ)
−
The normal-state chemical potential at finite T (we use iωnZ0′(iωn)+µN0−ǫ−χ′0(iωn)#
T = 1.2 K for the tunneling inversion) is found by 1 ∞
+ dΩα2F(Ω)
comparison with the normal state at T = 0: 0 = 2ρ(E )
2T n[G(iωn) − G(iωn)|T=0] (note the T = 0 sum is F Zω−∞−Ω Ω
an approximation to the continuum integral). Finally, tanh +coth dǫρ(ǫ)
2T 2T
P
for the superconducting state, we use the comparison of (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)Z
1
the normal-state filling with the superconducting-state
×
filling to find µS: "(ω−Ω)Z′(ω−Ω)+µN −ǫ−χ′(ω−Ω)+iη
1
T −
0= π dǫρ(ǫ) (ω−Ω)Z0′(ω−Ω)+µN0−ǫ−χ′0(ω−Ω)+iη#
Z |ωmX|<ωc +Σ′(ω+iη), (18)
0
µ −ǫ−χ
S m
ω2 Z2 +(µ −ǫ−χ )2+∆2 Z2 where the Z and χ functions on the real axis are deter-
" m m S m m m
mined from equations analogous to Eqs. (15) and (16).
µ −ǫ+χ′
− N m . (13) For the superconducting state, calculations are per-
ωm2 Zm′2+(µN −ǫ+χ′m)2# formed relative to the normal state at T:
Thenextstepistocalculatetheself-energyonthereal Σ11(ω+iη)=T λ(ω−iωn)
axisusinganexactanalyticcontinuationtechnique.21We
|ωnX|<ωc
begin with the normal state at T = 0. The self-energy
iω Z +µ −χ
n n S n
satisfies × dǫρ(ǫ)
ω2Z2+(µ −ǫ−χ )2+∆2Z2
Z " n n S n n n
1 ∞
Σ′0(ω+iη)= 2πρ(E ) dω′λ(ω−iω′) − iωnZn′ +µN −χ′n
F Z−∞ρ(ǫ) ωn2Zn′2+(µN −ǫ−χ′n)2#
× dǫ 1 ∞
+Z 1 iω′Zω0′(diΩω′α)2+Fµ(ΩN)0−dǫǫ−χ′0(iω′) + 2ρ(EF)Zω−∞−dΩΩα2F(Ω) ω−Ω
ρ(EF)Z0 ρZ(ǫ) tanh 2T +coth 2T dǫρ(ǫ)
× , (14) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)Z
(ω−Ω)Z0′(ω−Ω)+µN0−ǫ−χ′0(ω−Ω)+iη (ω−Ω)Z(ω−Ω)+µS −ǫ−χ(ω−Ω)
(ω−Ω)2Z2(ω−Ω)+[µ −ǫ−χ(ω−Ω)]2+∆(ω−Ω)2Z(ω−Ω)
" S
whereη →0+. Thisisaself-consistentequation,because
(ω−Ω)Z′(ω−Ω)+µ −ǫ−χ′(ω−Ω)
the second integral contains the self-energy on the real − N
axis from the definitions (ω−Ω)2Z′2(ω−Ω)+[µN −ǫ−χ′(ω−Ω)]2#
4
+Σ′(ω+iη), which is the(1s9u)perconducting gapfor Nb3Sn at low tem-
perature. If we knew the correct Eliashberg function,
for the diagonal self-energy and then steps (i)-(vii) would be all that we need to calcu-
late the RDOS from Eq. (22). But we need to find the
Σ12(ω+iη)=T [λ(ω−iωn)−µ∗] best fit α2F(Ω) that is consistent with the experimen-
|ωnX|<ωc tal RDOS. To do this, we simply employ the McMillan-
∆ Z
× dǫρ(ǫ) n n Rowell procedure for the tunneling inversion. We start
+Z 1 ωn2∞Znd2Ω+α(2µFS(Ω−)ǫ−χn)2+∆2nZn2 wtioitnhaladgeureivsastfiovreαo2fFho.w(vcihiia)nNgeexstinwethcealEculilaasthebtehrge ffuunncc--
2ρ(E ) tion affect the RDOS. The functional derivative is found
F Z−∞
ω−Ω ω−Ω byaddingasmall-weightGaussiantoα2F(Ω)centeredat
tanh +coth dǫρ(ǫ)
2T 2T agivenfrequencyΩ0,repeatingsteps(ii-vii)todetermine
