Ginzburg-Landau theory and effects of pressure on a two-band superconductor : application to MgB2

6 Pages·0.14 MB·English

by  J. J. Betouras

Ginzburg-Landau theory and effects of pressure on a two-band superconductor :
application to MgB
2
∗
J. J. Betouras, V. A. Ivanov , F. M. Peeters
Department of Physics, University of Antwerpen (UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium
3 (February 2, 2008)
0
Wepresentamodelofpressureeffectsofatwo-bandsuperconductorbasedonaGinzburg-Landau
0
free energy with two order parameters. The parameters of the theory are pressure as well as tem-
2
perature dependent. New pressure effects emerge as a result of the competition between the two
n
bands. The theory then is applied to MgB2. We identify two possible scenaria regarding the fate
a
of the two σ subbands under pressure, depending on whether or not both subbands are above the
J
Fermienergyatambientpressure. ThesplittingofthetwosubbandsisprobablycausedbytheE2g
4
distortion. IfonlyonesubbandisabovetheFermienergyatambientpressure(scenarioI),applica-
2
tionofpressurediminishesthesplittinganditispossiblethatthelowersubbandparticipatesinthe

superconductivity. The corresponding crossover pressure and Gru¨neisen parameter are estimated.
]
n In the second scenario both bands start above the Fermi energy and they move below it, either
o by pressure or via the substitution of Mg by Al. In both scenaria, the possibility of electronical
c topological transition is emphasized. Experimental signatures of both scenaria are presented and
-
r existing experiments are discussed in thelight of thedifferent physical pictures.
p
u PACS numbers: 74.20.De, 74.25.Dw, 74.62.Fj, 74.70.Ad
s
.
t
a
I. INTRODUCTION lifting its two-fold degeneracy at the Γ-point of the BZ.
m
In general,measurementsof the effect of pressureon the
-
d electronic structure are based on the Raman technique,
n The discovery of superconductivity in the material therefore only the influence on the E2g mode can be
o MgB2 [1] initiated intensive recent theoretical and ex- traced. Highpressureexperimentsupto15GPa[6]have
[c perimental interest. The possibility of a high critical revealeda large increase of the out of phase E2g phonon
temperature in a class of materials which are chemi- mode. This leads to a suppression of the displacement

1 cally much simpler than the high-T cuprates and the of the boron atoms as well as of the deformation poten-
c
v occurenceoflargecriticalcurrentdensitiesposesomein- tial ∆ = B2u2+B4u4 where u is the displacement [10]
1
teresting new questions in the research of superconduc- which will be used below. Moreover, experiments under
9
4 tivity. MgB2isatypeIIsuperconductorinthecleanlimit pressure up to 40 GPa do not show any structural phase
1 [2]. The crystalbelongs to the space groupP6/mmmor transition [6,11–13]. Due to the anisotropy the hydro-
0 AlB2-structure where borons are packed in honeycomb static pressure decreases the inter-plane distance more
3 layers alternating with hexagonal layers of magnesium than the in-plane distance between adjacent borons.
0 ions. The ions Mg2+ are positioned above the centers of The interpretationofexperimentaldatafromdifferent
/
t hexagons formed by boron sites and donate their elec- spectroscopic methods [14–16], suggests the presence of
a
m tronstothe boronplanes. Theelectronicstructureis or- two different superconducting gaps. The specific heat
ganized by the narrow energy bands with near two-fold behavior [17], the low isotope effect [18], pressure effects
-
d degenerate(σ-electrons)andthewide-band(π-electrons) [19] and penetration depth data [20] provide evidence
n [3–5]. Without any of the lattice strain,the σ dispersion for a complicated superconducting order parameter in
co relationsareslightlysplittedduetothetwoboronatoms MgB2. Recent NMR measurements [21] of 11B as well
: per unit cell (the electronic analog of Davydov splitting as Hall measurements [22] are consistent with a σ-band
v
i for phonons). The σ portions of the Fermi surface (FS) drivensuperconductivityinMgB2 wheretheπ bandpar-
X consistofcoaxialcylinders alongthe Γ - Asymmetrydi- ticipates due to interband scattering. The use of two or-
r rection of the Brillouin zone (BZ), whereas the π- bands derparametersisjustifiedbyrecentexperimentalresults
a are strongly dispersive. on scanning tunneling microscopy [23].
