Electronic transport in MgB2, AlB2 and ZrB2 - A comparative study

Article abstract of DOI:10.1016/S0921-4534(02)02280-3

We report on the temperature dependence of the resistivity (ρ) and the absolute thermopower (S) of the polycrystalline title materials and of AlB₂ single crystals. For all samples ρ(T) exhibits a Bloch-Gru?neisen-like temperature dependence, with large characteristic temperatures θ_R (≈θ_D - the Debye temperature). At high temperatures the thermopower S(T) for ZrB₂ (n-type) is almost the mirror image of S(T) for MgB₂ (p-type) while S(AlB₂) is very small for all temperatures. The density of states distribution N(E) around E_F seems to play a dominant role in determining S(T) of these materials. ln(T) terms in the low-temperature ρ(T) and S(T) of ZrB₂ samples bear evidence for weak localization in 2D.

Full text of DOI:10.1016/S0921-4534(02)02280-3

Physica C 384 (2003) 1–10
www.elsevier.com/locate/physc
Electronic transport in MgB , AlB
2
2
and ZrB ––a comparative study
2
*
B. Fisher , K.B. Chashka, L. Patlagan, G.M. Reisner
Physics Department and Crown Center for Superconductivity, Technion, Haifa 32000, Israel
Received 24 July 2002; received in revised form 1 October 2002
Abstract
We report on the temperature dependence of the resistivity (q) and the absolute thermopower (S) of the polycrys-
talline title materials and of AlB
single crystals. For all samples qðTÞ exhibits a Bloch–Gr u€ neisen-like temperature
dependence, with large characteristic temperatures h ––the Debye temperature). At high temperatures the
(ꢀh
thermopower SðTÞ for ZrB (n-type) is almost the mirror image of SðTÞ for MgB (p-type) while S(AlB ) is very small
seems to play a dominant role in determining
2
R
D
2
2
2
for all temperatures. The density of states distribution NðEÞ around E
F
SðTÞ of these materials. lnðTÞ terms in the low-temperature qðTÞ and SðTÞ of ZrB
2
samples bear evidence for weak
localization in 2D.
ꢀ
2002 Elsevier Science B.V. All rights reserved.
PACS: 74.70.Ad; 72.15.Jf; 72.15.Rn; 71.20.Gj
Keywords: Superconductivity; Thermopower; Weaklocalization; Density of states
ꢀ
ꢁ
5
1
. Introduction
T
DqðTÞ ¼ q À q ¼ A
J ðh =T Þ
ð1Þ
0
5
R
h
R
Since the discovery of superconductivity in
MgB [1], many studies of its normal state trans-
where q is the residual resistivity, A is a constant
0
2
and h
to h
R
is a characteristic temperature, usually close
port properties have been carried out. Experi-
mental results [2–7] show that its resistivity qðTÞ
exhibits a Bloch–Gr u€ neisen-like (BG) temperature
dependence typical for phonon scattering in a
simple metal with a high Debye temperature, su-
perimposed on a constant (residual) resistivity q0.
The textbookBG formula is [8,9]:
D
––the Debye temperature deduced from spe-
5
R
x
0
z
Àz
cific heat; J ¼ z dz=½ðe À 1Þð1 À e ފ where
5
x ¼ T=h . In a generalized BG formula (BG(n))
R
the power 5 and J are replaced by n and J . At low
5
n
n
temperatures DqðT Þ / T . For high quality MgB
2
samples (with low q ) n is closer to 3 [3–7].
0
Therefore, in Refs. [3–5] the BG(3) formula was
fitted to the experimental qðTÞ. DqðT Þ expressed
by the BG(3) formula may be associated with in-
terband electron–phonon scattering [10]. The fitted
hR values for BG(3) are higher than those for
BG(5) and also larger than most hD values ob-
tained from heat capacity measurements.
*
Corresponding author. Tel.: +972-4-8293652; fax: +972-4-
221514.
E-mail address: phr06bf@physics.technion.ac.il (B. Fisher).
8
0
921-4534/02/$ - see front matter ꢀ 2002 Elsevier Science B.V. All rights reserved.
