Physics and Astronomy Publications
Physics and Astronomy
7-1976
Thermoreflectance of V, Nb, and paramagnetic Cr
J. H. Weaver
Iowa State University
David W. Lynch
Iowa State University
, dlynch@iastate.edu
C. H. Culp
Iowa State University
R. Rosei
Istituto di Fisica dell'Universita di Roma
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Thermoreflectance of V, Nb, and paramagnetic Cr
Abstract
Thermoreflectance measurements of V, Nb, and Cr films were performed near 80 and 340 K to investigate the
electronic properties of these metals. The thermoreflectance spectra ΔRR were Kramers-Kronig analyzed to
determine the temperature dependence of the dielectric functions. The results were compared with previous
derivative measurements of Mo and with static measurements of the optical reflectance of the bcc metals.
Interpretation of the structures was guided by recent band calculations. It was shown that the observed
features reflect the behavior of large volumes of the Brillouin zone rather than well-defined singularities.
Keywords
Ames Laboratory, Kramers-Kronig, Brillouin zone
Disciplines
Condensed Matter Physics | Physics
Comments
This article is from
Physical Review B
14 (1976): 459, doi:
10.1103/PhysRevB.14.459
. Posted with permission.
PHYSICAL REVIEW
B VOLUME14,
NUMBER 2 15JULY
1976
Tbermoreflectance
of V,
Nb,
andparamagnetic
Cr
J.
H. WeaverAmes Laboratory-ERDA and Department ofPhysics, Iowa State University, Ames, Iowa 50010
and Synchrotron Radiation Center, ~~ University ofWisconsin-Madison, Stoughton, Wisconsin 53589
D.
W. Lynch and C. H. Culp~Ames Laboratory-ERDA and Department ofPhysics, Iowa State University, Ames, Iowa 50010 R.Rosei~
Istituto diFisica dell'Universita di Roma, Roma, Italy (Received 23 January 1976)
Thermoreflectance measurements ofV, Nb, and Crfilms were performed near 80and 340Ktoinvestigate the electronic properties ofthese metals. Thethermoreflectance spectra
6
R/R were Kramers-Kronig analyzed todetermine the temperature dependence ofthe dielectric functions. The results were compared with previous derivative measurements ofMo and with static measurements ofthe optical reflectance ofthe bcc metals. Interpretation of the structures was guided by recent band calculations. It was shown that the observed features reflect the behavior of large volumes of the Brillouin zone rather than well-defined singularities.
INTRODUCTION
The last few years have seen considerable
inter-est
in the electronic properties of the transitionmetals.
'
Numerous studies have been reportedwith the predictable outcome that our
understand-ing of the metals has increased dramatically. The contributions of optical studies have been
funda-mental.
For
many of the transition metals,first
generation optical experiments in which the
reflec-tivity was measured over
a
wide energy rangehave been completed,
'
'
and it has been shown that,while the low-energy optical properties (g v
s
10 eV) vary significantly with the filling of the d bands, the high-energy behavioris
very similar from metal to metal. The low-energy structurecan be attributed to transitions within the
hy-bridized sp and d bands within which the Fermi
level
lies.
With the horizontal progressionacross
the periodic table, the average energy of the d bands drops with
respect
tothe sp bands and theFermi level
rises
within them.'
Accompanying this, the width of the dbandfirst increases,
reaching
a
maximum with the Cr group, thende-creases
until, at the noble metals, the d bandsare
completely filled and lie ~2 eV belowE~.
Sim-ilarly, with progression from 3d to 4d to 5dmetals, the dbandwidth
increases.
Suchsyste-matics
are
particularly interesting, and theinter-play between theory and experiment has proved
mutually beneficial. Calculations of the optical
absorption from
first
principles (though generallywithout consideration ofwave functions and matrix
elements)
are
becoming more reliable.To better understand the low-energy features
ob-served in the
bcc
transition metals, we haveex-ploited thermoderivative techniques and measured the thermoderivative of the reflectance, the
ther-moreflectance hR/R. The technique
is
nota
newone, and it has seen wide application
for
the noblemetals,
"
semiconductors,"
and insulators,"
but few of the transition metals have been so studied."