The excited phonon modes in MgB2 present a sharp Some microscopic theories [9,24] are based on σ band
cut-off at about 100 meV. The optical modes (B1g, E2g, or σ π band scenaria for superconductivity whereas
−
A2u and E1u) are practically non-dispersive, along the others [25] are concentrated on a π band superconduc-
Γ A direction. Various Ramandata [6–8]show a small tivity. A consensus is formed though, that the driving
−
spread of the E2g frequency around 74.5 meV in differ- band for the superconductivity is the σ one. The cal-
ent ceramic samples of MgB2. The strong deformation culated spectral functions [26] and the analysis of the
potential ∆ of the in-planeE2g mode [9,10]causes a sig- reflectancemeasurements[27]showthepossibilityofdif-
nificant energy splitting of the σ band around 1.5 eV, ferent superconducting mechanisms beyond the conven-
1
tionalelectron-phononBardeen-Cooper-Schrieffer(BCS) are explained using a different approach and the effects
pairing. The complicated nature of superconductivity of pressure are discussed in more detail and compared
in MgB2 does not allow at present to accept the con- with very recent experiments. The work is organized as
ventional superconducting mechanism. Therefore, it is follows. In section II the GL description is presented, in
appropriate to construct a phenomenologicaltheory. Section III the pressure effects, in connection with the σ
Since MgB2 presents anexample of a two-bandsuper- band, are discussed and a general discussion with some
conductor [28] may serve as a paradigm to investigate concluding remarks are given in Section IV.
new effects. The purpose of this article is to present
a phenomenological Ginzburg-Landau (GL) theory for a
two-bandsuperconductorappropriateto MgB2 basedon II. GINZBURG-LANDAU DESCRIPTION
twoorderparametersand,inthatframework,to provide
aphysicalpictureofpressureeffectsonMgB2. Thestudy
of pressureeffects is one way to investigate(i) the differ- We introduce the GL free energy functional with two
ent topology of the two bands and (ii) the question of order parameters, appropriate for MgB2, assuming that
the participationofthe twoσ subbands in the supercon- both the order parameters belong to the Γ1 representa-
ductivity. Abriefaccountofa partofthis workwaspre- tion of the point group of MgB2 crystal. The two order
sentedelsewhere[29]. Inthepresentarticlemissingterms parameters are labeled by the corresponding band (σ or
in the GL functional are restored, non-adiabatic effects π) of MgB2, without loss of generality :
1 1
F = d3r Π~ψ 2+α ψ 2+β ψ 4+ Π~ψ 2+α ψ 2+β ψ 4+r(ψ∗ψ +ψ ψ∗)
Z {2m | σ| σ| σ| σ| σ| 2m | π| π| π| π| π| σ π σ π
σ π
1
+γ1(ΠxψσΠ∗xψπ∗ +ΠyψσΠ∗yψπ∗ +c.c.)+β|ψσ|2|ψπ|2 + 8π(∇~ ×A~)2}, (1)
where Π~ = i~~ 2e/cA~ is the momentum operator,A~ GLfunctional,Eq. (1),thetermwhichcouplestheorder
is the vecto−r po∇te−ntial and α = α0 (T T0 ). It is parameters, in second order, with the strain tensor ǫ to
σ,π σ,π − cσ,π
possible to take into account the large anisotropy of the first order, having already specified the symmetry of the
two order parameters (the σ is almost two-dimensional) order parameters :
by rescaling of the axis according to the effective masses
(directional dependence effective mass). In the present Fstrain = C1(Γ1)[δ(ǫxx+ǫyy)+ǫzz]ψσ 2
work, since we are focusing exclusively on pressure ef- −C2(Γ1)(ǫxx+ǫyy+ǫzz)ψπ| 2 |
fects, the derivative terms are not important. − | |
′ ∗
−C3(Γ1)[δ (ǫxx+ǫyy)+ǫzz](ψσψπ +c.c.), (2)
In Eq. (1) we have used the fact that the mixing r-
termfavorsthecouplingoflinearcombinationofthetwo ′
order parameters with phase difference 0 if r <0 or π if where C1,2,3(Γ1) are coupling constants and δ, δ are
r > 0 between them [30], therefore a term ψ2∗ψ2 +c.c. given in terms of the elastic constants of the material.
