Specific heat of MgB$_2$ in a one- and a two-band model from first-principles calculations

9 Pages·0.21 MB·English

by  A. A. Golubov

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Specific heat of MgB in a one- and a two-band model from
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n first-principles calculations
a
J

1
3 A AGolubov , J Kortus ,O VDolgov ,O Jepsen , Y Kong ,O K
† ‡ ‡ ‡ ‡
Andersen , B JGibson , K Ahn and R KKremer
]
n ‡ ‡ ‡ ‡
UniversityofTwente,DepartmentofAppliedPhysics,7500AEEnschede,TheNetherlands
o
†
c Max-Planck-Institutfu¨rFestko¨rperforschung,Heisenbergstr.1,D-70569Stuttgart,Germany
- ‡
r E-mail:[email protected]
p
u
s
Abstract. The heat capacity anomaly at the transition to superconductivityof the layered
.
t
a superconductorMgB2iscomparedtofirst-principlescalculationswiththeCoulombrepulsion,
m µ∗, as the only parameter which is fixed to give the measured T . We solve the Eliashberg
c
- equationsforbothanisotropicone-bandandatwo-bandmodelwithdifferentsuperconducting
d
gaps on the π-band and σ-band Fermi surfaces. The agreement with experiments is
n
o considerablybetterforthetwo-bandmodelthanfortheone-bandmodel.
c
[