PII: S0921-4534(02)02280-3

2
B. Fisher et al. / Physica C 384 (2003) 1–10
A number of studies were devoted to the
thermopower of MgB [4,11–15]. The absolute
thermopower ðSÞ of MgB is positive for all tem-
peratures above T and increases quasilinearly with
T in the range 40 6 T 6 150 K; the extrapolated
value of S at T ¼ 0 has a small but finite negative
value and above ꢁ150 K S becomes nonlinear
and exhibits a tendency towards saturation. The
quasilinear SðTÞ was analyzed in terms of diffusion
the tensorÕs components were calculated instead of
using MottÕs approximation; an isotropic relax-
ation time, independent on energy, was assumed in
that workand a strong phonon drag contribution
was included. The calculated SðT Þ, representing an
average over all directions is in semiquantitative
agreement with the experimental SðTÞ and repro-
duces the main features of the dependence on
temperature and on Al-doping. This shows that in
2
2
c
thermopower ðS
D
Þ dominated by holes; the nega-
2
undoped MgB the contribution of the r bands to
tive intercept of S at T ¼ 0 was attributed to
phonon drag [11,15] and the deviation from lin-
earity at high temperature to electron-like states
adding a negative contribution at high tempera-
tures. In the earlier publications the linear portion
of SðTÞ was interpreted in terms of MottÕs formula
for diffusion thermopower of a degenerate electron
gas [8,9]:
S exceeds by far that of the p bands. SðTÞ in MgB
2
may be regarded as fairly simple in spite of the
materialÕs complex electronic structure and com-
plications mentioned below:
• The scattering mechanisms dominating trans-
2
port in each of the bands of MgB are presently
under debate [18,20–22]. The role of the elec-
tron–phonon interaction in the thermopower
ꢂ
ꢃ
2
2
B
2
2
p k T o ln rðEÞ
p k T
B
S
D
¼ À 3
¼ À
n
ð2Þ
D
of metals at T < h , is not trivial and has been
jej
oE
3 jejE
F
E¼EF
the subject of many studies that did not yet lead
to a consensus [23,24]. MgB (and many other
where rðEÞ, would be the conductivity measured
2
diborides) have very high Debye temperatures,
typically between 500 and 1000 K [25], thus
the interesting regime for their thermopower is
for E
F
at E, n ¼ ½o ln ðrðEÞÞ=o ln EŠ and E
F
is
EF
measured from the band edge. For spherical Fermi
surfaces n ¼ 1 for scattering by impurities (con-
stant mean-free-path) and ¼3 for quasi-elastic
phonon scattering [8,9]. For holes, the corre-
sponding values of n are negative. In metals with
more complicated Fermi surfaces n deviates
strongly from these ideal values and its sign may
D
T < h where the phonon scattering is inelastic.
•
All measurements reported so far have been car-
ried out on polycrystalline samples. In such
samples it is difficult to distinguish between the
contributions of impurity scattering in the bulk
and grain-boundary resistance to the residual
resistivity (thermopower is insensitive to grain-
boundaries). Moreover, a material formed of
anisotropic, randomly oriented crystallites ap-
pears as if it is multiphase. The thermopower
of such a system may be affected by the anisot-
ropy of q, of the heat conductivity and of S [26].
1
be unrelated to the type of the carriers. The
2
Fermi energy of carriers in MgB , estimated [11]
by fitting MottÕs formula with n ¼ À1 to the linear
portion of SðTÞ was found in fair agreement with
that obtained from band structure calculations.
Good agreement between the fitted E and band
F
calculation is claimed also in Ref. [15], although
different parameters were employed in the latter
The solution of the theoretical problems as well
as the growth of MgB single crystals (SCs) (with
interpretation. MgB
1
2
is a multi-band system [16–
9] with complicated Fermi surfaces. In Ref. [4]
2
controlled impurity content) may take some time.
At this stage, it was interesting to find out if the
the Seebecktensor S
pairs of bands (r and p) that contribute to the
conductivity of MgB . The general expressions for
D
was calculated using the two
relative simplicity of SðTÞ for MgB
2
is accidental
2
or if it is common among the diborides. In par-
ticular we looked for a correlation between the
density of states (DOS) distribution function NðEÞ
which is known for many diborides and SðT Þ. For
spherical equi-energy surfaces and elastic scatter-
1
Even the sign of S for Cu, Ag and Au (which are simpler
than MgB2 in all respects) was not understood for many
decades.