σ π This will lead to a change of the bare critical tempera-
is incorporated into the β-term of the free energy [31].
tures T0 and consequently of the actual critical tem-
If r = 0 then Eq. (1) is the free energy for two bands cσ,π
perature T . The physical reasoning behind this is that
without Josephson coupling between them. The quartic c
the material will change in such a way as to gain con-
β-term is the only one which mixes the two gaps and
densation energy by enhancing the density of states in
is unimportant in this case. The onset of the supercon-
the direction where the superconducting gap is larger.
ducting state in one band does not imply the onset in ∗
The term proportional to ψ ψ +c.c. in the F will
the other. This correspondsto the case ofV =0 in the σ π strain
sd renormalize the mixing coupling r to r˜. We model the
two-bandBCStreatment[32]. Ifr =0,whichis thecase
6 corresponding differences by taking a linear dependence
ofMgB2, then the pair transferingtermis presentin the ofthe two bare T ’s with pressureT =T0 η p ,
GL functional and it means that the onset of supercon- where η = ∂Tc0 (p = 0)/∂p andcσ,wπe wilclσd,πis−cusσs,πthe
ductivity in one band implies automatically the appear- σ,π | cσ,π |
validity of the assumption for the MgB2 later.
ance of superconductivity in the other. There is a single
observed T which is a function of the bare ones T0 . Analyzing the equations which result from the mini-
c cσ,π
mization of the GL free energy we get :
In MgB2 the compression due to pressure is anisotropic
[6,33,34]. According to Ref. [6] the compressibility along
the c-axis is almost twice larger than that of the plane απασ =r˜2. (3)
compressibility. Therefore application of uniaxial pres-
sureonsinglecrystalswillaffectdifferentlythetwogaps. This gives the pressure dependence of the superconduct-
FollowingOzaki’sformulation[35,36],wemayaddtothe ing critical temperature:
2
1
T (p)= T0 +T0 (η +η )p the modification due to the particular physical situation
c 2(cid:2) cσ cπ− π σ (cid:3) by writing:
1
+ [T0 T0 +(η η )p]2+a2 1/2, (4)
2{ cσ− cπ π − σ } α =α0(T T0 +η p)+α1(p p )Θ(p p ),
σ σ − cσ σ σ − c − c
where a2 = 4r˜2/(α0α0). Deviations from a straight (5)
π σ
line at moderate values of pressure can be attributed to
whereΘ(x)istheHeavisidestepfunction,α0 andα1 are
the two bands. From the above formula the inequality σ σ
positive constants. This choice reflects the fact that at
dT /dp < 0 is always true as long as the renormalized
c
a certain crossoverpressure p the second band starts to
mixing parameter r˜ is real, as a consequence of the ini- c
participateinsuperconductivityaswellandthatinitially
tial assumption on the pressure dependence of the bare
it is an increasing function of pressure. We obtained the
critical temperatures. In the case of MgB2 we can safely
above formula by requiring the continuity of the coeffi-
consider that η >η .
π σ
cient α at p .
σ c
After the energy difference between the two σ bands
III. NONADIABATIC EFFECTS almost disappears, then they both follow the same re-
duction ofthe bareT ’s with pressure. The approximate
c
We now wish to address an issue specific to MgB2 value of the crossover pressure can be estimated as fol-
which can be also realized in other multiband supercon- lows. Theenergydifferencebetweenthe twosubbandsis
ductors. Thephysicalsituationwewouldliketoquestion approximated by δE (1 n)√∆, where the fraction
≃ −
isthesplittingofthetwoσsubbandsatambientpressure of superconducting electrons is 1 n 0.03 [3], i.e. the
− ∼
and their participationin superconductivity with the in- carrier density per boron atom in MgB2. The approxi-
creaseof pressure. Due to contradictoryexperimentalas mate value of pc is half the value of the pressure which
well as theoretical results we proceed by distinguishing suppresses the deformation potential ∆. The deforma-
two different scenaria. The first one addresses the split- tionpotentialinturn,canbeestimatedbytheexpression
ting of the two sigma subbands at ambient pressure as pcΩ (1 n)√∆ where Ω 30˚A3 is a unit cell volume
shownbyfirstprinciblecalculations[9,10,37]whenoneof of M∼gB2 −and ∆ = 0.04eV2∼is the deformation potential
the subbands is below the Fermienergy E . The second for a boron displacement u 0.03˚A [10]. Using these
F ∼
scenarioaddressesthe opposite situation where both the parameters, we get a crossover pressure of pc 15GPa.