×∆(cid:20)(ω−Ω(cid:18))Z(ω−(cid:19)Ω) (cid:18) (cid:19)(cid:21)Z the RDOS, and calculating the functional derivative by
/ {(ω−Ω)2Z2(ω−Ω)+[µ −ǫ−χ(ω−Ω)]2 taking the difference of the new RDOS with the RDOS
S
+∆(ω−Ω)2Z(ω−Ω)2}, (20) for the original Eliashberg function. This step (viii) is
repeatedforeachfrequencyinthe discreteα2F to deter-
for the off-diagonal self-energy. minethefunctionalderivativematrix. Thenextstep(ix)
The tunneling conductance satisfies is to choose the weights for adjusting α2F to fit the ex-
perimentalRDOSbetter. Sincethe functionalderivative
[dI/dV]S ω matrixmayberank-deficient,weuseasingular-valuede-
(ω)=Re . (21)
[dI/dV]N " ω2−∆2(ω)# compositiontodeterminetheupdatedweights. Oncethe
updated α2F(Ω) is found, (x) we apply an exploratory-
The reduced density of statesp(RDOS) is the ratio of the data-analysis robust smoother followed by a Hanning
tunnelingconductancetotheBCStunnelingconductance smoother. Steps (viii-x) are repeated until the updated
minus one, which becomes α2F(Ω) ceases to produce a RDOS that is closer to ex-
periment. WeforcetheEliashbergfunctiontobepositive
RDOS(ω)=Re ω2−∆20 −1, (22) everywhere and to be quadratic for Ω < 2.5 meV. The
" pω2−∆2(ω)# stepsizeforthediscretefrequenciesatwhichα2F iseval-
uated is 0.45 meV. The algorithm is depicted pictorially
where ∆0 is the superconpducting gap, defined as the so- in Fig. 1.
lution to the equation Re∆(ω)=ω (at ω =∆0). There is a numerical problem with this algorithm. In
To solve all of these self-consistent equations and to general, the weights for the shift in α2F are strongly os-
invert the tunneling conductance to get the Eliashberg cillatory, which forces the Eliashberg function to have
function, we begin with the LDA band-structure elec- large-amplitude narrow-width peaks. Since we do not
tronic density of states as input. We take the band to expecttheEliashbergfunctiontohavesuchsharppeaks,
consist of the states that contain nine electrons below they need to be smoothed away in the next update. Be-
the Fermi level and four electrons above, corresponding cause the functional derivatives are not calculated with
to approximately±0.8eVaboutE . Hence the electron thissmoothingprocedure(andthereisnoobviouswayto
F
filling satisfies ρ = 9. Our first step (i) is to determine incorporate the smoothing into the derivatives), we are
e
the band-structure Fermi level E at T = 0 by solv- limitedinhowcloselywecanreproducetheexperimental
F
ing Eq. (11). Then we take two initial guesses for µ . RDOS, even though we have the full freedom to adjust
N
(ii) For each µ , we self-consistently solve for Eqs. (1) α2F.