B. Fisher et al. / Physica C 384 (2003) 1–10
3
ing this correlation is straightforward, but for real
metals, even elemental and monovalent, it is not.
tion [36]. Early [35,37] and recent band structure
calculations [38–40] show that its E lies in a
‘‘pseudogap’’; close inspection of the figures of
Refs. [35,38–40], reveals that E lies slightly above
F
Except for MgB
gonal structure are either non-superconducting or
low-T superconductors [27]. Although many have
2
, all known diborides of hexa-
F
c
the dip, that is, where dNðEÞ=dE > 0. Supercon-
been investigated in the past, we found in the lit-
erature detailed results of SðTÞ only for TiB [28].
ductivity at 5.5 K was recently reported for a
sintered ZrB sample [41] but it was later attri-
buted to a minority phase [42]. According to
2
2
It is more complicated than SðTÞ for MgB , ex-
2
hibiting a sign change from p-type at low-T, to n-
type at high-T, and rather poor reproducibility
Ref. [39], the low value of NðE
electron–phonon coupling (obtained from specific-
heat measurements) seem inconsistent with T
5:5 K. In our investigation we found marks of
superconductivity in a porous ZrB sample sin-
F
Þ and the weak
from sample to sample. E
F
of TiB
2
lies at a mini-
c
¼
mum of the DOS, in a ‘‘pseudogap’’ [25,29]. Our
choices of diborides for electronic transport mea-
surements were AlB and ZrB . Following Mg
2
2
2
tered at low temperature. The resistivity and ther-
mopower of samples sintered at high temperatures
exhibit logarithmic temperature dependencies,
typical for weaklocalization in 2D.
which is divalent, Al is trivalent and Zr––tetrava-
lent. As summarized below, these materials are
interesting for additional reasons.
Thirty years ago it has been pointed out that
although AlB is the prototype structure for many
We tried to grow crystals of all three materials;
2
2
so far only some of our AlB SCs have reasonable
diborides, nothing has been reported on its elec-
trical properties [30]. Room temperature (RT)
resistivity and Hall effect measurements were re-
ported in that work; the RT Hall constants of
several samples were p-type and consistent with
p 6 1 hole/unit cell. It seems that since then not
much has been published on this topic. X-ray
spectroscopy results published before [31] and af-
ter [32,33] the discovery of superconductivity in
dimensions for resistivity and thermopower mea-
surements. It will be shown however that the
thermopower of polycrystalline AlB
that of AlB SCs. The results for our MgB samples
2
is close to
2
2
agree with the earlier results and confirm the weak
sensitivity of its SðT Þ to microstructure. The ap-
parent systematics of SðTÞ as function of valence is
one of the significant findings of this work: while
the SðTÞ graphs for ZrB
almost mirror images of those for MgB
(p-type), S(AlB ) is very low at all temperatures.
2
samples (n-type) are
MgB
lies in a pseudogap. This is in contrast with the
2
show that as in the case of TiB
2
, E
F
of AlB
2
2
samples
2
large DOS at E in MgB . Polarization-dependent
F
2
X-ray emission and absorption in SCs of AlB2
showed that E lies well above the r band. Al can
substitute Mg to form Mg1ÀxAl in a wide range
of concentrations [34]. This substitution leads first
to a decrease in T and then to the disappearance
of superconductivity, consistent with expectations
from electronic structure calculations [16]. For low
values of x the thermopower increases with x [11];
F
2. Experimental
2.1. Sample preparation and characterization
MgB . Polycrystalline MgB was prepared via
x
B
2
c
2
2
various protocols. Here we report on two very
different types of MgB : (a) non-annealed (NA)––
2
this is consistent with the shift of E towards the r
band-edge.
formed by RT compression of Alpha-Aesar MgB2
powder and (b) hot pressed (HP)––annealed for
several hours at 1000 ꢁC under a pressure of 20
F
2
ZrB is remarkable: it is a hard, refractory
2
material, its melting point (ꢁ3000 ꢁC) is much
higher than that of its constituents, and it is a very
good conductor although its electronic structure is
not that of a metal [35]. The Hall constants ob-
tained in the sixties were consistent with a very
low, temperature independent electron concentra-
MPa.
2
The hot-pressed annealing of MgB2 and sintering of ZrB2
at high temperatures has been carried out at the Ceramic and
Silicate Institute, Technion City.