∼
σ subbands are above E at ambient pressure. This estimate shows,that a realistic applied pressurein-
F
fluences drastically the electronic structure and the FS
topology, restoring the degeneracy of the two subbands
A. Scenario I : one σ sub-band above E at ambient at the Γ-point which are initially splitted. Superconduc-
F
pressure. tivity in the second subband may occur at lower values
of the estimated p due to the fact that both subbands
c
In particular in [37] the splitting of the two subbands are affected. The elimination of the energy difference
wasstudiedindetailandcomparedwithsimilarsituation around the Fermi energy will also occur at lower values
in AlB2. The conclusion is that due to the E2g phonon duetothecorrugationoftheσ portionsoftheFS,theal-
mode,thetwoσbandssplitnearbyΓpointandthelower readyexistingstrainsinthematerialandtheanisotropic
band completely sinks below the Fermi energy. This is compressibility. Theseconsiderationsmaketheabovees-
the case for MgB2 and also for the heavily hole-doped timate an upper limit of the crossover pressure pc. The
graphite. Moreover, experimental results [33,38] showed degreeto whichshearstressesofthe sampleandthe sur-
a kink in the superconducting critical temperature as a rounding fluid under pressure affect the data of pressure
function of pressure at approximately 6-8 GPa and also measurements is still under investigation [19].
a kink in the volume dependence of Tc for Mg10B2 at The microscopic nature of this crossover is caused by
around 20 GPa and Mg11B2 at around 15 GPa [7]. the change of the FS. If the topology of the FS changes
The physics we discuss here, is the change in the elec- under pressure (this subsection) or due to substitutions
tronic properties of the material under pressure and its in the composition (next subsection), a van Hove sin-
influence on the superconducting state. The band which gularity at critical energy E in the electronic density
c
is below the Fermi level at ambient pressure [9] is possi- of states ρ(E) is manifested in an electronic topological
ble to overcome the energy difference and to get above transitionofthe5/2kind[40]. SincedT /dpisessentially
c
the Fermi level at a certain value of pressure (termed as proportional to the derivative dρ(E )/dE , it develops
F F
crossover pressure), restoring the degeneracy of the two singularities 1/√E E , provided that the strength
F c
∝ −
σ bands at point Γ. This is a non-adiabatic effect as dis- of carrier attraction varies slightly with pressure [41].
cussedin[37]. Tounderstandandillustratethiseffecton In Fig. 1 using a model calculation, we illustrate the
the superconducting state we need to consider the pres- expectedbehaviorofT andtheformoftheorderparam-
c
sure dependence of the coefficient α as usual [39] with eter as a function of pressure at fixed temperature and,
σ
3
schematically,thekinkofT atp duetothechangeofthe substitution of Mg by the higher valent Al was used to
c c
slopefromEq. (5). Thepredictedkinkclosetop canbe providethenecessarychangeoftheFermienergyviaelec-
c
detected directly in a penetration depth experiment un- tronconcentrationandthelatticeparametersinthecom-
der pressure. The chosen parameters are such that they pound Mg1−xAlxB2. This is in essence equivalent to a
respect the relation η >η , the fact that the supercon- change of Fermi surface topology as in the pressure ex-
π σ
ducting density coming from the σ band is higher than periments. As it was shown in [44] by increasing the Al
the one coming from the π band (α >α ) andthat the content(equivalentlybyincreasingthepressure),atopo-
σ π
driving band for superconductivity is σ (T0 >T0). logical crossover occurs at some point and the negative
σ π
The contribution on the Gru¨neisen parameter : slopeofT asafunctionofAlcontentxdecreasesfurther
c
giving a different picture from that of Fig. 1.
dlnω Bdω
G=B = , (6) TomodelthisinthelanguageoftheGLmodel,oneba-
dp ω dp
sicallyhas to substitute the pressurepand the crossover
where B is the bulk modulus and ω is the phonon fre- pressure p with the Al concentration x and and the
c
quency, can be approximately obtained if we observe crossover concentration x respectively, requiring again
c
that δE ~ωE2g then G B/2pc. The bulk modu- continuity at xc:
≃ ≃
lus is measured to be B 114 GPa [42] and our estimate
≃
for the Gru¨neisen parameter is G 3.8. Experimentally α =α0 (T T0 +η x)+α1 (x x )Θ(x x )
G=2.9 0.3 as reported in [6] for≃the measured Raman σ σ,x − cσ σ,x σ,x − c − c(7)
±
active E2g phonon mode. However, as indicated in Ref.