N
and (2) at T = 0. Once those functions are known, (iii)
we use a one-dimensional root-finder to solve Eq. (12)
to determine µ . The next step is to determine the su-
N
perconducting chemical potential µ and the Coulomb
S
pseudopotentialµ∗. Wefirstchooseaguessforµ∗. With
µ∗ fixed, we choose two initial guesses for µ and (iv)
S
self-consistentlysolveEqs.(8),(9),and(10). Oncethose
functions are determined, (v) we use a one-dimensional
root-finder to solve Eq. (13) and find µ . (vi) Now the
S
real-axis equations for both the normal state at T = 0,
Eqs. (15) and (16), and the normal and superconduct-
ing states at T, Eqs.(18), (19), and (20), are solved self-
consistently. Once the gap function ∆(ω) is known, we
can find the superconducting gap ∆0. (vii) Steps (iv),
(v), and (vi) are repeated for different values of µ∗ un-
til we find the particular value where ∆0 = 3.15 meV,
5
(a) energy-dependentNb3SnDOSisplottedasadottedline,
Eq. (9) and corresponds to λ = 1.42, with a maximal error of
0.004, and a root-mean-square error of 0.0015. The fit
(i) EF
with a nonconstant DOS is about four times worse than
µ* (vii)
withtheconstantDOS.Notefurther,thatthemaineffect
Eqs. (1) and (2)
of the sharp peak in the DOS is to reduce the overall
Steps (iv)
(v) and (vi) scaleofα2F fornearlyallfrequenciesexceptthehighest,
(ii) χ 0’ ( i ω) Z ’0 ( i ω) whereitisstronglyenhanced. Qualitatively,thetwoα2F
χ ’ (ω) Z ’ (ω)
0 0 curves are very similar in shape.
Eq. (10) (vi)
χ(ω) Z (ω) ∆(ω)
µ
(iii) N
Eqs. (13) (14)
Eqs. (6-8) (16) and (17)
χ ∆ µ
(iv) n Z n n S (v)
Eq. (11)
(b)
(vii-a) α2 F (Ω)+ g( Ω − Ω 0 )
(ii-vii)
α2 F (Ω) (x)
new
(vii-b) ∆(ω) µ * R D O S (ω)
compute
functional smooth
derivative
(vii-c) δδαR2 D F O(ΩS)(ω) δα2 F (Ω) (ix)
Singular Value Decomposition
FIG.2. Eliashberg functions for a tunneling inversion in
FIG.1. Block diagrams of thealgorithm employed in the lead with (solid line) constant density of states and (dotted
modifiedMcMillan-Rowelltunnelinginversionwithanoncon- lines)thenonconstantdensityofstatesofNb3Sn. Thegeneral
stant electronic density of states. In panel (a), we show the shapesofthetwocurvesaresimilar. Thevaluesofλare1.56
mainalgorithmusedtodeterminetheRDOS.Steps(i-vii)are and 1.42 for the constant and nonconstant density of states
described in detail in the text. Panel (b) shows the McMil- cases, respectively.
lan-Rowell strategy for updating the Eliashberg function in
thetunneling inversion, with details described in thetext. The quality of the tunneling data for lead is so good
that one can actually see that the tunneling inversion
works better with a constant DOS than with a noncon-
III. RESULTS
stant DOS. What is interesting is that one can get rea-
sonableresultsusingthe wrongnonconstantDOSdueto
We first consider a test case, where the experimental the freedom allowed in choosing the function α2F.
tunnelingdataforleadisusedtoperformbothaconven- We next turn to the results for the tunneling inver-
tional and a nonconstant DOS tunneling inversion. In sion of Nb3Sn. The extracted Eliashberg functions are
both cases we use the same code to perform the analysis shown in Fig. 3. The thick curve shows the noncon-
as summarized in Fig. 1. In the constant DOS case, we stant DOS results, while the thin curve is for the con-
usetheDOSforlead(2.5×10−4 statesperspinpermeV stantDOScase. Thepropertiesoftheseelectron-phonon
per unit cell) and choose a band that contains two total spectralfunctionsaresummarizedinTableI.Incontrast
electrons(includingspin)andρ =1. Inthenonconstant to the results for lead, where the Eliashberg function is
e
DOS case, we use the electronic DOS for Nb3Sn (see in- reducedin the nonconstantDOS analysis,it is enhanced
setto Fig.5). Thisis ofcourseanartificialproblemthat for Nb3Sn. The main difference, aside from an overall
is presented for illustrative purposes. scale factor, is a large enhancement for Ω > 20 meV,
The results for the best fit Eliashberg functions are particularly in the highest energies, where the constant
given in Fig. 2. The solid line results from a tunneling DOSα2F isstronglysuppressed. Notethatthemaximal
inversion done with a constant DOS, and is consistent allowedphononfrequencyistheDFT-calculated38meV
withearlieranalysesoftheleadtunnelingdata(λ=1.56, in both cases; the fitting procedure sharply suppresses
maximal error of 0.002, and a root-mean-square error theα2F athighenergyfortheconstantDOScalculation.