4
B. Fisher et al. / Physica C 384 (2003) 1–10
ZrB . One set of samples was prepared by the
standard solid state synthesis from the elements in
stoichiometric ratio. The bar-shaped samples were
2
closed cycle refrigerator or on a cold finger of a
He cryostat. The thermopower measurements on
polycrystalline samples were carried out in a He
cryostat from 5 K up to RT. In a separate system,
that may be adapted to small and fragile samples
annealed twice at 1500 ꢁC in flowing Ar5%H
2
with
intermediate grinding. They were very porous and
brittle but on one of them (sample S1) we were
able to carry out transport measurements. Rigid
samples were obtained by sintering of pressed
2
and can be heated from liquid N temperature up
to 400 K, we measured the thermopower of AlB2
SCs and of the polycrystalline samples (the latter
above RT). The thermal emf was measured be-
tween the copper wires of two copper-constantan
thermocouples with their junctions embedded in
the copper heat sinks bridged by the samples; S
was obtained by adding S(Cu) to the measured
thermal emf. For some unknown reason, it was
very difficult to obtain good thermal contacts to
2
powder at 1800 ꢁC in Ar atmosphere for 1 h. The
raw materials for the high temperature sintering
were (a) the powder prepared before from the el-
ements (sample S2) and (b) commercial (Art) ZrB
2
powder with a small boron deficiency (of 0.23% of
the total mass) and carbon and oxygen impurities
(
0.6% and 1.4% of the total mass, respectively). We
attempted to grow SCs by slow cooling of molten
aluminum solutions of ZrB [43]. The dominant
the ZrB samples in the system for high tempera-
2
2
ture thermopower measurements. Good thermal
contacts were obtained once for sample S2 but
above 340 K they were lost and the measurements
became irreproducible.
products of two such attempts were mm size
rectangular platelets later identified via X-ray dif-
fraction (XRD) as Al
mm size hexagonal crystallites were later identified
as ZrB , however these were too small for trans-
3
Zr SCs. A minority of sub-
2
AC susceptibility of ZrB was measured in a
2
home-built susceptometer based on Oxford Te-
slatron, with a frequency of 1 KHz.
port measurements.
AlB . Sub-mm up to mm size, ꢁ10 lm thickSCs
2
were grown by slow cooling of molten aluminum
solutions, following a protocol similar to those of
Refs. [30,33]. Slightly larger and thicker (20–40 lm
thick) crystals were obtained (unintentionally) as a
by-product of hexaboridesÕ SC growth [44]. It is
3. Experimental results
3.1. Temperature dependence of resistivity
easier to grow SCs of AlB
2
from molten aluminum
solution than to sinter pellets of high purity pow-
For all samples investigated in this work qðTÞ
exhibited a BG-like temperature dependence. We
analyzed the experimental traces in terms of the
textbookBG formula with n ¼ 5 (see Eq. (1)). The
der. This is due to the decomposition of AlB into
2
AlB12 þ Al in the absence of excess Al [45]. Here we
report on a NA, polycrystalline sample––formed
by RT compression of AlB powder (Aldrich 99%);
it was relatively rigid and robust.
R
fitting parameters q , A and h for each sample are
0
2
given in Table 1. We also added the qðRTÞ=q
0
data. Values of h
literature, are shown in the last column of Table 1.
(No datum was found for AlB ). The reduced re-
D 2 2
for MgB and ZrB , from the
The XRD patterns for all polycrystalline sam-
ples show no foreign phases except for traces of
MgO in the HP MgB samples and of ZrO in
2
sistivities of all samples DqðTÞ=Dqðh Þ are plotted
2
2
R
ZrB2 S1. The diffraction pattern of the ZrB2
samples grown at 1800 ꢁC showed no traces of
impurities.
as function of the reduced temperature T =h , in
R
Fig. 1. All data fall close to the solid line which
represents the theoretical BG function. Since for
all samples q is much larger than Dq(T < 100 K)
0
2.2. Resistivity, thermopower and AC susceptibility
measurements
the fit is biased by the high temperature data.