[6], foranisotropiccrystalsit wouldbe moreappropriate
where the extra substrcipt x denotes that the numerical
to scale the frequency shift in the Gru¨neisen parame-
coefficients are different from Eq. (5). The critical tem-
ter with the variation of the lattice constant a such as
perature as a function of the Al content is illustrated in
G = dlnω/3dlna [43]. Then, the Gru¨neisen parameter
Fig. 2. Thephysicsissimilartothetopologicaltransition
for the MgB2 takes the value 3.9 ± 0.4. There is an ex- describedinscenarioIwherenowinsteadofpressure,the
cellentagreementwiththeaboveestimateanditjustifies
changingparameteristhe composition. Itisamorecon-
our approach.
trolledway to change the topologyof the FS. In fact, an
abrupt topological change in the σ-band Fermi surface
was found at x=0.3 in Ref. [44]. When the σ bands are
filled in Mg1−xAlxB2 at x 0.6, superconductivity diss-
≈
apears. It is worth noticing that in this compound, the
impurity scattering broadens the van Hove singularities
at the saddle points E of the FS and smears the change
c
of slope of T at E . Therefore the kink of T in Fig. 2
c c c
will be less pronounced (see also the experiment in Ref.
[44]).
We emphasize here the need of de Haas-van Alphen
(dHvA) data for Mg1−xAlxB2, especially close to x=0.3.
The pressure derivative of the superconducting critical
temperatureasafunctionoftheconcentrationdT (x)/dp
c
can be also conclusive since it can clarify the correlation
between pressure, Al concentration and the kink at x .
c
FIG.1. The form of the order parameter as a function of
pressure, where the kink at pc is shown. Inset: the behavior
of the critical temperature Tc(p). The chosen parameters
are (in reduced units): T=0.8 T0cσ, T0cπ=0.7 T0cσ, ησ= 0.2
T0cσ/pc, ηπ= 0.5 T0cσ/pc,α0σ=2, α0π=1, α1σ= 0.1.
B. Scenario II : both σ sub-bands above E at
F
ambient pressure.
FIG. 2. Schematically the form of the superconducting
Some recent experimental and theoretical studies [44] critical temperature Tc(x) in the second scenario. x is the
lend support to a second scenario. More specifically, the concentration of Al in a Mg1−xAlxB2-like system.
4
IV. DISCUSSION superconductsaboveanappliedpressureof12GPawith
T 14K [49].
c
∼
One proposed experiment which can potentially show
Interestingly enough, there is an obvious discrepancy
the different role of the two bands (σ and π) is the de-
fromthetwopicturesofthe previoussection,whichcalls
tection of the Leggett mode or the internal Josephson
for more experimental data. Observation of quantum
oscillations in dHvA experiments provides the informa- current [50]. Moreover an upward curvature in Hc2 is
also known to be a signature of multicomponent super-
tion about the bands, the effective mass of the carriers
conductivity.
and the shape of the Fermi surface in MgB2. In Ref.
[45], only the smaller cylindricalFermi surface along the Insummary,wepresentananalysisofpressureeffects,
c-axis was observed (with a hole band mass -0.251 m , withintheGLtheory,ofatwo-bandsuperconductorand
e
where m is the electron mass). The same authors re- applyittoMgB2. Wemakepredictionsfornon-adiabatic
e
ported the observation of only the larger of the two σ effects anddiscussdifferent experimentsfromwhichcru-
tube-like Fermi surfaces (with a band mass -0.553 m ) cial information can be extracted on the physics of the
e
in Ref. [46]. Although the estimated parameters of the two participatingbands σ and π as wellas the moredel-
MgB2 are in agreement with the LDA band calculations icate questions on the two σ subbands.
[47]future dHvA experiments should clarifythe detailed We are grateful to Daniel Agterberg, Christos
observation of the σ portions of the FS. Also note that Panagopoulos, Robert Joynt, Manfred Sigrist, Stefan-
angle resolved photoemission spectroscopy experiments Lu¨dwig Drechsler and the anonymous referee for very
[48]detectedonebandalongthe Γ K symmetrylineof useful comments and constructive criticisms. This
−
the BZ. work was supported by the Flemish Science Foundation
There area few points in orderto discuss further with (FWO-Vl), the Inter-University Attraction Poles Pro-
respect to applicability of the theory developed here. gramme(IUAP)andtheUniversityofAntwerpen(UIA).
Early pressure experiments [13] demonstrated the over-
all decrease of T with a pressure increase which was at-
c
tributed to the loss of holes. In two of the samples of
[13]there isa lineardependence ofT onpressureandin
c
two others a weakquadratic dependence. We stress that
the samples were polycrystalline and the experiment is
∗
On leave from N. S. Kurnakov Institute of General and
effectively under hydrostatic pressure. Also the degree
InorganicChemistryoftheRussianAcademyofSciences,
of nonstoichiometry was not known. The almost linear
31 Leninskii prospekt,117907, Moscow, Russia.
dependence for a wide range of pressures, makes the GL
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