Phonon dispersion and lifetimes in MgB2

4 Pages·0.26 MB·

by  Abhay Shukla

Phonon dispersion and lifetimes in MgB
2
Abhay Shukla,1 Matteo Calandra,1 Matteo d’Astuto,2 Michele Lazzeri,1 Francesco Mauri,1 Christophe
Bellin,1 Michael Krisch,2 J. Karpinski,3 S. M. Kazakov,3 J. Jun,3 D. Daghero,4 and K. Parlinski5
1Laboratoire de Min´eralogie-Cristallographie, case 115, 4 Place Jussieu, 75252, Paris cedex 05, France
2European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble cedex, France
3 3Solid State Physics Laboratory, ETH, CH-8093 Zu¨rich, Switzerland
0 4INFM - Dipartimento di Fisica, Politecnico di Torino, C. Duca degli Abruzzi 24, 10129 Torino, Italy
0 5Institute of Nuclear Physics, ul. Radzikowskiego, 152, 31-342, Cracow, Poland.
2
(Dated: February 1, 2008)

n
a We measure phonon dispersion and linewidth in a single crystal of MgB2 along the Γ-A, Γ-M
J andA-LdirectionsusinginelasticX-Rayscattering. WeuseDensityFunctionalTheorytocompute
theeffectofbothelectron-phononcouplingandanharmonicityonthelinewidth,obtainingexcellent
7
agreementwithexperiment. AnomalousbroadeningoftheE2g phononmodeisfoundallalongΓ-A.
] The dominant contribution to the linewidth is always theelectron-phonon coupling.
n
o PACSnumbers: 63.20.Dj,63.20.Kr,78.70.Ck,71.15.Mb
c
-
r
p The discovery of 39 K superconductivity in MgB2 [1] ray scattering experiments have recently become avail-
u hasledtoin-depthstudyofthematerialandapicturehas able. The crystalusedin ourexperiment wasgrownat a
s emergedofaphonon-mediatedsuperconductorwithmul- pressure of 30-35 kbar. A mixture of Mg and B was put
.
t tiple gaps [2, 3, 4], moderate electron-phonon coupling intoaBNcontainerinacubic anvildevice. Thetemper-
a
◦
m (EPC)[3,5,6,7],andanharmonicity[3,4,8,9]. However aturewasincreasedduringonehourupto1700−1800 C,
no measurements exist concerning either phonon disper- keptstablefor1-3hoursanddecreasedduring1-2hours.
-
d sion or the evolution of phonon lifetimes over the Bril- Asaresultplate-likeMgB2crystalswereformedofwhich
n louin Zone (BZ), due to the absence of large single crys- we used a sample of about 400 x 470 x 40 µm3, with a
o ◦
tals. Neutron scattering on powder samples [8, 10] has measured in-plane mosaicity of 0.007 . The beam inci-
c
[ been limited to the determination of phonon density of dent on the sample was obtained from a high-resolution

states. Raman spectroscopy, which is restricted to the Silicon backscattering monochromator using the (8 8 8)
2
v BZ center, has shown that the optical mode with E2g reflectionatanincidentenergyof15.816keV.TheX-ray
4 symmetry,correspondingtoin-planedistortionsofthe B beam was focused onto the sample by a toroidal mirror
6 hexagons, is strongly damped [11, 12, 13, 14]. into a spot of 270 x 90 µm2(horizontal × vertical), full
0 Phonon damping can be caused by (i) EPC medi- width at half maximum (FWHM). Slits before the sam-
9
ated phonon decay into electron-hole pairs [15], or (ii) ple further limited the vertical beam size to 30 µm. The
0
phonon-phonon interaction due to anharmonicity [16]. scatteredphotonswereanalyzedinenergybyfivespheri-
2
0 The linewidth (the inverse of the lifetime) of a given calsiliconcrystalanalyzersoperatingat the same reflec-
/ phonon is the sum of both contributions. Direct deter- tion order and mounted in pseudo Rowland circle geom-
t
a mination of the contribution of each phonon mode to etry. The total energy resolution was 6.1 meV FWHM,
m
EPC [15] from the measured linewidth is only possible as determined by a fit to a Lorentzian lineshape. The
- if the anharmonic contribution is negligible [17] and is momentum transfer Q was selected by rotating the 7 m
d
n seemingly questionable for MgB2 where many calcula- longanalyzerarmaroundthesampleposition,inthehor-
o tions [3, 4, 8] suggest strong anharmonic effects. izontal plane, which also contained the linear x-ray po-
c In this work we present the first measured phonon larization vector of the inci−d1ent beam. The momentum
: resolutionwas set to 0.04˚A in the horizontaldirection