Both MgB
2
HP and NA samples were super-
conducting with T
c
¼ 38:3 and 36 K, respectively
The resistivity of bar-shaped samples was mea-
sured by the standard four-probe method in a
in spite of their different resistivities: q ðNAÞ=
0
q ðHPÞ ꢁ 140 (!) and AðNAÞ=AðHPÞ ¼ 8. The
0

B. Fisher et al. / Physica C 384 (2003) 1–10
5
Table 1
Fitting parameters of the BG formula to qðTÞ for various samples of MgB
2
, AlB
2 2
and ZrB
Sample
Type
Relative density qðRTÞ=q0
q0 (X cm)
A (X cm)
h
R
(K)
D
h (K)
À5
À3
À4
À3
a
MgB
MgB
2
HP
NA
Hot pressed
NA
0.8
0.8
3.0
1.2
1:12 ꢂ 10
4:09 ꢂ 10
862 ꢃ 18
797 ꢃ 13
750–1050
2
1:54 ꢂ 10
3:16 ꢂ 10
b
À4
À4
À4
À3
À4
À3
À4
À3
AlB
AlB
AlB
AlB
2
2
2
2
SC1
SC2
SC3
NA
Single crystal
Single crystal
Single crystal
NA
1.4
1.4
1.4
1.1
1:89 ꢂ 10
1:41 ꢂ 10
1:35 ꢂ 10
2:11 ꢂ 10
6:20 ꢂ 10À4
7:71 ꢂ 10À6
1:05 ꢂ 10À5
9:63 ꢂ 10
1:11 ꢂ 10
6:20 ꢂ 10
2:17 ꢂ 10
5:67 ꢂ 10À3
2:70 ꢂ 10À4
1:49 ꢂ 10À4
749 ꢃ 14
956 ꢃ 13
706 ꢃ 16
524 ꢃ 20
b
c
0.65
0.55
d
ZrB
ZrB
ZrB
a
2
2
2
S1
S2
S3
Sintered
1.7
3.9
2.2
712 ꢃ 16
666 ꢃ 4
678 ꢃ 8
585, 630
(
1500 ꢁC)
Sintered
1800 ꢁC)
Sintered
1800 ꢁC)
0.65
0.72
(
(
See Refs. [4,48] and references therein.
b
c
Obtained unintentionally during growth of CaB
6
following the protocol in Ref. [44].
Obtained following the protocol in Ref. [30].
See Tables 3 and 4 in [25] and references therein.
d
2
The RT resistivities of our AlB SCs are larger
than those cited in Ref. [30] by factors of 3–9.
For all three SCs, qðRTÞ=q ¼ 1:4; their resistivity
0
seems to be dominated by impurity scattering. For
the porous, NA sample this ratio is only 1.1. In
this case, a large part of q is probably due to grain
0
boundaries. The values of the fitted h
over a range that is much wider than those for
MgB or ZrB . This is in spite of the fact that
R
are spread
2
2
qðRTÞ=q for the MgB NA and AlB NA samples
0
2
2
are comparable and the porosity of the ZrB S1
2
sample is larger than that of the AlB NA sample.
2
We looked therefore for a process that may com-
pete with electron–phonon scattering leading to a
BG type qðTÞ. Such a process is the scattering
from vibrating impurities. This mechanism known
as ‘‘electron–phonon–impurity-interference’’ was
analyzed in detail by Reizer and Sergeev [46]. Its
signature is the behavior of qðTÞ at temperatures
Fig. 1. DqðTÞ=Dqðh
R
Þ versus T=h
R 2 2
for various MgB , AlB and
ZrB samples. The solid line represents BG function (Eq. (1)).
2
The fitting parameters for each sample are shown in Table 1.
2
far below h : Dq ¼ q À q ¼ q BT where B is a
D
Int
0
0
large values of q indicate that at least for the
constant. The reduced resistivities DqðTÞ=Dqðh Þ
R
0
NA sample the resistivity is dominated by grain-
boundaries. The fairly large ratio of AðNAÞ=
AðHPÞ indicates that a large part of the grain
boundaries of the NA sample probably blocks the
current. The values of hR for these two samples
calculated from the BG formulae BG(5) (Eq. (1))
and BG(3), and the Reizer and Sergeev (RS) for-
mula [46] are plotted as function of the reduced
R
temperature T=h , in Fig. 2(a). It can be seen that
5
3
up to T ꢀ 0:1h
, DqðTÞ / T and T according to
R
2
(
and of additional samples investigated by us) are
the BG formulae and / T according to the RS
formula; with increasing T the slope drops grad-
ually towards 1 for the first two cases and 0 for the
close and fall within the range of hD values ob-
tained from lattice measurements.

6
B. Fisher et al. / Physica C 384 (2003) 1–10
ZrB
2
sintered at 1500 ꢁC was very porous and
brittle. However it was possible to carry out
both resistivity and thermopower measurements
on such a sample. Upon increasing the sintering
temperature of ZrB
2
from 1500 to 1800 ꢁC, q0
dropped by about two orders of magnitude and
A––by more than one. High temperature sintering
heals most of the grain boundaries, as hot pressing
2
does in the case of MgB . The material sintered at
the higher temperature is very rigid although its
density is much lower than the ideal density (see
relative densities of samples S2 and S3 in Table 1).