We find, in the present study, that the derivative
spectra
oftransition metals donotpossess
the sharp and distinctive features observed whendealing with semiconductors since critical-point
contributions
are
minimal. Instead, the broadstructures clearly
arise
from large volumes ofk space. Further, since the interband absorption
persists
to very low photon energies and generally involves states of hybridized angular momentum, line shapes cannot be calculated without detailedwave functions and matrix elements.
EXPERIMENTAL TECHNIQUE
The thermoreflectance of V, Nb, and
Cr
has beenmeasured near 80and 340K between
0.
5 and 5 eV. The general techniques ofthermoderivativespectroscopy"'"
have been described in detailelsewhere and need be only briefly reviewed here.
The samples were opaque films evaporated by
electron-beam bombardment in vacua of
-10
'
Torr.
They were approximately 3&&25 mm, wereevaporated onto quartz substrates, and had 2—5-Q
electrical resistance.
After evaporation, eachwas quickly attached tothe tip of
a
cryofinger-sample-chamber, and the chamber was evacuated
(working pressure
-10
'
Torr).
The thermal excursion of
-1
K deg was providedby
a
unipolar 7-Hz square wave with-2-W
powerdissipation. Conventional light sources (W- iodine
460
J
H~ D~~
I
INCH,
CUL
ANDR.
RQSE
CLEAVER
I I I
14
„ochromator
(Leiss
lamp and
1000-(pbS cell from
0.
-WXe
arc),
monocr
.
5double-prism), anand detectors
t
3 eV EMI 6256B7102from 1to e
-in amplifier, an e
a
ted for 100
sec.
Daa
wh
asis
with the en 'P01
nature ofthe spec ru
dictated by the na u
d with a
thermo-ature was measure w'
ered adjacent o
couple In- solder
resents only an erature thus repr
measured temper
pie temperature. to the actual samp
approximation o
RESULTSAND DISCUSSION
ermoreflectance
spectra are
The measured thermore ec a
1-3.
Also shown 1n1
'k'
St
en from our
ear
ier
for Mo as taken
r
the temperature
ex-ectra scale
with emeasured spec
bR/R
are
shown ingn
cu,
a
itudes ofu
hitisun
er
arbitrary units, thoug ' '
r
w
arts
in10'.
Signgn p
ra are
complica eThe hR/R
spectra
of the dielectric
func-h
t
t
dypa
of
dt
1t't
is
the more fun amrd
tive of E» b&»
is
h
a
straightforwartermined throug
a
&R/R. S h
i
analysis ofle
set
of dielectricfunc-quires
a
reliablese
oanalysis requi
1)+ne,
](~R/R)-
[n(e,-1
—e,
66,
=2[0 e,
— +Plr
art
of 4R/R andw ' e causal counterpart o
1
t
ft
oob-com lex index o
r
'.
r
we have used dielec
ri
2$3 the analysis, we
easurements. '
tained in our previo us static me
2IOK I 8OK
~
f—340K(/)
Z
-80K
CL I0
CCIK
I I Ct p I Cl0
/ iJ540K
0
I I I
I
2
3
4
PHOTON ENERGY (eV)
fNb at approximatetely80
FIG.2. Thermore flectance o a
0.6eV represen ts 340K. The dashed curve below
rs-Kronig analys is.
e used for our
Kramers-the extrapolation use o
ers-Kronig analysis o
ofRE),
stI the usual Kram
is
well knowna
f
the extrapolationsa
on the nature o e e
ers-dependent up
th derivative Krame
high and low en
er
gy.. For
eis
muchless
1
sis
the dependenceis
the shapesof ~,
are
reli-severe.
' Ac cordingly, esre
were no der1vative data below' o gi
analysesfor Cr,
, anmers-Kron1g an
4and 5. A
sos
0
shown in
Figs.