v dispersion curves and linewidths (where possible) along −1
i three directions in the BZ, Γ-A, Γ-M and A-L. We cir- and0.07˚A intheverticaldirection. Thefollowingmea-
X
surements were performed at a temperature of 300K: i)
cumventedtheproblemofsamplesizebyusinghighreso-
r Q=(1 2 ξ), in almost transverse configuration along the
a lutioninelasticscatteringofafocusedandintenseX-Ray
Γ-A direction, i.e. with Q·q ≈ 0, q=(0 0 ξ) being the
beam at the European Synchrotron Radiation Facility
phonon wavevector, ii) Q=(1-δ 1+ξ 0), while nearly fol-
(beamlineID28),atechnique[18]successfullyusedinsin-
glecrystallinesamplesandinparticularforthemeasure- lowing the Γ-M direction (0 ≤ δ ≤ 0.05). iii) Q=(1-δ1
ment of high energy optical modes [19]. To understand 2+ξ 0.5+δ2), while nearly following the A-L direction
the mechanisms governingthe measured phenomena, we (0 ≤ δ1,δ2 ≤ 0.04). The choice of BZ points measured
was dependent on a series of conditions including the
calculated phonon dispersion, the contributions of EPC
need to optimize the structure factors, single (Γ-A) or
and anharmonicity to the linewidth and the structure
multi-analyzer(Γ-M,A-L)measurementmode andspec-
factors using Density Functional Theory (DFT).
trometer and time limitations.
Small single crystalsof MgB2, suitable for inelastic X-
2
E
1u
Stokes
acoustic
s) s)
nit nit
u u
y y
ar transverse ar
bitr acoustic bitr
ar ar
y ( anti−Stokes y (
sit sit
n n
e 50 75 100 e 50 75 100
nt nt
i i
E
2g A
2u acoustic E
2g
−25 0 25 50 75 100 −25 0 25 50 75 100
Energy [meV] Energy [meV]
FIG. 1: (color online) Energy loss scan in almost transverse FIG.2: (coloronline)Energylossscan,measuredatQ=(0.97
geometry measured at Q=(1 2 0.3) corresponding to 0.6 Γ- 2.29 0.54) corresponding closely to 0.58 A-L, with the least-
A.Thedata, normalized totheincidentflux,areshown with squaresfitandtheab-initiospectrumcalculatedat0.58A-L.
theleast-squares fit (dashed line) and theab-initio spectrum
withandwithoutbroadeningduetoexperimentandelectron
phononcoupling(solidlines). Thebroadpeakcorrespondsto zone boundary. At the point measured nearest to M
the damped E2g mode and is shown in greater detail in the ((1.05 1.45 0), not shown) the E2g and E1u modes are
inset.The peak at zero is due todiffuse scattering.
comparable in intensity but only separated by about 1
meV according to our calculation and we do measure a
single peak only somewhat broader (FWHM ≈ 10meV)
In Fig. 1 we show data taken at the 0.6 Γ-A point in thantheexperimentalresolution,indicatingreducedE2g
theBZ.Theacousticmodeaswellasthelowerenergyop- linewidth. The proximity of the E1u mode however pre-
tical mode (E1u) are visible as resolution-limited peaks. vents a firm conclusion in this regard. We mention that
Mostimportantly,abroadpeakisobservedathigheren- the calculated structure factors and energies show excel-
ergyloss,correspondingtotheE2g mode. Weperformed lent quantitative agreement with our measured data of
least square fits to sums of Lorentzian functions with which we have shown only two examples.
FWHMcorrespondingto theexperimentalresolutionfor Similar analysis was done for several points along the
the resolutionlimitedpeaksandafree parameterforthe threedirectionsinordertoexperimentallydeterminethe
strongly damped phonon. These yield the dispersion as phonon dispersion and the linewidths. The difference in
well as the linewidth variation over the BZ. Despite sta- calculatedphononenergiesbetweenthe measuredpoints
tistical limitations (3-6 counts per minute on this peak and corresponding points exactly along Γ-M and A-L
along Γ-A) and tails of the peaks from the stronger, low (δ,δ1,δ2 = 0) is less than half a meV in all cases. We
energyphonons,thepeakenergyaswellasthe linewidth can thus compare the experimental phonon dispersion
can be estimated with reasonable confidence. withthetheoreticalcalculationalongthehighsymmetry
Fig. 2 shows a similar energy loss scan at (0.97 2.29 lines, as shown in the bottom panel of fig. 3 (circles).
