Table Of ContentTheoretical de Haas-van Alphen Data and Plasma Frequencies of MgB and TaB
2 2
S. Elgazzar1, P.M. Oppeneer, S.-L. Drechsler, R. Hayn, H. Rosner†
Institut fu¨r Festk¨orper- und Werkstofforschung, P.O. Box 270116, D-01171 Dresden, Germany
†Dept. of Physics, University of California, Davis CA 95616, USA
2
0
The de Haas-van Alphen-frequencies as well as the effective masses for a magnetic field parallel to the crystallo-
0
graphicc-axisarecalculatedwithinthelocalspindensityapproximation(LSDA)forMgB andTaB . Inaddition,
2 2 2
we analyze the plasma frequencies computed for each Fermi surface sheet. We find a large anisotropy of Fermi
n
velocities in MgB in difference to the nearly isotropic behavior in TaB . We compare calculations performed
a 2 2
within the relativistic non-full potential augmented-spherical-wave (ASW) scheme and the scalar-relativistic full
J
potential local orbital (FPLO) scheme. A significant dependence for small cross sections on the bandstructure
6
method isfound. Thecomparison with thefirst available experimentaldeHaas-vanAlphen-databyYelland et al.
1
(Ref. 19) shows deviations from the electronic structure calculated within both L(S)DA approaches although the
] cross section predicted by FPLO are closer to the experimental data. The elucidation of the relevant many-body
n
effects beyond the standard LDA is considered as a possible key problem to understand the superconductivity in
o
MgB .
c 2
-
r 71.18.+y; 74.70.Ad; 74.25.Jb
p
u
s
at. I. INTRODUCTION 12–14 and for which single crystals can be produced15.
m It should be noted that stochiometric TaB2 is non-
superconducting (at least down to 1.5 K12,16,17) and an-
- The unexpected discovery of superconductivity in
d other phase, unidentified yet, should be responsible for
MgB 1, with a surprisingly high transition tempera-
n 2 the observed superconductivity with a T of 9.5 K18.
ture T 40 K, has raised considerable interest in c
o c
∼
c the clarification of its electronic structure and its rela-
[ tionship to the mechanism of superconductivity. The
II. METHODS
present theoretical approaches to this challenging prob-
2
lem can be divided roughly into three scenarios em-
v
Very detailed insight into the electronic structure
0 phasizing: (i) a conventional electronic structure and a
can be gained from de Haas-van Alphen (dHvA) mea-
3 moreorlessstandardelectron-phononinteractionwithin
2 a (multiple) wide-band model2−7, (ii) the presence of surements. To the best of our knowledge there is as yet
1 strong electron-electron8 or magnetic interactions9, and only one very recent measurement devoted to MgB219
0 (iii) strong nonadiabatic10 or polaronic11 effects. In andwearenotawareofanydHvA-experimentsforTaB2.
2 With the aim to initiate possible experiments, we pro-
the present paper we shall mainly consider the first ap-
0 vided calculated dHvA-frequencies and effective masses
proach which is based on calculations of the electronic
at/ structure within the local (spin) density approximation for the two materials under consideration20. From that
data one can extractinformation about the Fermi veloc-
m (L(S)DA) to the density-functional theory. The L(S)DA
bandstructurecalculations2−7revealedfoursheetsofthe ities averaged over the extremal orbits. We discuss also
-
the possible enhancement of the effective masses by the
d FermisurfaceofMgB witharemarkablehighanisotropy
2
n ofFermivelocitiesforσ-holes. Thatanisotropyisimpor- electron-phonon coupling which is expected to be quite
o different for the two materials. Information about the
tant to explain the significant anisotropy of the upper
c Fermi velocities can furthermore be obtained by means
critical field H (T) in terms of a two-band Eliashberg
: c2
v model5−7. The relatively high value of H (0) is related oftheplasmafrequenciesdetectablefromtheDrudepart
c2
i of the optical conductivity. We compare also the plasma
X to the strong electron-phonon interaction of the σ-holes
on the tube-like Fermi surface sheets7. Details of the frequencies computed for each Fermi surface sheet and
r find a large amount of anisotropy in MgB in difference
a electronic structure are important for a precise under- 2
to TaB .