1 the tempera
t
urre
modulat1on'
r
contribution too e
d
is
difficult to remove. low energy andis
i iderivative feature.
to obscure the lowest-energy eriv
1 t' ns have been Numerous recent band calcu
a1o
CA
I-z
2-K o Kl tr) I— 2—
Cr-80K——Cr-340K
CL p g
J
&-2 fta
I 2PHOTON ENERGY (eV)
-4—
I I
2 4
PHOTON ENERGY (eV)
t a roximately 80
fl tance ofVa pp
FIG.
1.
Ther more ecdetermined near
ctrum was also e
tdb
th; i ' at
1.
30eV is indica eK; its maximum a
Ref.
ceof Cr and Mo (from
14
THERMORE
FLECTANCE
OF
F V,
V NbNb,
AND PARAMAGNE
TIC
Cr
V)
2
0
ml I
~,
I ~ I ~i I IH 6 N Z I" A P D N D P F H
FIG.6. Ener
(Ref. 15). gy band
structure ofNb from Matthecss'
I I I I
I 2 5 4
PHOTON ENERGY (eV)
FIG,4. TemTemperature derivative of&2for Nb and V
CC
CQ
0
CV
0
I I I I
I P.
PHOTON ENERGY (eV)
FIG.5. TTemperature derivative of& for Cr and Mo.
performed' for V, Nb, and Mo, and these h
shown ener
an
ese
havegenerally good agreement on many of the important
features.
The details still remainde-pendent upon the choice of starting potential
(atom-ic
configuration) and the handling ofthein eraction. Relativistic effects in Nb and Mo
are
also important and tend to lift band degeneracies
a
ongcertai
1'Nb V and M
g n lines. Because of the sim'la1ar1
't'
les
1nofNb
, V, and Mo, we reproduce here only theb
ans
dNb as calculated by
Mattheiss" (Fig. 6).
Much ofthe theoretical interest in
Cr
has beenbecause of ituse o
i
s antiferromagnetic ordering(T
=312K
for
for bulkCr).
16 Recent calculations of thege1cp
aseof
Cr
band structure of the paramagn
t'
hthe
have been performed by Rath and C 11
e derivative
spectra
ofCr
mill be interpreted in terms of th
of-atomic-o ose self-consistent
linear-co
mb'1nat'
1on-o-a
omic-orbitals paramagnetic bands8(Fi
lg..
71sInore e owest-energy structure in V and Nb
the
prominently seen in nB/ft thann in'
6
g,because of efree
carrier
contribution to4e
.
U on co from 340too 800 K, the features sharpen and shift toslightly lower energy
(Figs.
1 and2}.
Themaxi-mum in Vshifts from
1.
33 (340K)to1.
30 (210K1.
67eto
1.
27 eV (80 K), and in Nb from1.
70(340K toeV(80
K}.
The other structures sh8 ow 1ess
epen ence on the ambient temperature. Th
ener
ure.
elow-gy der1vative feature corresponds to the
max-imum in
a,
at1.
6 eV (V)and2.
4 eV(Nb)or
in the81 seems
o the origin of the structure
as
Z-
Zst ll
reasonable. Harmon" has shown that the wavefunctions along the Z direction
are
of mixed andd
character
er,
witha
greater admixture ofp in the lower Z band. Sucha
m1x1ng~ ~ reduced the expectedweakness of
d-to-d
transitions. Further, thein-volvement of
states
near and atE t
ends to enhance the temperature dependence of eoe,
.
.
I
t
is
' reasonablea
e spreading ofthe dbandincreases
theen-ergy separation of the bands and moves the
struc-ture to higher energy for Nb than for V.
The energy bands of Cr and Mo have been shown to be very similar in the absence ofsp
in-or
b't
1p
i
1ng and magnetic ordering. In addition the have the samme general featuresas
those of V and1lonq ey
J.
H.
%'EAVER,
D.
W.
I
YNCHsC.
H.
CUI
P,
ANDR.