0.54)closetothe0.58A-Lpoint. Astrongacousticmode The measured intrinsic linewidth of the E2g branch,
is seen at 30 meV. The peak at 50 meV corresponds to showninthetoppaneloffig.3,isstronglyanisotropicin
theA2u branchandthe oneat65meVtoanotheracous- the BZ. Along Γ-A it is particularly large (ranging from
tic branch. Finally two resolved features are seen at 85 20 to 28 meV), while near L and probably near M it is
and 97 meV. These are the two E2g modes which in this below the experimental resolution.
region of reciprocal space are well separated from other Electronic structure calculations [20] were performed
modes. Though, given the statistics, it would be haz- using DFT in the generalized gradient approximation
ardous to estimate a linewidth, the comparison between [21]. We used norm conserving pseudo-potentials [22].
the experimental and ab-initio spectra suggests that the For Mg we used non-linear core corrections [23] and
linewidths of both the E2g modes are resolution limited we treated the 2s, 2p levels as core states. The wave-
and so the damping is much less than that along Γ-A. functions were expanded in plane waves using a 35 Ry
As for the measurement nearly along Γ-M, the structure cutoff. The calculations were performed with the exper-
factor for the optical modes strengthens only near the imental crystal structure, namely a=3.083˚A and c/a=
3
In the calculations we used N = 303 inequivalent k-
30 k
V] points and, in Eqs. (1) and (3), we substituted the δ
e
m20
functions with Gaussians. The electronphonon coupling
h [
dt10 λqν is obtained from the linewidth [15] as:
wi
0 γqν
λ = , (2)
B qν 2πN(0)ω2
1g qν
80
V] E2g N(0) = 0.354 states/(MgB2 eV spin) being the density
e of states at the Fermi level.
m60
y [ A The second contribution to the linewidth is given by
c 2u
n the anharmonicity in the crystal potential. At lowest
ue40
q E orderforthemodeν ofazonecenterphonontheFWHM
e 1u
fr linewidth is [16, 26, 27]:
20
π¯h ∂3E ID +IA
0 Γ0ν = X| |2 qµην qµην (3)
M Γ A L 8Nq q,µη ∂u0ν∂uqµ∂u−qη ω0νωqµωqη
E being the total energy, n the Bose occupa-
FIG. 3: (color online) Bottom: Experimental (circles) and qµ
theoretical phonon dispersion (solid line) in MgB2 along Γ- tion for mode µ at wavevector q, IqDµην = (nqµ +
A and Γ-M and A-L. In the region near the M point, the n + 1)δ(ω −ω −ω ) describes the decay in
qη 0ν qµ qη
probabledetectionoftheE2g modeisindicatedwithasquare the two phonons µ and η, and IqAµην = 2(nqµ −
symbol (see text). The crosses indicate a parasite signal of n )δ(ω −ω +ω ) describes the η-phonon absorp-
qη 0ν qµ qη
unknown origin. Top: Intrinsic linewidth of the E2g mode. tion and the µ-phonon emission.
The experimental linewidth (circles) is large along Γ-A and
We computed the anharmonic linewidth at the high-
below the experimental resolution (dashed line) near L and
symmetry points Γ, A, M. For the calculation at A we
M.Thetheoreticalresult(diamonds)fortheelectron-phonon
coupling contribution to the linewidth is also shown. Along considera1×1×2supercellwith6atoms,whilefortheM
A-L and Γ-M where the E2g mode is non-degenerate both pointweusea2×2×1cellwith12atoms. Thethirdorder
theoretical values are shown, when different. E2g linewidth matriceswereevaluatedusinglinearresponsetheoryand
decreases progressively from A to L and experimentally the the 2n + 1 theorem for metals [28]. The anharmonic
two branches are resolved for the point nearest to L.
contribution was evaluated at 0K and 300K.
At Γ the anharmonic linewidth is largest for the E2g
mode and equal to 0.16 meV at T = 0K and 1.21 meV
1.142. The harmonic phonon frequencies were computed
atT =300K. Boththe values arenegligibleifcompared
in the linear response [24]. We used a 16 × 16 × 16
with the experimental Raman linewidth of roughly 40
Monkhorst-Pack grid for the electronic BZ integration
meV [13], suggestingthat the mainsourceof broadening
and first order Hermite-Gaussian smearing [25] of 0.025
is the electron-phonon interaction.