standingofthepairingmechanism,like,e.g.,bycompar- 2
The band structure calculations have been per-
ing the calculated “undressed” electronic effective mass
formed by means of two computational codes: (i)
with its actual dressed value due to the electron-phonon
the fully relativistic augmented spherical waves method
coupling. On the other hand, it is interesting to com-
(ASW)21 and (ii) the scalar relativistic full potential lo-
pare MgB with relatedisostructuralcompounds having
2
cal orbital scheme (FPLO). In case (i) we used the von
much lower transition temperatures or even showing no
Barth-Hedinparametrizationoftheexchange-correlation
superconductivityatall. We choosehereTaB forwhich
2
potential. A k-mesh of 148 special k-points in the irre-
band structure calculations have been reported recently
1
ducible Brillouin zone part was chosen. For details of due to the spin-orbit coupling in a fully relativistic cal-
case (ii) see e.g. Ref.7,12. culation. Instead we are forced to ascribe these effects
Both compounds crystallize in the hexagonalspace mainly to a slightly different crystal field potential in
group P6/mmm with lattice constants a = 3.083 ˚A, spherical or full-potential descriptions, which affects the
c = 3.5208 ˚A and a = 3.082 ˚A, c = 3.243 ˚A18 for relative position of boron derived π and σ bands. The
MgB and TaB , respectively. The resulting ASW band spin-orbitcoupling ofthe Ta 5dstatesin TaB results in
2 2 2
structureforMgB issimilarwiththosepublishedinthe differences compared to scalar relativistic calculations,
2
literature2−7 and it is shown for the sake of complete- especially between Γ and A (compare Fig. 2 and Fig. 3
ness in Fig. 1. The relativistic band structure of TaB of Ref. 12). Although these differences should not af-
2
is presented in Fig. 2. We have checked our results by fect the absence of superconductivity in this compound,
comparingthemwiththescalarrelativisticfull-potential theyleadtoconsequencesfortheFermisurfacetobedis-
localorbital(FPLO)methodrepeatingpreviouscalcula- cussed below. To predict the dHvA-frequencies F for a
tions (MgB 5−7, TaB 12) and we found some differences magnetic field applied, say along the crystallographic ~c
2 2
for both compounds. In MgB these differences are not direction,wecalculatetheareaAofanyextremalorbitΩ
2
according to the Onsager relation (written in cgs units)
¯hc ¯hc
F = A= d2k . (1)
10 2πe 2πeZ
Ω
It is convenient to measure F in units of the applied
5 magnetic field and A in units of squared wave vectors,
e.g.,inkT(1kT=107G)andthecross-sectionAinnm−2,
)
V respectively. Then Eq. (1) reads simply
e 0
(
y F =0.104867 A . (1′)
g ×
r
e −5 The effective mass of the correspondingorbit is givenby
n
E
¯h dk ∂ǫ
k
m= , with ~v = . (2)
−10 2πI ~vk k ∂~k
| |
From the dHvA-frequency F and the effective mass m
−15
Γ K M Γ A H L A one can introduce an effective Fermi velocity
FIG.1. The calculated energy bandsof MgB2. The Fermi ¯h A 1 2h¯eF
v = = , (3)
energy is at zero energy. f mrπ mr c
which corresponds to the hypothetical value for a cir-
cular orbit with constant ~v . This expression is fre-
| ~k|
5 quently used to estimate the Fermi velocity from dHvA
data. Alternatively, one can also average 1/~v around
k
| |
the circumference and define
0
¯hL ¯h
V) v˜f = = dk. (4)
e −5 2πm 2πmI
(
y Obviously, both velocities defined by Eqs. (3) and (4)
g
er −10 coincide for a circular orbit. For an orbit with an ideal
n
E hexagon shape vf/v˜f = π/2√3 0.9523 is estimated
q ≈
from Eqs. (2,3). Both of these quantities contain aver-
−15
ages of point properties of the Fermi surface around a
cyclotron orbit.