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2 Submittedto: J.Phys.: Condens. Matter
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PACSnumbers:74.25B,74.80Dm,74.72.-h
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Specificheatof MgB fromfirst-principlescalculations 2
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1. Introduction
The nature of the superconducting state in MgB has been characterized by a broad range of
2
experimental and theoretical methods and many basic properties have been unambiguously
established since the discovery of the 40K superconductor MgB by Nagamatsu and
2
collaborators[1].
While electron-phonon coupling as the underlying pairing mechanism has been
pinpointed by a large B isotope effect on T proving B related vibrations to be essential
c
[2, 3] the nature of the order-parameter (viz. the superconducting gap) has remained a
matter of debate. The order-parameter has been intensively investigated by tunnelling and
point contact spectroscopy [4–16] as well as by high-resolution photoelectron spectroscopy
[17,18]. While these techniques show an energy gap in the quasiparticle spectrum most
likely of s-wave symmetry the magnitude of the gap, ∆(0), itself remained an open
question: tunnelling experiments initially revealed a distribution of energy gaps with lower
boundary 2∆ (0)/k T 1.1 and upper boundary 2∆ (0)/k T 4.5. These values
1 B c 2 B c
≈ ≈
are either considerably lower or distinctly larger than the weak coupling BCS value of
2∆(0)/k T =3.53 and these controversial findings have been discussed in terms of gap
B c
anisotropy or more recently attributed to the presence of two gaps or multiple gaps [12,18].
The analysis of the electronic Raman continuum of MgB by Chen et al. [13] also pointed
2
to the presence of two gaps with gap values within the limits indicated by the tunnelling
experiments[19].
While these experiments employ surface sensitive techniques to determine the gap
properties, evidencefor multigapbehaviouremerges also from methodslike heat capacity or
µSR measurementsprobing true bulk properties [20]. In the early heat capacity experiments
the typical jump-like anomaly is seen at T the magnitude ∆C /T of which amounts at best
c p c
to only about 70-80% of the value 1.43γN(T ) predicted by weak-coupling BCS theory
c
[2,21–25]. γN(T ) is the Sommerfeld constant in the normal state which was obtained from
c
heat capacity measurements in high magnetic fields and which was determined to be 2.7-
3 mJ mol−1K−2. The shape of the heat capacity anomaly compares reasonably well with
BCS-type behaviour assuming 2∆(0)/k T =3.53 with appropriately adjusted magnitude.
B c
An improved fit of the detailed temperature dependence of the heat capacity anomaly was
obtained when calculating the heat capacity within the α-model [26] assuming a BCS
temperature dependence of the gap but with an increased ratio 2∆(0)/k T =4.2(2) [21].
B c
This result matches very well with the upper limit of the gap value consistently found in
the tunnellingexperiments and was suggested as an evidence that MgB is in the moderately
2
strong coupling limit. More recently, the excess heat capacity observed close to T /4 by
c
Bouquet et al. [22] and Wang et al . [24] has been attributed to a second smaller gap. Fits
with a phenomenological two-gap model assuming that the heat capacity of MgB can be
2
composed as a sum of the two individual heat capacities gave very good description with
gapvaluesof2∆ (0)/k T =1.2(1)and2∆ (0)/k T 4[27]. Recentmuon-spin-relaxation
1 B c 2 B c
≈
measurementsofthemagneticpenetrationdepthareconsistentwithatwo-gapmodel[28].
Theoretically multigap superconductivity in MgB was first proposed by Shulga et al.
2
Specificheatof MgB fromfirst-principlescalculations 3
2
to explain the behaviour of the upper critical magnetic field [29]. Based on the electronic
structure the existence of multiple gaps has been suggested by Liu et al. in order to explain
themagnitudeofT [30]. TheelectronicstructureofMgB containsfourFermisurfacesheets
c 2
[31]. Two of them with 2D character emerging from bonding σ bands form small cylindrical
FermisurfacesaroundΓ-A.Theothertwooriginatingfrombondingandantibondingπbands
have 3D character and form a tubular network. Liu et al. from first-principles calculations
of the electron-phonon coupling conclude that the superconducting gap is different for the
individualsheetsandtheyobtaintwodifferentorderparameters,alargeroneonthe2DFermi
surface sheets and a second gap on the 3D Fermi surfaces, the latter was estimated to be
approximatelya factorofthreereduced compared totheformer[30].
In the present paper we calculate the specific heat capacity from the spectral Eliashberg
function α2(ω)F(ω) first in the one-band model using the isotropic α2(ω)F(ω) as given by
Kong et al. [32]. Then we calculate the heat capacity in a two-band model by reducing the
16 Eliashberg functions α2(ω)F (ω) appropriate for the four Fermi surface sheets into four
ij ij
Eliashbergfunctionscorrespondingtoaneffective-two-bandmodelwithaσ-bandandπ-band
only. From the solution of the Eliashberg equations we obtain a superconducting gap ratio
∆ /∆ 2.63 ingoodagreementwiththeexperimentaldata. Thetwo-bandmodelexplains
σ π
≃
the reduced magnitude of the heat capacity anomaly at T very well and also reproduces the
c
experimentalobservedexcessheat capacity at lowtemperatures.
2. Theory
2.1. One-Band Model
First we discuss the specific heat in the isotropic single band model with a strong
(intermediate) electron-phonon interaction (EPI). In the normal state and in the adiabatic
approximation the electronic contribution to the specific heat is determined from the
Eliashbergfunctionα2(ω)F(ω)by theexpression[33]
Cel(T) = (2/3)π2N(0)k2T (1)
N B
∞
1+(6/πk T) f(ω/2πk T)α2(ω)F(ω)dω ,
B B
×(cid:20) Z (cid:21)
0
where N(0) is a bare density of states per spin at the Fermi energy. The kernel f(x) is
expressedin termsofthederivativesofthedigammafunctionψ(x)
f(x) = x 2x2Imψ′(ix) x3Reψ′′(ix). (2)
− − −
At low temperatures the specific heat has the well known asymptotic form: Cel(T
N →
0) = (1 + λ)γ T, where λ = 2 ∞dωω−1α2(ω)F(ω) is the electron-phonon coupling
0 0
constant, and γ = 2π2k2N(0)/3 isRthespecific heat coefficient for noninteracting electrons.
0 B
Athighertemperaturesthespecificheatdiffersfromthistrivialexpression(see,thediscussion
inreference [34]).
In the superconducting state an expression for the specific heat obtained by Bardeen
and Stephen [35] which is based on an approximate sum rule has often been used. We shall
Specificheatof MgB fromfirst-principlescalculations 4
2
however use an exact expression for the thermodynamical potential in the electron-phonon
systemwhich is based on theintegrationoftheelectronic Green’s function overthecoupling
constant
1
dx
Ω = Ω(0) +Ω(0) +T tr Σˆ(x)Gˆ(x) (3)
el ph Xωn Z0 x h i
where x is dimensionless, Gˆ(x) and Σˆ(x) are the exact electron Green’s function and the
self-energy, respectively, for a coupling constant of x λ. The functions Ω(0) and Ω(0)
· el ph
are the thermodynamic potentials for noninteracting electrons and noninteracting phonons,
respectively. Some further arithmetics leads to the expression for the difference in free
energies, F and F , ofthesuperconductingand normalstate[36]
N S
FN −FS = ωc  |ωn|(ZN(ωn)−1)− |ω2nω|n2+[(√ZωSn2(ω(Zn)S)(2ω−n1))]+2+ϕϕ2n2n , (4)
− πN(0)T nX=−ωc +ωn2Z√S(ωωn2n()Z(ZS(Sω(nω)n))2−+1ϕ)2n+ϕ2n 
 