Ry. The dynamical matrix at a given point of the BZ
The results of the calculation of the two contributions
wasobtained froma Fourierinterpolationofthe dynam-
tothelinewidthatAandMareshowninTableI. Atthe
icalmatricescomputedona6×6×4phononmesh. The
resulting phonon frequencies are shown in fig. 3 and are A point the linewidth of the E2g mode due to electron-
phononscatteringisverylargewhiletheanharmoniccon-
in good agreement with a recent calculation [11]. The
tribution is more than an order of magnitude smaller.
agreement with experiment is remarkable.
For sizable values of the EPC and at low temperature
The contribution to the FWHM linewidth γ at mo-
qν
mentum q for the ν phonon mode due to the electron- the anharmonicity is negligible showing that a measure-
ment of the linewidth at q for the mode ν is essentially
phonon interaction can be written as [15]:
equivalent to determining λ . This is unexpected since
qν
4πω
γqν = Nkq X |gkνn,k+qm|2δ(εkn)δ(εk+qm) (1) etoarblieerimthpeoorrteatnictalfowroarkdi[ff3,er4e,n8t]beusttimrealatetsedanqhuaarnmtiotyn,ictihtye
k,n,m
frequency shift.
where the sum is extended over the BZ, Nk is the Since the anharmonic contribution is negligible at Γ,
number of k-points in the sum, and εkn are the en- A, M, and computationally demanding, we only evalu-
ergy bands measured with respect to the Fermi level ated the EPC linewidth for the other points along the
at point k. The matrix element is gkνn,k+qm = three directions. Using the calculated phonon frequen-
hkn|δV/δu |k+qmi/ 2ω , where u is the ampli- cies,displacementsandlinewidthswecomputethestruc-
qν p qν qν
tude of the displacement of the phonon ν of wavevector ture factors for one phonon processes using the X-ray
q, ω is the phononfrequencyandV isthe Kohn-Sham formfactors,obtaining goodagreementwith experiment
qν
potential. as shownin Figs.1,2. Inthe top panelofFig. 3 we show
4
samples, be measured for all modes over the whole Bril-
M A
louin zone so as to extract the anisotropic Eliashberg
Γ0ν Γ3ν00 γν λν Γ0ν Γ3ν00 γν λν coupling function.
0.00 0.12 0.00 0.00 0.00 0.17 0.00 0.00
0.00 0.15 0.01 0.01 0.00 0.17 0.00 0.00
0.02 0.12 0.06 0.02 0.00 0.63 0.08 0.05 We acknowledge illuminating discussions with R. S.
0.12 0.48 1.13 0.20 0.00 0.63 0.08 0.05 Gonnelli, P. Giannozzi, M. Xu, and F. Sette. The cal-
culations were performed at the IDRIS supercomputing
0.06 0.25 0.00 0.00 0.02 0.22 0.84 0.28
center. M.C.wassupportedbyaMarieCurie Fellowship
0.07 0.33 2.34 0.30 0.02 0.20 0.08 0.02
of the EuropeanCommission, contract No. IHP-HPMF-
0.26 0.42 1.06 0.06 0.10 2.13 20.35 2.83
CT-2001-01185.
0.45 0.69 1.21 0.07 0.10 2.13 20.35 2.83
0.47 0.72 0.08 0.00 0.13 0.23 0.05 0.00
TABLEI:Calculatedlinewidths(meV)duetoanharmonicity
at0K(Γ0ν),300K(Γ3ν00)andelectron-phononinteractionγν at [1] J. Nagamatsu et al.,Nature(London) 410, 63 (2001).
MandAforallmodes. Phononfrequenciesincreasefromtop [2] S.V. Shulga et al. cond-mat/0103154.
tobottom. λν istheelectron-phononcoupling. E2g modesin [3] A.Y.Liu,I.I.MazinandJ.Kortus,Phys.Rev.Lett.87,
boldface. 087005 (2001).
[4] H. J. Choi et al., Nature (London) 418, 758 (2002), H.