−20Γ K M Γ A H L A Information about the Fermi velocity is also con-
tained in the plasma frequencies ω (α=x(y) or z)
p,α
e2 d3k
(ω )2 = δ(ǫ E )(v )2, (5)
p,α ǫ◦ Z (2π)3 k− f k,α
FIG.2. The same as in Fig. 1 for TaB .
2
wherewewilldistinguishthecontributionsfromdifferent
Fermi surface sheets in the following (denoted by ωn ).
p,α
2
Analogously to Eq. (5), components α of the Fermi ve- MgB (seeFig.3)consistsoffoursheetsandthereare6
2
locities v = <v2 > can be introduced which are extremalorbitsforamagneticfieldin~cdirection. Wewill
α α FS
averaged overpselected sheets or over the whole Fermi denote them by numbers 1 to 3 centeredaroundΓ and 4
surface (FS) to6centeredaroundAwithincreasingareaineachcase.
Soonecanseethatorbits1and4(2and5)belongtothe
d3k smaller (larger) tube, where the remaining orbits 3 and
(v )2 = δ(ǫ E )(v )2. (6)
α Z (2π)3 k− f k,α 6areratherlargeandbelongtothe ring-likepartsofthe
Fermi surface. The electrons on orbit 6 have the largest
The ratio of these Fermi velocity components related to Fermi velocity and v of orbit 3 is smaller compared to
f
the mostanisotropicσ-tubes withthe strongestel-phin- thatoforbit6due tothe largereffective mass. The elec-
teractionatthesametime canbeusedtodescribeupper trons on the tubes are slow in general,with the minimal
bounds for the anisotropyparameterγ ofthe upper crit- velocity on orbit 2. The ratio of Fermi velocities for the
ical field Hc2, twodifferentgroupsofelectronsisroughly2. Inaddition
thereisaratherdifferentstrengthoftheelectron-phonon
γ =Hab/Hc < <v2 > /2<v2 > , interactiononbothtypes ofFermisurfacesheets: strong
c2 c2 f,xy tubes f,z tubes
q onthe tubes andweakonπ-derivedrings. This confirms
if that anisotropy is the predominant one as suggested theintroductionofatwo-bandmodelproposedforMgB2
by several microscopic calculations5. firstby Shulga et al.6 andelaboratedlater on in Ref. 22.
Therelativedifference(v v˜ )/v doesnotexceed
f f f
−
7percentanditisnoteworthyonlyforthoseorbitswhich
III. MAGNESIUM DIBORIDE have a pronounced hexagonal form than a nearly circu-
lar one (for example orbit 3 is such a nearly circular or-
bit). Accordingtothetwo-bandmodel6 wewouldexpect
Let us first discuss the results for MgB . The band
2
a different electron-phonon mass renormalization (1+λ)
structure near the Fermi energy E is depicted in Fig.
f
forelectronsonthetubes(withλ 1.2 1.5)andonthe
1. The corresponding dHvA-data for a magnetic field ≃ ···
large orbits 3 and 6 (λ 0.3). The comparison with the
directedalongthec-axisareshowninTableI.(Notethat ≃
experimentally determined cross sections shows signifi-
in ourpreviouspaper20 the numbers forthe orbits 2 and
cant deviations from our L(S)DA predictions. The mi-
4 were mistakenly interchanged.) The Fermi surface of
croscopic reason for that unexpected deviation remains
unclear at the moment. Standard electron-phonon in-
teraction does not affect the cross-section. It might ex-
plainonlytherenormalizationoftheFermivelocitiesand
of the masses. The cross-sections might be affected by
strong electron-electron interaction and/or the presence
ofelectronicinstabilitiescompetingwithsuperconductiv-
ity which might result in local changes (destruction) of
the Fermi surface by a partial dielectrization (gapping)
from a charge density or spin density wave23. The ex-
FIG.3. (Color) The calculated Fermi surface, distribution periments yield a much narrower smaller tube than the
of Fermivelocities, andextremal orbits(pinklines, thenum- L(S)DAcalculationpredict. Whilethisalreadyindicates
bers denote the orbits that are explained in the text) (mag- a discrepancy between theory and experiment, more se-
netic field kc-axis) of MgB2. The Fermi velocities expressed riousmightbethefactthatthewider,secondtubeisnot
by different colors (blue-slow, green-medium, read-fast) are observedintheexperimentatall. Wouldthisfactbecon-
given in units of 105m/s as shown in thelegend. firmedalsoforcleanersamples,i.e.themissingcrosssec-
tions with lowerFermi velocities could not be attributed
to the residual impurity scattering19, it would imply the
failure of the L(S)DA to describe correctly MgB .