2
The resistivities of ZrB samples S2 and S3 are
larger but of the same order of magnitude as that
reported in Ref. [41]. qðTÞ of these samples ex-
hibited a shallow minimum around 40 K which is
barely resolved from the noise in the case of S2.
No drop in qðTÞ, that could be associated with
transition to superconductivity, was detected in
this sample down to T ¼ 4:75 K. The clear mini-
mum of qðTÞ of sample S3 seen in Fig. 3 was ob-
tained under very slow cooling. Attempts were
made to fit several functions to the regime where
the temperature coefficient of resistivity is nega-
Fig. 2. (a) DqðTÞ=Dqðh
R
Þ versus T=h
R
given by the BG func-
tions with n ¼ 5 and 3 and the RS functions, (b) q versus T for
sample AlB
2
NA, with the BG(5) and RS functions fitted to the
for the
plot of the experimental data. Note that the values of h
R
fitted BG(5) and RS functions differ by more than a factor of
two.
tive. The best fit to the experimental data for
2
7
6 T 6 40 K, and a very good one ðR ¼ 0:997Þ,
was obtained for the logarithmic function. This
behavior is the signature of weaklocalization in
2D [47]. The coefficient of lnðTÞ seems to support
third. Around T=h
/
D
¼ 1=3 the BG ðn ¼ 5ÞDqðT Þ
T ; if the temperature dependence of the resis-
tivity is minor due to a large q , and h is un-
2
2
À5
this possibility: Drsquare ¼ cDq=q ꢀ 8 ꢂ 10 lnðTÞ
À1
ꢀ
X , where c (¼ 3:53 A) is the lattice parameter
0
D
known, it is hard to distinguish between the BG
and RS temperature dependencies (or their com-
bination). The analysis of qðTÞ for the AlB NA
2
sample in terms of the two alternative formulae is
a good example for this uncertainty. Fig. 2(b)
shows the two lines fitted to the experimental plot.
R
The fitted h value for the RS formula is more
than twice larger than that for the BG formula.
Dq(T < 100 K) for the three SCs was plotted as
2
5
function of T and as function of T and linear
2
functions were fitted to the plots. The R of the
lines fitted to the Dq versus T plots were higher
2
but not enough to prove that the low-T resistivity
is governed by the RS mechanism. The spread of
the h
mechanisms probably compete at low tempera-
tures; this implies that h (AlB ) is about 1000 K.
R
values among the SCs indicates that the two
2
Fig. 3. The shallow minimum in q versus T for sample ZrB S3
at low temperatures. The solid line represents the fitted loga-
rithmic function that indicates weaklocalization (see text).
R
2

B. Fisher et al. / Physica C 384 (2003) 1–10
7
of ZrB
2
; the coefficient of lnðTÞ in Drsquare is
2
very close to the value of G
0
¼ 2e =h (¼ 7:75 ꢂ
À5
À1
1
0
X ), of the order of the theoretical coeffi-
cient. The origin of the weakdisorder in S2 and S3
may be carbon impurities from the graphite cru-
cible or deviations from stoichiometry. As shown
below, the thermopower of the ZrB samples S2
2
and S3 provides further support for weaklocal-
ization. The shallow minimum discussed above
had a negligible effect on the fitting parameters of
the BG formula. It can be seen that h
from sample to sample; it is much larger than that
found in Ref. [41] but closer to h (see Table 1).
R
varies little
D
A narrow step was observed around 5.4 K in
the AC magnetic susceptibility of sample S1; it
was about 0.3% of the step found (around 87 K)
for an YBCO sample of comparable volume. A
similar step was observed around 3 K for the
unprocessed commercial powder. AC susceptibil-
ity showed no marks of a transition to super-
conductivity in S2 and S3 and in the powder
synthesized by us.
Fig. 4. S versus T for samples of MgB
2
, AlB
2
and ZrB
2
on
which resistivity was measured (see Fig. 1 and Table 1). The
solid line that passes through data points for the MgB HP
sample represents S ¼ À1 þ 0:034T lV/K and the two solid
lines that pass through data points for ZrB S1 and S2 represent
S ¼ À0:025T and S ¼ À0:032T lV/K, respectively. Low tem-
S1 is shown on an expanded scale in the
2
2
3
.2. Temperature dependence of thermopower
, AlB and ZrB
2
perature SðTÞ for ZrB
2
inset. The low temperature range of SðTÞ for S2 is replotted in
Fig. 4 represents SðT Þ for MgB
2
2
Fig. 5.
samples on which qðTÞ was measured. The most
prominent feature of this figure is that SðTÞ for
MgB and ZrB
as mirror images of one another with SðTÞ of AlB
2
2
lie on both sides of S ¼ 0 almost
fact that the residual resistivities of these two
samples are different by more than two orders of
magnitude. This emphasizes the weaksensitivity of
the thermopower to grain-boundaries and poro-
sity. The difference between SðTÞ for the HP and
NA samples is larger at high temperatures where
SðT Þ is non-linear and tends to saturate.