of 0.0004). The tunneling-inversion result based on the The general shapes of these curves are similar to those
6
found previously5 but the overallscaleis larger. The en-
hancement occurs because we are unable to reproduce
the full overswing. In order to compensate for this, λ is
increasedandthequalityofthelow-energyfitisreduced.
Note that we fit all experimental points, including those
in the overswing region (ω >35 meV)—nevertheless, we
are unable to get “good” results for the extracted α2F
in the sense that the bestfit α2F is unable to accurately
reproduce the tunneling data over the entire experimen-
tallymeasuredrange(duetotheinabilitytoproducethe
full overswing).
There are, nevertheless, a number of promising fea-
tures of this calculation. If we compare Fig. 3 to the
dataforlow-Tc(disordered)Nb3Snjunctions,wefindthe
broad low-energy peak and shoulder (present from 4 to
8meV)liesafewmeVlowerinthehigh-T materialthan
c
the low-T one5 (whereitlies at10meV).This isthe ex-
c
pected phonon softening that leads to the high T of the
c
A15 compounds that is evident also in second-derivative
data24 on Nb3Sn, where the low-Tc material has a large FIG.3. Eliashberg functions for a tunneling inversion in
peak at 10 meV which disperse to two peaks (one at 6 Nb3Snfornonconstant densityofstates (thick)andconstant
density of states (thin). In the nonconstant density of states
and one at 8 meV) in the high-T material. Another
c
analysis, thespectral weight is enhanced at high energies.
strong indication of enhanced electron-phonon coupling
is the pronounced softening of the 10 meV shoulder ob-
Nextweconsiderthereduceddensityofstatesforboth
servedin the phonon density of states of Nb3Sn on cool-
casesinFig.4. Thethickdashedcurveistheexperimen-
ing down from room temperature to 4.2 K as described
by Schweiss et al.25 Hence the extracted α2F displays tal data,5 the thin dashed line is the horizontal axis, the
thicksolidlineisthenonconstantDOScaseandthethin
the expected phonon-mode softening, with the expected
solidlineistheconstantDOSresults. Therearethreeim-
energy scales. Another test is to compare the extracted
α2F to the phonon DOS, F(Ω), measured with neutron portantpointsto noteaboutthese curves;(i)the fits are
scattering25. Inactuality,neutronscatteringweightsthe poor at the lowest energies, (ii) the nonconstant density
of states produces a more rapid “overswing”and return-
phonons by the neutron scattering cross section for each
to-zero at about 40 meV, and (iii) the amplitude of the
nucleus forming the generalized DOS G(Ω). Nb has a
overswing is enhanced in the region from 40−60 meV.
cross section approximately twice as big as Sn, but as
Asneitherofthefitsisparticularlygood,wecannotcon-
the atomic masses of Nb and Sn are similar, we expect
clude from this work that the inclusion of nonconstant
both atoms to be in motion for most phonon modes.
density of states effects alone is sufficient to explain the
Hence these modes can be excited by interactions of
A15 tunneling data. But we do see that the incorpo-
neutrons with either Nb or Sn nuclei. This averaging
ration of a nonconstant density of states definitely pro-
effect implies that the phonon density of states F(Ω)
vides improvements. It is not clear what else is needed.