2
We consider the elucidation of the responsible
many-body effects which are missing in the standard
LDA approach to MgB , i.e. treating it as a simple sp-
2
metal, as a possible key problem for our physical under-
standing of the electronic structure of MgB and its re-
2
markable superconductivity as well. Anyhow, in view of
thegreatimportanceoftheexperimentaldHvAdata,the
FIG. 4. (Color) The same as in Fig. 3 for TaB . Note confirmation of a more or less unusual electronic struc-
2
thatforvisibilitypurposesAisatthecenterofthehexagonal ture inMgB2 by other groupsusingdifferent singlecrys-
prism and the Γ point is the midpoint of the lower and the tals would be highly desirable.
upperhexagon. The experimental determination of the effective
3
masses by dHvA measurements and its comparisonwith differences that affect the Fermi surface: First, a split-
the calculatedelectronicvaluesgivesimportantinforma- ting of bands along the high symmetry direction Γ A,
−
tion on details of the electron-phonon coupling. If the which leads to a lifting of band degeneracy at the Fermi
assignmentproposedinRef.19iscorrect,stronglyrenor- level. Second, also at the K-point a small band shift
malizedFermivelocitiesbyafactorof2to3shouldbeex- occurs. The first spin-orbit modification of the Fermi
plained. Comparingthecalculatedandmeasuredmasses surface makes one Fermi sheet disjointed along Γ A.
−
for the smaller tube, one arrivesat a slightly largerlocal The calculated areas, effective masses and Fermi veloc-
λ 1.27 comparedwith λ 1 obtained by Choi et al.22 ities of TaB are collected in Table IV and one can see
2
≈ ≈
within full anisotropic (harmonic) Eliashberg theory. that any kind of anisotropies of Fermi velocities is much
Ifthecorrespondingcouplingconstantforthelarger smaller than for MgB . According to the calculation of
2
tubewouldbealsoenhanced,possiblecontributionsfrom the electron-phonon coupling constants12 we would ex-
the coupling of a soft mode should be considered. Oth- pectarathersmallrenormalizationofeffectivemassesin
erwise it would be unclear, why for a non-enhanced TaB (λ 0.05 0.2) again in difference to MgB . The
2 2
≃ ···
Coulombrepulsion(measuredbytheCoulombpseudopo- plasma frequencies are collected in Table V. Since there
tential µ∗) or the absence of any other enhanced pair- areno analogousFS sheets to the two tubes ofMgB we
2
breaking, the critical temperature is not higher than 40 do not obtain a correspondinganisotropyin ωn with
p,x(y)
K.Inthis contextthe recentexperimentalobservationof respect to ωn . This can be understood due to the loss
p,z
asoft-modenear17meVinRamanspectroscopy24,point ofthequasitwo-dimensionalityoftheBσ-statesinTaB
2
contactspectroscopy25 as well as in neutronscattering26 due to the hybridization with the Ta 5d states.