 
where Z(ω ) is a normalization factor, ϕ = ∆ /Z(ω ) is an order parameter, and ∆ is the
n n n n n
gapfunction.
Thespecificheat at temperature, T,iscalculated according to:
∆C (T) = T∂2(F F )/∂T2. (5)
el N S
−
The specific heat jump ∆C (T ) at T = T is determined by the coefficient β =
el c c
T ∆C (T )/2in F F = βt2, wheret = (T T)/T .
c el c N S c c
− −
2.2. Two-Band Model
We have calculated the 16 Eliashberg functions α2(ω)F (ω) where i and j label the four
ij ij
Fermi surface sheets and thereafter combined them into four corresponding to an effective
two-band model which contains only a σ- and π-band. Their respective densities of states
at the Fermi energy have values of N (0) = 0.300 states/cell eV and N (0) = 0.410
σ π
·
states/cell eV. Similar coupling constants λ ,λ ,λ , and λ which are required for a
σσ ππ σπ πσ
·
two-band model were calculated earlier in reference [30]. The procedure of reducing the
16 Eliashberg functions of the real 4 band system due to the 4 different Fermi surface sheets
to an effective two-band model with only four coupling constants λ is an approximation
ij
which is based on the similarity of the two cylindrical and the two three-dimensional sheets
of the Fermi surface requiring the same physical properties in both σ-bands or both π-bands.
Moredetailscan befoundelsewhere[37].
The four Eliashberg functions α2(ω)F (ω) for the effectivetwo-band model are shown
ij ij
in figure1. Themostsignificantcontributioncomes from thecouplingofthebondstretching
phonon modes to the σ-band. The coupling constants corresponding to the superconducting
Eliashberg functions have been calculated to be: λ = 1.017,λ = 0.448,λ = 0.213,
σσ ππ σπ
andλ = 0.155. Thesmalldifferencetothevaluesgiveninreference[30]maybeattributed
πσ
tothedifferentfirst-principles methodsused inthecalculationoftheEliashbergfunctions.
Specificheatof MgB fromfirst-principlescalculations 5
2

