PHYSICAL REVIEW B 71, 134409 共2005兲

Finite-size effects on transverse magnetoresistance of NbSe3Yamaguchi Takahide, Shinya Uji, Kengo Enomoto, Takako Konoike, Taichi Terashima, and Mitsuka Nishimura
National Institute for Materials Science, Tsukuba 305-0003, Japan

Taku Tsuneta, Katsuhiko Inagaki, Satoshi Tanda, and Kazuhiko Yamaya
Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
共Received 17 November 2004; published 12 April 2005兲
A charge-density-wave material NbSe3 with thickness less than 1 µm shows a distinct peak in the lowtemperature transverse magnetoresistance when the magnetic field is applied along the c axis. The resistancepeak is less pronounced as the field is tilted away from the c axis to the a* axis. The peak height also decreases
as the temperature increases, although the magnetic field at the peak shows no appreciable dependence on
temperature. The resistance peak is attributed to surface diffuse scattering of normal carriers which remain
even below the lower charge-density-wave transition temperature.
DOI: 10.1103/PhysRevB.71.134409

PACS number共s兲: 75.70.⫺i, 72.15.Nj, 73.23.⫺b, 71.45.Lr

Niobium triselenide 共NbSe3兲 is a quasi-one-dimensional
conductor having a ribbon shape which extends along the baxis and has the wider transverse dimension along the c axis.
It exhibits two charge-density-wave 共CDW兲 transitions at
T1 = 144 K and T2 = 59 K, but remains metallic at T ⬍ T2.1 It
has a large electric conductivity along the b axis even underthe electric field E lower than the threshold ET for the CDW
sliding. Below about 10 K, the magnetoresistance exhibits
the Shubnikov-de Haas 共SdH兲 oscillation. Extensive studies
on the SdH oscillation for different orientations of the magnetic field have shown that there exist small pockets of theFermi surface in the shape of nearly an ellipsoid, indicatingthat the nesting of the Fermi surface is imperfect.2–6
Recently, interesting features in NbSe3 crystals with reduced cross sections have been reported. The threshold electric field ET increases with decreasing thickness and/or widthof the NbSe3 crystals.7,8 It is also found that the resistanceratio R共300 K兲 / R共4.2 K兲 decreases rapidly with decreasing
thickness below about 1 µm, which is attributed to the surface scattering of normal carriers at the low temperature.7
Measurements9 on NbSe3 crystals with cross-sectional areas
reduced to 1 ␮m2 by a plasma-etching process show that thephase slip voltage10 significantly decreases when the spacing
between current probes is smaller than a few µm. Furthermore, metal-oxide-semiconductor field-effect transistor
共MOSFET兲 like devices are fabricated of very thin NbSe3
crystals with a thickness of much less than 1 µm.11 Despite
the growing interest in electrical transport properties on
NbSe3 crystals with such small cross sections, systematic
magnetoresistance measurements of them are still lacking.
In this study, we have measured the low-temperaturetransverse magnetoresistance of NbSe3 crystals with a thickness of less than 1 µm and found that the magnetoresistancecurves have a maximum at a low magnetic field 共B ⬍ 1 T兲 in
addition to the SdH oscillation at B 艌 2 T. We show that theresistance maximum is attributed to the finite-size effect on
the ballistic transport of the normal carriers, namely, theirsurface diffuse scattering.
NbSe3 crystals are placed on silicon substrates and morethan four gold wires are attached to a crystal by silver paste.
1098-0121/2005/71共13兲/134409共4兲/$23.00