is ofconsiderableinterest. Especiallywithrespectto the The hole Fermi sheet around A shown in the mid-
assignmentofthatboson-mode,sincestandardharmonic dle panel of Fig. 4 resembles a feature calculated within
phononcalculationsareunabletoexplainthepresenceof an augmented-plane-wave code for ZrB 30 The exper-
2
such a low-lying mode4. Notably, also from recent stan- imentally observed dHvA-frequencies of the central ε-
dard tunneling experiments27 (unfortunately with a cut orbit31 (corresponding to orbit No. 2 in Fig. 4) near
off at 26 meV, i.e. just slightly above the low-frequency 1.8 kT and 1.5 kT for the similar system TiB 32, re-
2
anomalies) significant deviations from the standard har- spectively, are significantly smaller than 3.48 kT in our
monic phonon calculations have been reported. FPLO-calculation for TaB . The opposite behaviour is
2
The anisotropy of Fermi velocities of MgB2 is even observed for the µ orbit (orbit No. 6 in Fig. 4) where
morevisibleinourcalculatedplasmafrequencies(seeTa- the experimental v−alues for ZrB and TiB read 2.46
2 2
ble II), which are similar to the results of Refs. 4,3,28,29 kT and 3.44 kT compared with our FPLO calculated
(see Table III). In our previous calculation20 a scaling 3.44 kT for TaB . Further theoretical and experimental
2
factor of c/a was missing for ωpx(y). Here we have cor- studies shouldbe awaitedto answerthe important ques-
rected the present results for that factor. It is another tion whether there are significant deviations even from
kind of anisotropy: the z-component of vf is about 6 full-potential L(S)DA-calculations only for the peculiar
timessmallerthanthex(y)componentsforthetwotubes MgB or also for other diborides with weak electron-
2
(sheets1and2). Theanisotropyratiovpn,x(y)/vpn,z (vαn ∝ phonon interaction.
ωn ) is much smaller for the sheets 3 and 4 and its
| p,α|
average Fermi velocity is larger. The total plasma fre-
quency, however, shows only a small anisotropy of 1.04 IV. SUMMARY AND CONCLUSIONS
between the z and x(y) components. In Table III we
compare the anisotropy in the components of the total In summary, we have calculated the dHvA-
plasma frequency as obtained by various groups. The frequencies and effective masses as well as the plasma
obtained numbers are generally in good agreement with frequencies of MgB and TaB which would be worth to
2 2
one another. be investigated experimentally in detail. First experi-
The anisotropyofFermivelocities is muchless pro- mental de Haas-van Alphen data for MgB point to sig-
2
nouncedinTaB2 (TablesIII andIV).Ourfully relativis- nificant deviations. The comparison of the experimental
tic Fermi surface coincides almost with that published masseswithL(S)DAcalculationcouldgiveimportantin-
in12 and consists of three sheets. The extremal orbits formation on the electron-phonon coupling. According
are characterized as follows (s. Fig. 3): numbers 1 to 3 to Refs.2,7,12 we expect remarkable differences between
are centered around the A point and correspond to the MgB and TaB : Large enhancement for one group of
2 2
three different sheets. Orbit 4 is centered around K and electronsinMgB andasmallenhancementfortheother
2
orbit5correspondstothesmallFermisurfacepartinthe electrons in MgB as well as in TaB . We found a re-
2 2
neighborhoodof K. There is one maximal orbit(number markableanisotropyofFermivelocitiesinsuperconduct-
6) with nontrivial value kz (0.20615×2π/a). Since Ta ing MgB2 which is in difference to the more isotropic
is a 5d element, the influence of the spin-orbit coupling behavior of non-superconducting TaB .
2
is more pronounced in TaB2 than in MgB2. With re- Note added.After finishing the present second version of
spect to the scalar relativistic calculations12 we find two our work we have learnt on a closely related work by
4
Mazinand Kortus33 who also stressthe necessity of full- 10Boeri L., Bachelet, G.B., Cappelluti, E., and Pietronero,
potential calculations for MgB and arrive essentially at L., cond-mat/0112075 (2001); Cappelutti, E., Ciuchi, S.,
2
very close numerical values for the cross sections and Grimaldi, C., Pietronero, L., and Str¨assler, S., cond-
the masses under consideration (see Table I). Remain- mat/0105351.
ing slight numerical differences, comparison with other 11Alexandrov,A., cond-mat/0103006.
full-potential bandstructure codes, and somewhat differ- 12 Rosner, H., Pickett, W.E, Drechsler, S.-L., Handstein,
ent physicalinterpretationwill be discussedelsewhere34. A., Behr, G. Fuchs,G., Nenkov,K., Mu¨ller, K.-H.,and
Eschrig, H.,Phys. Rev. B64, 144516 (2001).