ROSE
I
-2
z
4J
FIG.7. Energy bands inparamagnetic Cr from Bath
and Calla@ay (Bef.17).
bands has an important influence on the low-energy
optical properties. The Ztransitions identified
as
being responsible for thefirst
structure in bR/Afor
V and Nbare still
possiblefor
Cr
and Mo,though they might be expected to appear at higher
energy and be weaker since the x'espective
Fermi
levels
are
no longer involved, but additionalstruc-ture due to transitions along
4
also come into play. The degeneracy of the6,
bands inCr is
lifted by relativistic effects in Mo and
%,
Rnd theoptical features associated with the
62(6,
}-&,
(4„&,
)transitionsreflect
the changes. InCr,
only
a
single featureis
observed inflor
AR/A;in Mo,
static
measurements failed to resolve the splitting ofthe initialstates,
'
but the derivativespectra
clearly showed the separation'(see Fig.
3);
in W, the spin-orbit splittingis
-0.
5eV, andstatic
measurements4 were able to resolvestruc-tures
in ARnd &,.
Hence, the assignment ofthefirst
structure inCr
as4,'
4,
seemssecure.
The identification ofthe second derivative
struc-ture in V and Nb and the third in
Cr
and Mois
dif-ficult, but large volumes ofk space
are
certainlyinvolved. Intheir xecent relativistic-augmented-plane-wave calculations for Mo, Koelling et gl.
"
concluded that the experimental feature in
z,
at4.
1eV was built up from transitions between bands alongI,
6,
and Z.Cr
would be expected to havean analogous structure in &» but the extrapolation
of those transitions to the
Cr
bands of Rath andCallaway
is
thwarted: Examination shows the band separation atE
to be about 2 eV, at Gto
be2.
6 eV, and along Zto be about2.
3eV. Few new "high-symmetry*' candidates present themselves. Transitions like P~—P,
are
allowed by selection rules, have large joint densities ofstates,
andwould probably have non-negligible matrix
ele-ments since
P,
wave functionsare
=100% d-likeand those at
P,
are
20%p-like,
80%d-like.
"
Nevertheless, though the energy difference
is
ap-proximately
correct
inCr(-3.
5 eV), itis
too great to account for the features in the otherbcc
metals,based upon existing energy bands
for
those metals.Probably, then, the
P
transitionsare
of minimalimportance. Transitions like
4,
,
—
b,,
can also be discounted by analogy to Mo and%' since, if suchtransitions were evident in
Cr,
they could beex-pected to
carry
over to Mo and %'and result inpRlx'8 ofstructul es& the sepRx'Rtlon ofwhich would
grow with the spin-orbit splitting. No such
fea-tures were observed.
The recent calculations of Pickett and Allen"
indicate that the d bands in the txansition metals
Nb and M0 tend to be flatter array from symmetry
lines than along them. Their calculations showed
that major contributions to the g, structure in Nb
near
4.
3 eV and in Mo near4.
1eVarose
from theregion around k = (—,
',
—,',
—,'}.
Though there have beenno calculations of
a
similar nature fox V andparamagnetic
Cr,
one might expect that theex-perimental features in
Cr
near3.
5 eV(4e,
)or3.
3ev (o),
and in Vnear2.
5ev
(ae,
)or 2.
35(&,)could be explained by transitions in that part of
the Brillouin zone. The consequence of the
Pic-kett-Allen calculations was that structures which
is
observed in the experimentalspectra
canarise
from off-symmetry parts of the zone.
It
is clear
from our measurements that the de-rivative features in the transition metals Rrebroad, thereby indicating that large volumes of
k space
are
contributing tothestructures.
Itis
also apparent that the last word
is
not inas far
as calculationsare
concerned, and identifications suchas
those put forth tentatively hereare
sub-ject
to improvement. This refining must comeafter
good agreement with experimental datais
achieved; such agreement must result from the inclusion ofmatrix elements in the calculations.
The derivative
spectra
reported here provide yet anotherset
of experimental data to which thetheorist must compare his calculations.