2
lying in the middle, with almost zero values for all
temperatures.
Looking for systematics in SðTÞ for these ma-
terials was rewarding. The details, specific for each
system, will be discussed prior to drawing con-
clusions on the systematics.
Up to RT, SðT Þ for the AlB
2
, NA and SC
In general there is good agreement between
samples, lie within ꢃ1 lV/K. Above RT, S for the
polycrystal remains zero and that of the SCs in-
creases up to 1.5 and 2.5 lV/K. As in the case of
SðTÞ of the MgB
2
HP sample and most of the
earlier results in the overlapping ranges of tem-
perature. The solid line through the low-T data
points represents S ¼ À1 þ 0:034T lV/K. In Refs.
MgB
S(AlB
that at high temperature, the sign of S for the AlB
, the microstructure (or composition) affects
2
2
) mainly at high temperatures. It is notable
[
4,11,13–15] the RT values of S range between 7
2
and 8.7 lV/K, those of dS=dT in the linear regime
range between 0.035 and 0.042 lV/K . There is
also agreement on the negative extrapolated value
SC samples ðþÞ is the same as that of the RT Hall
constant [30]. NðEÞ in the pseudogap is probably
non-symmetric. With increasing temperature the
2
of S at T ¼ 0. The SðTÞ data for the NA MgB
2
increasing DE window around E
more hole-like than electron-like states leading to
positive S and R
F
may contain
sample are very close to those of the HP sample in
the regime where SðTÞ is linear; this, in spite of the
.
H

8
B. Fisher et al. / Physica C 384 (2003) 1–10
SðT Þ for the ZrB
2
S1 (sintered at 1500 ꢁC) and
S2 (sintered at 1800 ꢁC) are quasilinear over wide
ranges of temperature. The straight lines that pass
through the data points in this region represent
S ¼ À0:025T lV/K and S ¼ À0:032T lV/K, for S1
and S2, respectively. The reproducibility of SðT Þ
for samples prepared differently is poorer than for
MgB . Above ꢁ150 K for S1 and ꢁ250 K for S2,
2
the corresponding SðTÞ become nonlinear exhib-
iting a tendency towards saturation, very similar to
that of S(MgB
2
). SðT Þ of S1 at low temperatures is
shown on an expanded scale in the inset of Fig. 4.
A minute but reproducible deviation from the
straight line is observed around 20 K; it resembles
what one would expect for phonon drag thermo-
power in a dirty metal. At 5 K S ꢀ À0:01 lV/K,
i.e. zero within the experimental uncertainty as
expected for a superconductor. However, a thin,
percolating superconducting filament could ac-
count for this result.
Fig. 5. S versus T for the nonsuperconducting ZrB
samples plotted on a linear-log scale. q versus T for sample
ZrB S3 plotted on linear-log scale is shown in the inset.
2
S2 and S3
2
NðEÞ [9]. Suppose that this relation, with NðEÞ
2
and the diffusivity D ð/ v ðEÞsðEÞÞ replaced by
averages over equi-energy surfaces may be ex-
tended to our systems. The question ‘‘which of
the two terms d ln NðEÞ=dEj and d ln DðEÞ=dEj
Below ꢁ50 K, SðT Þ for ZrB
2
S2 deviates from
the straight line, crosses the S ¼ 0 line at T ¼ 38 K
and increases steeply with further decrease of T .