2.0
lss=1.017,
lpp=0.448,
1.5 lsp=0.213,
lps=0.155
w) ss
F( 1.0 pp
2 a sp
ps
isotropic
0.5
0.0
0 200 400 600 800
-1
w (cm )
Figure 1. The four superconducting Eliashberg functions α2F(ω) obtained from first-
principlescalculationsfortheeffectivetwo-bandmodelandtheisotropicEliashbergfunction
fortheone-bandmodel.Thecouplingconstantoftheisotropicone-bandmodelhasavalueof
λ =0.87.
iso
Besides the spectral functions we need to know the the Coulomb matrix element
µ . With the help of the wavefunctions from our first-principles calculations we can
ij
approximately calculate the ratios for the µ-matrix [37]. The σσ-, ππ- and σπ-values were
in the ratio 2.23/2.48/1. This allows one to express µ∗ (ω ) by these ratios and one single
ij c
free parameter which is fixed to get the experimental T of 39.4 K from the solution of
c
the Eliashberg equations. The µ∗(ω ) matrix elements determined by this procedure are
c
µ∗ (ω )=0.210,µ∗ (ω )=0.095,µ∗ (ω )=0.069,andµ∗ (ω )=0.172.
σσ c σπ c πσ c ππ c
Using our calculated Eliashberg functions on the imaginary (Matsubara) axis together
with the above matrix µ∗(ω ) we obtain the gap values ∆ = lim ∆ (iπT) 7.1meV, and
ij c σ T→0 σ ≃
∆ 2.7meV,whichcorrespondsto2∆ /k T =4.18and2∆ /k T =1.59. Thetemperature
π σ B c π B c
≃
dependence of the superconducting gaps is shown in figure 2. The filled circles (squares)
display the gap for the 2D σ- (3D π-) band. Due to interband coupling between the bands
both gaps close at the same critical temperature. For a comparison also the BCS curve (line)
is shown for a single gap (one-band model) which closes at T =39.4 K. The corresponding
c
singleBCS gapwouldbe6 meV.
Theextensionofequation1 tothetwo-bandmodelgives
2π2
CN(T) = N (0)k2T +4πk [N (0)(I +I )+N (0)(I +I )](6)
el 3 tot B B σ σσ σπ π ππ πσ
whereI = ∞f(ω/2πk T)α2(ω)F (ω)dω (i,j = π,σ),andthefunctionf(x)isgivenby
ij 0 B ij ij
equation2. R
The generalization of the superconducting free energy (4) to the two band model is
straightforwardand theheatcapacity was obtainedaccording toequation5.
Specificheatof MgB fromfirst-principlescalculations 6
2

8 D s
D p
D
BCS
6
V)
e
m
D ( 4
2
0
0 10 20 30 40
T (K)
Figure 2. The temperature dependence of the the superconducting gaps from the solution
of the two-band Eliashberg equations. The values of the gaps at T=0 K were obtained
as ∆ (T =0) = 7.1 meV and ∆ (T =0) = 2.7 meV. The BCS value for the gap that
σ π
correspondstoT =39.4Kis6.0meVat0K.
c
3. ComparisonwithExperiment
For the comparison with experiment we have selected data obtained by our group [21] and
by Bouquet et al . [22]. The anomaly clearly visibleat T in the zero-field data is suppressed
c
by a magnetic field of 9 Tesla in both experiments. In figure 3 we display the difference
∆C = C (0Tesla) C (9Tesla). The anomalies at T detected by both groups clearly have
p p p c
−
a different magnitude, the one described in reference [22] amounts to 133 mJ/mol K at T
c
and represents the largest specific heat capacity anomaly reported for MgB so far [38]. The
2
∆C (T ) reported by our group is somewhat smaller, however, the shape of the anomalies
p c
close to T is very similar for both samples. In fact, fitting the anomalies with the α-model
c
revealedan identicalratio 2∆/k T =4.2 with∆=7 meVforboth samples[21,22,27].
B c
First we will try to discuss the experimental results in terms of a conventional one-band
model. The specific heat in MgB was calculated using the isotropic spectral Eliashberg
2
function α2(ω)F(ω) of Kong et al. [32]. This function yields an electron-phonon coupling
constant λ = 0.87 and together with a Coulomb pseudopotential of µ∗ = 0.1 yields T = 40
c
K. The calculated specific heat at T is γN(T ) = 1.94γ = 3.24 mJ/molK2 with γ =1.67
c c 0 0
mJ/molK2 from the band structure calculations of reference [31,32]. The specific heat jump
at T equals ∆C 196 mJ/mol K, which is a factor of 1.5-2 larger than the experimental
c
≃
values [38]. It corresponds to ∆C/(γN(T )T ) 1.51 compared to the BCS value of
c c
≃
1.43. The difference ∆C (T) = CS(T) CN(T) is shown in figure 3 (dashed-dot line)
el el − el
in comparison with the experimental data. Not only the size of the jump disagrees with the
experiment, but also the behaviour at low temperatures is different. The latter is connected
Specificheatof MgB fromfirst-principlescalculations 7
2