ACKNOWLEDGMENTS
The authors gratefully acknowledge
correspon-dence with
J.
F.
Alwardand%.
E.
Pickett.
We ax'e grateful tothemfor
sharing their resultsprior
to publication. The encouragement ofthe Directox'
ofthe Synchrotron Radiation Center,
E.
M. Rowe,THERMORE FLECTANC
EOF V,
Nb,
ANDPARAMAGNETIC
Cx'~Permanent address.
~Supported under NSFGrant No. DMB-7415089.
Gruppo Nazionale di Struttura della Materia (GNSM) of
&the Consiglio Nazionale delle Hicerche.
Present address: Interand Corporation, Chicago, Ill. 60611.
«For detailed reviews ofthe literature, see the
refer-ences cited in Befs.
2-8.
2J. H.%eaver, D.
%.
Lynch, and C.G. Olson, Phys.Bev.
8
7, 4311 (1973), Nb.3J.H.%eaver, D.
%.
Lynch, and C. G.Olson, Phys.Bev.
810,
501(1974),.V, Mo, Ta.4J.H.%eaver, C. G. Ol.son, and D.%.Lynch, Phys.
Bev.
812,
1293(1975), %.5J.H.Weaver, C. G. Olson, M.Piacentini, and D.%.
Lynch, Solid State Commun. 16, 163 (1975), Mo
thermoreflectance.
6D.W. Lynch, C. G. Olson, and
J.
H.Weaver, Phys.Hev.
8
ll,
3617(1975), Ti, Zr, and Hf.VJ.H. %'eaver, Phys. Rev.
8
11, 1416 (1975), Bh, Pd,Ir, Pt; and
J.
H.%eaver andB.
L.Benbow, i&id. 12, (1975), Pd.J.
H.%eaver and D.%.
Lynch, Phys. Hev.8
7, 4737(1973)and
J.
H.Weaver and C.G.Olson. (unpublished),Y.
~L. Hodges, R.E.%atson, and H. Ehrenreich, Phys.
Hev.
8
5, 3953(1972).«OB.Hosei and D.
%.
Lynch, Phys. Bev.8
5, 3883(1972), Al, Au, and Cu;
B.
Bosei, C. H. Culp, andJ.
H.Weaver, ibid. 10, 484 (1974),Ag; C.G. Olson, M. Piacentini, and D.%.Lynch, Phys. Hev. Lett. 33,644 (1974), Au; %.
J.
Scouler, ibid. 18, 455(1967),Au.See, for example, M. Cardona, Modulation
Spectr0s-eopy (Academic, New York, 1969);and B.Batz,
Sengi-conduetors and Semimetagg, edited by
B.
K.%illardsonand A. C.Beer(Academic, New York, 1972) Vol. 9,
p. 316.
«2J. Hanus,
J.
Feinleib, and%.
J.
Scouler, Phys. Bev. Lett. 19, 16 (1967), Ni; and D.%.Lynch,B.
Bosei,and
J.
H. Weaver, Solid State Commun. 9, 2195(1971),Ni. Seealso Ref. 5.
«3B.Bosei, Phys. Bev.
8
10, 501 (1974).'4A. Balzarotti,
E.
Colavita, S.Gentile, and B.Bosei, Appl. Opt. 14, 2412 (1975).«5L.
F.
Mattheiss, Phys. Bev.8
1, 373(1970).«6For results of optical measurements of
antiferromag-netic Cr and Cr alloys, see L.W. Bosand D.%.
Lynch, Phys. Bev.
8
2, 4567(1970);A. S.Barker andJ.
A.Ditzenberger, ibid. 1, 4378(1970); D.%.
Lynch,R.Bosei, and
J.
H.Weaver, Phys. Status Solidi A27, 515(1975).ath and
J
CaI,laway Phys Bev8
8 5398(1973) B.N. Harmon (private communication).«BD.D.Koelling,
F.
M. Mueller, and B.%.
Veal, Phys.Bev.
8
10, 1280(1974).0%.E.Pickett and