13 Stein, I.R., and Ivanovski,A.L., cond-mat/0109445.
ACKNOWLEDGMENTS 14 Singh, P.P., cond-mat/0104580 .
15 Otani, S., Korsukova, M.M., and Miksuhashi, T., J. of
Crystal Growth 194, 430-433 (1998).
WethankS.V.Shulga,I.Mazin,andA.Carrington 16 Leyarovska, L., and Leyarovska, E., J. Less Common
for critical remarks and discussions. We are indebted Metals 67, 249 (1979).
to H. Eschrig, M. Richter, V.D.P. Servedio, W. Pickett 17 Gasparov, V.A., Sidorov, N.S., Zvor’kova, I.L., and Ku-
and J. An for discussions. This work was supported by lakov, M.P., JETP Letters 73, 532 (2001).
theEgyptianMinistryofHigherEducationandScientific 18 Kaczorowski, D., Zaleski, A.J., Zogal, O.J., and Klamut,
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Meeson,P.J.,Lee,S.,YamomotoA.,andTajima,S.,cond-
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5
TABLEI. TheLSDAdeHaas-vanAlphendataofMgB (magneticfieldkc−axis). Availableexperimentalvaluesaccording
2
to the assignment of Ref. 19 have been added for the sake of comparison. Calculated quantitities without explicit label have
been computed within the ASWscheme.
orbit FASW FFPLO Fexp A/ABZ m/m◦ mexp/m◦ vf vf,exp v˜f
[kT] [kT] [KT] % [105m/s] [105m/s] [105m/s]
1 0.99 0.79 0.54 1.98 0.25 0.57 7.9 3.2 8.5
2 2.16 1.63 4.3 0.55 5.4 5.8
3 32.88 34.2 65.6 1.70 6.8 7.2
4 2.31 1.84 1.53 4.6 0.31 0.70 9.8 3.3 10.5
5 4.31 3.44 8.6 0.61 6.8 7.3
6 30.20 30.6 60.2 0.92 12. 12.8
6
TABLEII. Squaredplasma frequencies of MgB in [eV2].
2
OneeVfortheplasmafrequenciescorrespondsto80.4·km/s.
FS sheet ωp2,x(y) ωp2,z vx(y)/vz
1 (tube) 9.639 0.242 6.31
2 (tube) 9.444 0.323 5.40
3 (ring) 8.620 22.053 0.63
4 (ring) 23.042 24.6698 0.97
All 50.75 47.290 1.04
TABLE III. Comparison of several calculated results for
the two components of the total plasma frequency ωpx and
ωpz of MgB2 (in eV).
ωp,x(y) ωp,z Reference
7.12 6.88 thiswork
7.21 6.87 Ref. 3
7.02 6.68 Ref. 4
7.13 6.72 Ref. 28
7.04 6.77 Ref. 29
TABLEIV. Calculated de Haas-van Alphen data of TaB .
2
orbit FASW FFPLO AASW/ABZ m/m◦ vf v˜f
[KT] [KT] [km/s] [km/s]
1 2.08 2.50 0.041 0.221 1320 1410
2 2.99 3.48 0.059 0.356 979 1050
3 8.40 8.31 0.167 0.677 864 926
4 13.26 13.84 0.264 0.558 1320 1410
5 1.05 1.00 0.021 0.236 869 891
6 1.15 3.44 0.023 0.632 339 365
TABLEV. Squared plasma frequencies ωpx, ωpz, and the
anisotropy of averaged Fermivelocities for TaB (in eV2).
2
FS sheet ωp,x(y) ωp,z vx(y)/vz
1 68.125 84.779 0.90
2 15.149 17.049 0.94
3 3.243 3.254 1.00
All 86.52 105.08 0.91
7