J. Choi et al.,Phys.Rev.B 66, 020513 (2002)
q 0.2 Γ-A 0.5 Γ-A 0.6 Γ-A 0.8 Γ-A 1.0 Γ-A [5] J. M. An and W. E. Pickett, Phys. Rev. Lett. 86, 4366
λexpt. 2.5±1.1 2.6±0.6 2.3±0.5 3.6±0.7 3.6±0.8 (2001), J. M. An et al. cond-mat/0207542
qν
λtheo. 3.32 2.80 2.77 3.12 2.83 [6] J. Kortuset al.,Phys.Rev. Lett., 86, 4656 (2001).
qν
[7] Y. Kong et al. Phys.Rev. B, 64, 020501(R) (2001).
[8] T. Yildirim et al. Phys. Rev.Lett. 87, 37001 (2001).
TABLE II: Experimental and theoretical λqν of each of the [9] L. Boeri et al.,Phys.Rev. B 65, 214501 (2002)
two degenerate E2g modes along Γ-A. [10] R. Osborn et al. Phys.Rev.Lett. 87, 17005 (2001)
[11] K. P. Bohnen, R. Heid and B. Renker, Phys. Rev. Lett.
86 5771 (2001).
the theoretical results for the E2g branch. The result is [12] A. F. Goncharov et al Phys. Rev.B 64, 100509 (2001)
[13] P. Postorino et al.Phys. Rev.B 65 020507(R) (2001)
consistentwith the experimental value whereverthe E2g [14] J.Hlinka et al. Phys.Rev.B 64, 140503(R) (2001)
branch is clearly visible namely along Γ-A and near the
[15] P.B.Allen,Phys.Rev.B6,2577(1972),P.B.Allenand
L point. Finally, the anharmonicity being negligible, we
R. Silberglitt, Phys.Rev. B 9, 4733 (1974).
used Eq. (2) to extract λqν of the E2g mode along Γ-A [16] J. Men´endez and M. Cardona, Phys. Rev. B 29 2051
using the measured linewidths and frequencies together (1984)
with the calculatedelectronic density of states. In Table [17] G. Grimvall, The electron-phonon interaction in metals,
II the experimental values are compared with the the- (North Holland, Amsterdam, 1981) p.201.
[18] T. Rufet al. Phys. Rev.Lett. 86, 906 (2001)
oretical predictions. The anomalously large EPC along
[19] M. D’Astutoet al. Phys. Rev.Lett. 88, 167002 (2002).
Γ-A is due to the nesting factor of the B bonding p
x,y [20] S. Baroni, et al. Rev.Mod. Phys. 73, 515-562 (2001).
Fermi surfaces, which are concentric cylinders centered
[21] J.P.Perdew, K.Burke,M.Ernzerhof,Phys.Rev.Lett.77,
on Γ-A [5, 6]. The E2g modes, which modify the B-B 3865 (1996)
distances, are the only ones with a sizeable matrix ele- [22] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993
ment, gν , between electrons on these surfaces. (1991).
kn,k+qm
In conclusion, we have measured phonon dispersion [23] S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B
26, 1738 (1982)
and linewidths in a sub-mm sized MgB2 crystal with in-
[24] P. Giannozzi et al., Phys.Rev.B 43, 7231 (1991)
elastic X-ray scattering confirming the power and ver-
[25] S. deGironcoli, Phys. Rev.B 51, 6773 (1995)
satility of this technique. Both acoustic and optical
[26] A. Debernardi, S. Baroni and E. Molinari Phys. Rev.
modes are detected and we find that the E2g mode is Lett. 75, 1819 (1995)
anomalously broadened along Γ-A but that this broad- [27] G.Lang et al. Phys. Rev.B 59 , 6182 (1999)
ening is not generalized over the Brillouin Zone. Our [28] M. Lazzeri and S.de Gironcoli Phys.Rev.B 65, 245402
Density FunctionalTheorycalculations ofthe dispersion (2002). For the electronic BZ integration we used a
14×14×8, 14×14×4, 7×7×8meshforthecalculation
and linewidth are in excellent agreement with experi-
at Γ , A, M, respectively. The third order derivative of
ment. They show that the dominant contribution to the thetotalenergywerecomputedwitha4×4×2,4×4×1
broadeningforallmodesistheelectron-phononcoupling, and 2×2×2 mesh of q points for Γ, A, M respectively
the anharmonic contribution being much smaller. Thus and Fourierinterpolated at thepointsqrequiredforthe
phononlinewidthinMgB2isadirectmeasureofelectron- BZ summation in Eq. (3). Weperformed the q BZ sum-
phononcouplingandcould,withtheavailabilityoflarger mation using a 203 grid of inequivalent points.