We took a great care in selecting crystals with well defined
facets using an optical microscope. Typical crystal dimensions are 0.2–0.6 µm along the a* axis, 1–3 µm along the caxis, and a few millimeters along the b axis. The crystal
dimensions are described in Table I. We measured the resistance using a standard ac lock-in technique with a small excitation current so that E ⬍ ET. The samples were cooled
down by a 3He or a 4He refrigerator equipped with a superconducting magnet.
Figure 1 shows typical magnetoresistance curves for athin crystal 共1兲 measured at several temperatures. We can see
the SdH oscillation at magnetic fields larger than about 2 T.
The dominant frequency of the oscillation is 28 T for themagnetic field parallel to the c axis. The angular dependenceof the oscillations was also measured, which showed that thefrequency increases monotonically to 95 T as the orientationof the magnetic field is rotated away from the c axis towards
a*. The values of the frequency indicate that the maximum
cross-sectional areas Saⴱb and Sbc of the Fermi surface in thea*b and bc planes are 2.7⫻ 1017 m−2 and 9.1⫻ 1017 m−2, respectively. The angular dependence of the frequency,
namely, that of the cross-sectional areas of the Fermi surface
is approximately the same as that obtained by othergroups.2–5 For B / / c, the carrier effective mass m* is estimated to be 0.20m0 共m0 is the free electron mass兲, and theTABLE I. Parameters of seven NbSe3 crystals. da* and dc are
the thickness along the a* axis and the width along the c axis,
respectively.

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Sampleda*共␮m兲dc共␮m兲R共300 K兲 / R共4.2 K兲

12
34
5
6
7

0.49
0.60
3.9
0.46
0.21
0.46
1.9

1.59
2.40
13.2
2.75
1.37
0.78
14.2

2966
110
64
256454©2005 The American Physical Society

PHYSICAL REVIEW B 71, 134409 共2005兲

TAKAHIDE et al.

FIG. 1. The magnetoresistance of a NbSe3 crystal with 0.49 µm
thickness 共sample 1兲 at several temperatures; T = 16.00 K, 12.00 K,10.00 K, 8.00 K, 6.00 K, 4.00 K, and 0.28 K from the top curve tothe bottom one. Inset: An enlargement of the low-magnetic-fieldregion. T = 8.00 K, 6.00 K, 4.00 K, and 0.28 K from top to bottom.
The arrows indicate the magnetic fields B = 0.55ប冑Saⴱb / ␲ / eda* and2.0ប冑Saⴱb / ␲ / eda*.

Dingle temperature TD is 1.5 K. These values are comparable
to m* of 0.30m0 共Ref. 2兲 and 0.24m0 共Ref. 3兲 and to TD of 1.7
K 共Ref. 2兲 and 2.3 K 共Ref. 3兲 in the literature. Besides theSdH oscillation at higher fields, there is a resistance maximum at a low field as shown in the inset of Fig. 1. The
magnetic field Bmax, where the resistance is at its maximum,shows no appreciable dependence on temperature, although
the zero-field resistance increases with increasing temperature.
Figure 2 shows magnetoresistance curves for two crystals
with different cross-sectional areas, which are measured at
the same time in the same cryostat. Two series of curves ineach figure correspond to two different sets of voltage
probes. The measured R共B兲 curves, in particular of the thicksample 共3兲, are not quite symmetric, most probably due tothe inclusion of Hall resistance. To remove the component of
the Hall resistance, we averaged the results for the two opposite polarities of the magnetic field. The R共B兲 curves forsample 3 have no peak structure and are similar for different
orientations of the magnetic field. On the other hand, theR共B兲 curves of the thin crystal 共4兲 have a clear maximum formagnetic-field directions close to the c axis. The peak structure becomes less evident as the field direction deviates fromthe c axis.
The resistance maximum can be explained by semiclassical theory treating surface diffuse scattering of carriers.
Within the Boltzmann equation approach, MacDonald andSarginson12 calculated the transverse magnetoresistivity ␳共B兲of a metal film with d / l 艋 1 共d is film thickness, and l is thebulk mean free path兲 under the condition that both the current and the magnetic field are in the plane of the film. Their

FIG. 2. The magnetoresistance of 共a兲 thick 共3兲 and 共b兲 thin 共4兲
crystals, measured at 0.28 K for the field orientations ␪ = 0°, 30°,60°, 90°, 120°, and 150°. The c axis corresponds to ␪ = 98° forsample 3 and ␪ = 36° for sample 4. The arrows indicate the magnetic
fields B = 0.55ប冑Saⴱb / ␲ / eda* and 2.0ប冑Saⴱb / ␲ / eda*.