and the generalized density of states G(Ω) should agree
closely25,26 for Nb3Sn. If we take the ratio of the tun- Partof the problem may be with the numericalinstabil-
ities of the tunneling-inversion algorithm. Alternatively,
neling data to the neutron data, we find a large peak in
α2(Ω) for energiesbelow 10meV. This agreeswith theo- there may be an intrinsic thin proximity layer that al-
reticalcalculations26,whichpredictalargeα2(Ω)forthe ways needs to be taken into account regardless of the
quality of the junction. Other factors that may be im-
low-energy phonon modes (below 10 meV).
portantinclude anharmonicity,anisotropy,and impurity
scattering. Itappearsunlikelythatdisorderistheexpla-
nation, since disorder tends to reduce the magnitude of
the RDOS, not increase it, as is needed. The proximity-
effect explanation is also hard to support, because the
constant DOS analysis found the proximity layer to be
vanishingly thin. If that conclusion holds true for the
nonconstantDOSanalysisaswell,thenthiswouldnotbe
aviableexplanationeither. Theeffectsofanharmonicity
atlowtemperatureshouldbeexplainablewithinaquasi-
harmonicapproximation,unlessthereisapreformed-pair
phase. Hence, we believe the most likely cause of the
discrepancy is from anisotropic effects. It is conceivable
7
that tunneling barriers grow differently on different in-
dexed surfaces. The strong dependence of the tunneling
conductanceonbarrierthickness thus couldleadto adi-
rectionalselectivityofthe tunneling currentandthereby
anisotropy effects would influence the tunneling current
eveninthecaseofpolycrystallinefilms. Incorporationof
such anisotropiceffects is beyondthe scope of this work.
FIG. 5. Density of states for Nb3Sn. The solid curve is
the band-structure density of states as calculated with den-
sity functional theory. The dashed curve is the quasiparticle
density of states calculated with the fit α2F. The chemical
potential lies at ω = 0. Inset shows the full band-structure
density of states, with the dashed box indicating the region
blownupinthemainfigure. Theunitsoftheverticalaxisfor
boththemainplotandtheinset arestatesperspinpermeV
FIG.4. Reduceddensityofstates[fromEq.(22)]fornon-
perunit cell.
constant (thick) and constant (thin) density of states. The
thick dashed line is the experimental data (scanned and dig-
InFig.6,weplotthe realandimaginarypartsofχfor
itized) and the thin dashed line is the horizontal axis. Note
(a) the normal state and (b) the superconducting state.
the poor quality of the fit at the lowest energies, and the
This function vanishes for the case of a constant density
sharper overswing at high energies for the nonconstant den-
of states. The chemical potentials are µ = 15.70 meV
sity of states case. The poor fit arises from the fact that λ N
and µ = 15.97 meV in the normal and superconduct-
mustbeincreasedinordertoproducetheoverswing,butthis S
reduces theagreement at low frequencies. The results shown ing states. The value of the real part of χ is of this
represent the best compromise for fitting the entire experi- order of magnitude. The normal and superconducting
mental spectrum. self-energies differ only at the lowest energies.
We conclude with a discussion of the properties of the
solutionstothemany-bodyproblem. Theelectronicden-
sity of states is shown in Fig. 5. The solid curve is
the band-structure density of states as calculated within
density-functionaltheory. Thedashedcurveisthequasi-
particle density of states in the normal state at T = 0
calculated with the fit α2F. It depicts the behavior ex-
pected. The density of states is unchanged at the chem-
ical potential (ω = 0) because the self-energy is mo-
mentum independent, the peak is narrowed by a factor
of about 3 (due to “1+λ” narrowing), and the overall
density of states is smoothed out due to lifetime effects.
In the inset, we show the full band-structure density of
statesusedinthecalculations. Thedashedboxindicates
the region shown in the main figure.
8
is larger for the latter. These functions vary from the
normaltothesuperconductingstateonlyatlowenergies
as expected. Note further, that the real part of Z can
dipbelow 1 for the nonconstantdensity ofstates case(it
never does for a constant density of states).