This behavior was exhibited by several samples
sintered at 1800 ꢁC and accompanies their qðT Þ
bearing the signature of weaklocalization. There
have been conflicting predictions for the thermo-
power in the weak-localization regime; recent re-
sults suggest a lnð1=T Þ dependence of S [49]. SðT Þ
for samples S2 and S3 below 40 K is plotted as
function of temperature on a linear-log scale in
Fig. 5. At least for S3 the logarithmic function
EF
EF
dominates d ln rðEÞ=dEj ’’ might be posed. The
EF
answer ðd ln NðEÞ=dEj Þ is simple in the case of
EF
AlB
values of dS=dT for MgB
2
for which dN=dEj ꢀ 0 and S ꢀ 0. The
EF
2
HP and ZrB
yield: d ln rðEÞ=dEEF ꢀ À1:4, þ1.0 and þ1.3 eV ,
2
S1 and S2
À1
respectively. Within the limited resolution of the
we checked if
available NðEÞ graphs around E
F
À1
ð1=NÞdN=dEj can reach )1.4 and þ1.3 eV , for
EF
MgB and ZrB , respectively. In Fig. 2 of Ref. [16]
2
2
NðEÞ of MgB has a steep negative slope around
2
2
fits remarkably well the experimental plot ðR ¼
E
F
where NðEÞ ꢁ 0:7; inspection of that graph
0
:998Þ. The inset of Fig. 5 shows qðTÞ for S3 (shown
À2
shows that dN=dE could be )1 eV . In Fig. 3 of
before in Fig. 3), replotted on a linear-log scale. To
our knowledge this is a rare example of evidence
for weaklocalization in 2D, based on both qðT Þ
and SðTÞ dependencies.
Ref. [38] NðEÞ of ZrB
2
has a steep positive slope
around E
F
where NðEÞ ꢁ 0:3; according to this
À2
figure dN=dE
E
F
could be þ0.4 eV ; however,
dN=dEEF in Fig. 1 of Ref. [39] seems larger. In spite
We now return to the gross features of SðTÞ and
show that these seem to be correlated with
of this uncertainty it seems that ð1=NÞdN=dEj
EF
plays an important role in SðTÞ of all three mate-
dNðEÞ=dEj . Inspection of the available NðEÞ
EF
rials.
graphs shows that dNðEÞ=dEj is negative for
EF
2 2
MgB [16,32,40,50], positive for ZrB [35,38–40]
and ꢁ0 for AlB
2
[31–33]. For spherical equi-energy
surfaces and elastic scattering this correlation is
2
4. Summary and final remarks
straightforward since rðEÞ / v ðEÞsðEÞNðEÞ where
vðEÞ is the energy dependent velocity, and the de-
Resistivity and thermopower measurements
were carried out on polycrystalline samples of the
2
rivative of v ðEÞsðEÞ has the same sign as that of

B. Fisher et al. / Physica C 384 (2003) 1–10
9
three title materials and on SCs of AlB
polycrystalline samples were either hot-pressed
2
. The
that for AlB
small at all temperatures.
2
dNðEÞ=dEj ꢁ 0 and SðEÞ is very
EF
(
MgB
2
HP), sintered at various temperatures
S1–S3) or just compacted at RT (MgB NA
NA). Single crystal growth was at-
tempted for all three materials but only the AlB
The results of this worksuggest that measuring
transport properties of additional diborides may
be rewarding. The first choices would be materials
for which calculated and/or experimental NðEÞ
(
ZrB
2
2
and AlB
2
2
crystals were large enough for both resistivity and
thermopower measurements.
around E results are available. It would also be
F
interesting to investigate diborides for which there
are conflicting reports about superconductivity.
For all samples qðTÞ exhibited BG-like tem-
perature dependencies. Fitting the textbookfor-
mula to the qðTÞ graphs for MgB
2 2
and ZrB gave
characteristic temperatures ðh Þ that vary little
R
Acknowledgements
from sample to sample and fall close to the re-
spective Debye temperatures obtained from ther-
We are grateful to Mrs. G. Bazalitsky for pro-
gramming the curve fitting of the BG and RS
functions. Helpful discussions with V.A. Gasparov
are gratefully acknowledged. This research was
supported by the Center of Absorption in Science,
Ministry of Immigrant Absorption, by the Pos-
nansky Research Fund for HTSC, and by the JCG
Research Fund.
mal measurements. Larger h values are obtained
from fitting to the experimental plots a generalized
R
BG function (BG(3)). A widespread of h
was obtained for the AlB samples. Using the case
of AlB NA sample having the largest q0 and
lowest fitted h , it was shown that the widespread
of h values among the AlB samples may be due
R
values
2
2
R
R
2
to competition between phonon scattering and
scattering from vibrating impurities [46].
MgB2 samples with resistivities different by
more than two orders of magnitude exhibited
superconductivity below only slightly different
transition temperatures. Marks of low-T super-
conductivity (at ꢁ5.5 K) were obtained in the AC
susceptibility and thermopower measurements in
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