calculation shows that the ␳共B兲 curve has a maximum. At
zero field, the resistivity is larger than the bulk resistivity
because of the surface diffuse scattering. The resistivity increases with increasing field as the carrier orbit is bent by theLorentz force and, consequently, the carriers are scattered
more frequently at the edges. At larger fields where the diameter of the cyclotron orbit is smaller than d, the resistivity
is close to the bulk value because the carriers do not reachthe edges unless they suffer internal collisions. As expected
intuitively, the maximum value of ␳ / ␳bulk is larger forsmaller d / l. An improved calculation13 shows that ␳共B兲
reaches the maximum when d / rC = 0.55, namely, B= 0.55បkF / ed, irrespective of the ratio d / l. Here, rC is thecyclotron radius and kF is the Fermi wave number. It alsoshows that the ␳共B兲 curve has a sharp bend at B

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PHYSICAL REVIEW B 71, 134409 共2005兲

FIG. 3. The magnetic fields Bmax and Bmin, where the magnetoresistance curves show a maximum and a minimum, plotted as afunction of ប冑Saⴱb / ␲ / eda*. The closed and open circles show Bmaxand Bmin, respectively. The solid lines have slopes of 0.55 and 2.0.

= 2.0បkF / ed, above which the resistivity approaches the bulk
value. Experimentally, such magnetoresistance peaks due todiffuse surface scattering of carriers have been observed inpure metal films14 and in narrow conducting channels of
two-dimensional electron gas.15
To compare the experimental results with the theory, it isnecessary to know the thicknesses d of the NbSe3 crystals.
We took pictures of the crystals by a scanning electron microscope 共SEM兲, typically about 16 pictures for each sample,
at different angles. The pictures were taken after the measurements to avoid possible damage caused by the electron
beam. The nominal error in length scale of the SEM pictures
is 10%. We plotted the widths of major facets seen in thepictures as a function of the angle, and fitted the data tocosine curves to determine the actual widths and directionsof the facets. The cross-sectional profiles estimated in thisway are depicted in the insets of Figs. 1 and 2, and 4. Someof the crystals do not have cross sections of simple parallelograms. For simplicity, we take the average thickness for such
crystals.
The arrows in Figs. 1 and 2 indicate the magnetic fields
B = 0.55ប冑Saⴱb / ␲ / eda* and 2.0ប冑Saⴱb / ␲ / eda*, where da* is
the thickness in the a* direction. The calculation in Ref. 13
assumes a spherical Fermi surface, but the Fermi surface ofNbSe3 is close to an ellipsoid in shape, as described above.
Since it is a formidable task to perform a calculation similar
to that in Ref. 13 by taking account of the shape of the Fermi
surface, we approximate kF as 冑Saⴱb / ␲ for B / / c. One cansee that the positions of the arrows are close to Bmax andBmin, where the resistance is at its maximum and minimum.
We plot Bmax and Bmin for seven crystals as a function ofប冑Saⴱb / ␲ / eda* in Fig. 3. The points for Bmax and Bmin lie
approximately on the straight lines with slopes of 0.55 and2.0, respectively. Note that the points showing Bmax = 0
共Bmin = 0兲 mean that the maximum 共minimum兲 is not observed in the R共B兲 curves of the crystals.
The absence of the resistance maximum and minimum forthe thick samples, 3 and 7, suggests that the bulk mean free

FIG. 4. The magnetoresistance of sample 6 at T = 1.72 K. This
sample has the narrowest width along the c axis. The upward arrows indicate the magnetic fields B = 0.55ប冑Saⴱb / ␲ / eda* and2.0ប冑Saⴱb / ␲ / eda*, while the downward arrows indicate
B = 0.55ប冑Sbc / ␲ / edc and 2.0ប冑Sbc / ␲ / edc.