FIG.6. Self-energy function χ(ω) plotted in the (a) nor-
mal and (b) superconducting states. These functions are
nearlyidentical, exceptforsmalldifferencesatthelowest en-
ergies.
TherenormalizationfunctionZ(ω)isplottedinFig.7,
for (a) the normal phase and (b) the superconducting
FIG.7. The renormalization function for (a) the normal
phase. The solid lines are for the nonconstant density
stateand(b)thesuperconductingstate. Thesolidcurvesare
of states case and the dashed lines are for the constant
foranonconstantdensityofstatesandthedashedcurvesare
density ofstatescase. Note howinthe nonconstantden- foraconstantdensityofstates. NotethatReZ candipbelow
sity of states case, the effective strength of the electron 1 for the nonconstant density of states case.
phonon coupling, measuredby Z (0)−1 is closer to 2.1
N
than the value of λ, which is 2.7. It is the former value Finally, the superconducting gap function ∆(ω) is
that is the true measure ofthe electron-phononcoupling shownin Fig. 8. The results for the nonconstantdensity
strengthwith a nonconstantdensity of states. Note that of states (solid line) and the constant density of states
the main differences between the nonconstant and con- (dashed line) are nearly identical at both low and high
stantdensityofstatescalculationsisthattheoverallscale energies. In the range from 30 meV to 130 meV, the
9
curves deviate from each other: in the nonconstant den- dependence of the density of states is enough to com-
sity of states case, the peak (real part) and dip (imag- pletely understand the tunneling data. It may be that
inary part) form faster than in the constant density of thetunnelingismodifiedbyanisotropiceffects,byanar-
statescase. Thisiswhatproducesthesharperoverswing row proximity-coupled layer or by other effects such as
inthe RDOSforthe nonconstantdensity ofstatescalcu- anharmonicity.
lation.
ACKNOWLEDGMENTS
We would like to acknowledge stimulating discussions
with M. Beasley, J. Carbotte, B. Klein, B. Mitrovi´c, E.
Nicol, and D. Rudman. We also want to thank D. Rud-
man for sharing his data with us. This work was sup-
ported by the National Science Foundation under grant
DMR-9973225.
TABLEI. CalculatedpropertiesoftheEliashbergfunction
extracted from thetunnelinginversion.
inversion λ µ∗ ωln A Tc error error
DOS [meV] [meV] [K] max. r.m.s.
nonconstant 2.738 0.286 7.072 13.652 19 0.033 0.0048
constant 2.501 0.210 6.415 11.130 23 0.034 0.0055
FIG.8. Superconductinggap function for (solid) noncon-
stantdensityofstatesand(dashed)constantdensityofstates.
Thetwo curvesarenearly identicalat low and high energies,
but differ from about 30 meV to 130 meV. The sharp over-
swing in the experimental RDOS data is better reproduced
bythemorerapidformation ofthehigh-energypeak(dip)in
thereal(imaginary)partofthenonconstant-density-of-states
gap function.
IV. CONCLUSIONS
We have performed a modified McMillan-Rowell tun-
neling inversion including the effects of the noncon-
stant electronic density of states near the Fermi level for
Nb3Sn. The Eliashberg function that is found by this
inversionhas a larger value of λ than generally accepted
forthismaterial,buttheeffectivevalueofλderivedfrom
therenormalizationfunctionismorereasonable. Ourex-
tracted α2F(Ω) also has a number of expected features
including largepeaksatabout6and8meVrepresenting
the soft phonon modes, and a sharply peaked α2(Ω) be-
low10meVwhichagreeswiththetheoreticalpredictions.
Even taking into account the energy dependence of the
density of states, we are unable to produce excellent fits
of the tunneling data, though we can better reproduce
the overswing observed at high frequencies. We believe
the problem is partially numerical, as the tunneling in-
version tries to force sharp spikes into α2F, but the fit
is still too poor at the lowest energies and in the over-
swing region to say that properly including the energy
10