200
one-band model
150 two-band model
)
K
ol
m 100
J/
m
(
p 50
C
D
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
T/Tc
Figure3.Experimentaldataoftheheatcapacitydifference∆C =C (0Tesla) C (9Tesla)
p p p
−
fromreference[21]( )andfromreference[22]( ). Thedashedlineisthetheoreticalresult
◦ △
of the one-bandmodeland the thick solid line correspondsto the two-band modelfrom the
solutionoftheEliashbergequations.Thetwo-bandmodelreproducesmuchbetterthespecific
heatjumpaswellasthelowtemperaturebehaviour.
withthefactthatatlowtemperatureequation(4)forasinglebandmodelleadstothestandard
exponential dependence CS T−3/2exp( ∆/T), while the experimental data show a more
∼ −
complicated behaviour. Clearly there exists a discrepancy between experimental data and a
theoreticalone-band model.
The solid line in figure 3 represents the theoretical results for the two-band model as
described above. The low temperature behaviour is in much better agreement with the
experiment. The specific heat jump is now significantly reduced in comparison with a single
band model and reproduces surprisingly well the experimental data of reference [22]. With
thedatagivenabovewe obtainfrom ourtheoretical calculation an electronicheat capacity in
the normal state of γN(T ) = CN(T )/T 3.24 mJ/mol K2, the same value as for the one-
c el c c ≃
band model. The absolute value of the specific heat jump in the two-band model is ∆C
≃
125 mJ/molK, correspondingto∆C/(γN(T )T ) 0.98which is nowsmallerthan theBCS
c c
≃
value.
Wewouldliketoemphasizeherethatnofittingisinvolvedinthetheoreticalcalculations.
The only free parameter which is in the Coulomb matrix elements is already determined by
theexperimentalT of39.4K.
c
Onecould expect that thedifference between thetheoretical results oftheeffectivetwo-
bandmodelandourexperimentaldatamaybeattributabletoadifferentamountofimpurities
in the samples compared to the samples of reference [22]. In the one-band model the critical
temperature T as well as the value and the temperature dependence of ∆C (T) are not
c p
affected by non-magnetic impurities (Anderson theorem). This is in complete contrast to the
Specificheatof MgB fromfirst-principlescalculations 8
2
situation for the two-band model, where both quantities are strongly dependent on interband
impurity scattering. Interband impurity scattering leads to averaging of the gaps and thus to
theincreaseof∆C /γN(T )T ratio. Ontheotherhand,duetodecreaseofT ,thespecificheat
p c c c
jump only depends weakly on the scattering strength. In order to investigatethe dependence
of T and of ∆C on the interband impurity scattering we included the effect of interband
c p
impurities in the Eliashberg equations. The results show that even for rather strong impurity
scattering1/2τ = 3πT 371 K, whichleads to adrasticchangeofthecriticaltemperature
c0
≃
(decreasing to T =29.4 K) and strong averaging of the gaps, the specific heat jump remains
c
practically unchanged ∆C 120 mJ/mol K. This corresponds to a ratio ∆C /γ(T )T
p p c c
≃ ≃
1.48, which is very close to the corresponding value of a single gap model. Therefore,
interband impurity scattering can explain the change of T in different samples, but is not
c
responsiblefortheobserveddifferent valuesofthespecific heat capacityanomalyat T .
c
We have shown that a complete theoretical calculation from first-principles using an
effective two-band model can explain the major features in the specific heat measurement
of MgB surprisingly well. The presented theoretical framework goes beyond a simple
2
phenomenological two-gap model because interband effects are included explicitely and
no fitting to experimental results has been performed. The reduced value of the heat
capacity anomaly at T as well as the low temperature behaviour are in excellent agreement
c
with experimental results. The same first-principles approach using exactly the same
Eliashberg functions and Coulomb matrix elements has been used in order to explain optical
measurements[37]and tunnellingexperiments[40]oftheinterestingsuperconductorMgB .
2
Acknowledgments
JK wouldliketothank theSchloeßmann Foundationforfinancial support.
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[38] It is interesting that experiments on different samples show similar values for the specific heat jump
∆C 113mJ/molK [2], ∆C 81mJ/molK [24], ∆C 133mJ/molK [22], ∆C 125mJ/mol
≃ ≃ ≃ ≃
K[23],∆C 115mJ/molK[25],∆C 92mJ/molK[39]
≃ ≃
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