path is shorter than the thickness da* of the samples.
McCarten et al.7 have estimated the bulk mean free pathalong the a* axis, la* to be 0.7 µm at 4.2 K from the dependence of the R共300 K兲 / R共4.2 K兲 on the crystal thickness inthe range 0.1–10 µm, assuming diffuse scattering at theedges. This value of la* explains the fact that the R共B兲 curvesof the samples with da* = 1.9 ␮m and 3.9 µm do not show amaximum if the quality of our samples is similar to theirs.
Note that the R共B兲 curve of the crystal 2 共da* = 0.60 ␮m兲shows only a minimum, but no maximum, which implies that
this crystal has a slightly shorter la*.
Another fact consistent with la* of 0.7 µm is that the resistance peak for the sample having da* of 0.49 µm is obscured above about 12 K 共Fig. 1兲. The bulk mean free path at
12 K is estimated to be about half of that at 4 K because theresistance of the thickest sample at 12 K is twice that at 4 K,
and the carrier concentration hardly varies in this temperature range according to the literature.6 Thus, the reduction of
the resistance peak with increasing temperature can be explained by decreasing la* less than da*.
The above discussion assumes that the surface scattering
is completely diffuse. However, there may be a certain probability p of specular scattering, which would not contributeto the magnetoresistance peak. Actually, the data of
McCarten et al. are also reasonably explained with p values
up to 0.8.7 The estimation of la* depends on the p value;la* ⬇ 10 ␮m 共at 4.2 K兲 for p = 0.8. If the specular scattering istaken into consideration, the magnetoresistance peak should
disappear when la* ⬍ da* / 共1 − p兲. From this point of view, ourdata can be also roughly explained with p of 0.8 and la* of 10µm at 4.2 K. A more reliable estimation of the p value needs
an accurate determination of la* from measurements on bulk
crystals.
The resistance peak has not been observed in the field
direction B / / a* as shown in Fig. 2. We cannot definitively

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TAKAHIDE et al.

account for this, but we presume that the bulk mean free pathalong the c axis, lc, is smaller than dc / 共1 − p兲, where dc is thewidth along the c axis. Figure 4 shows the magnetoresistanceof the sample 6, which has the narrowest width dc of 0.78 µm
along the c axis. The R共B兲 curve of this sample for the field
direction B / / a* exceptionally exhibits a kink around B⬇ 0.3 T. This feature implies that there is a small surfacescattering effect, and therefore lc is comparable to dc in thissample, i. e., ⬇1 ␮m, if p = 0. From the data of the Hall andSdH measurements of the bulk NbSe3 crystals, Ong6 has
obtained the carrier mobility ␮c of 1.8 m2 / Vs at T = 4.2 K
and a kc of 1.6⫻ 109 m−1. These values correspond to anlc共=បkc␮c / e兲 of 1.9 µm, which is consistent with the above
rough estimation. We should also note, however, that there
appears to be another kink around B ⬇ 1 T. Its origin is notclear at present.
The fact that the pocket of the Fermi surface has the shapeof an ellipsoid elongated to the c axis may also make theresistance peak for B / / c distinct, and that for B / / a* difficult
to be observed. This is because in the field B / / c, a larger

number of carriers have similar cyclotron radii, and hence
suffer the surface scattering most strongly at similar magnetic fields. In fact, the amplitude of the SdH oscillation for
B / / c is much larger than that for B / / a*, which can be understood in a similar way.
In summary, the transverse magnetoresistance curves ofNbSe3 crystals with a submicron thickness show a maximumat B ⬍ 1 T, in addition to the SdH oscillation at B 艌 2 T. The
resistance maximum can be explained by semiclassical theories in terms of surface diffuse scattering of the normal carriers. Further studies are in progress to investigate the finitesize effect on magnetoresistance in the nonohmic regime
E ⬎ E T.

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2The authors wish to thank Y. Ootuka for kindly allowing
the use of the SEM. This work was supported by a Grant-inAid for Scientific Research from the Ministry of Education,Culture, Sports, Science, and Technology 共Grant No.
15073225兲.

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