J Low Temp Phys (2016) 183:41–49
DOI 10.1007/s10909-016-1502-3

Chiral Unitary Quantum Phase Transitionin 2 H-Fe x TaSe2
Takuya Kanno1 · Takuya Matsumoto1 ·Koichi Ichimura1,2 · Toru Matsuura1,2 ·Satoshi Tanda1,2

Received: 18 August 2015 / Accepted: 12 January 2016 / Published online: 25 January 2016
© Springer Science+Business Media New York 2016

Abstract We have observed a metal-insulator transition of a quasi-two-dimensional
electronic system in transition metal dichalcogenide 2H -TaSe2 caused by doping
iron. The sheet resistance of 2H -Fex TaSe2 (0 ≤ x ≤ 0.120) single crystals rises
about 106 times with the increasing of x at the lowest temperature. We investigated
the temperature dependence of the resistance and found a metal-insulator transition
with a critical sheet resistance 11.7 ± 5.4 k. The critical exponent of the localization
length ν is estimated 0.31 ± 0.18. The values of the critical sheet resistance and ν
are accordant to those of the chiral unitary class (less than h/1.49e2 = 17.3 k and
0.35 ± 0.03, respectively). We suggest that 2H -Fex TaSe2 is classified as the chiralunitary class, not as standard unitary class.
Keywords Conductivity of disordered solids · Conductivity of transition metalcompounds · Quantum phase transition · Anderson localization

1 Introduction
The concept of disorder-induced localization of electrons was first proposed by Anderson [1,2]. An electron scattered by a disordered potential interferes with itself andconsequently localizes in space. In this situation, temperature and magnetic fielddependence of the conductance and critical exponents near metal-insulator transitions
are universal: independent of the kind of material [3]. They depend only on the dimen-

B Takuya Kanno

t_kanno@eis.hokudai.ac.jp

1

Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan

2

Center of Education and Research for Topological Science and Technology, Hokkaido University,
Sapporo, Hokkaido 060-8628, Japan

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sions and internal symmetries of single-particle Hamiltonians describing the systems.
The internal symmetries are time-reversal symmetry (TRS) and spin-rotational symmetry (SRS) [4,5]. For instance, quantum Hall state is a good example of an Anderson
localization state with the breaking of TRS (classified in unitary class). Furthermore,
the field of study of Anderson localization also involves that of topological insulators [6,7]. The classification of topological insulators in d dimensions is constructedby studying the localization problem in a (d − 1)-dimensional (surface of d dimensional systems)-disordered system. Identifying the universality class is regarded as animportant problem of modern condensed matter physics.
The scaling theory of Anderson localization predicts that all electrons in a disordered potential are localized in two-dimensional (2D) systems when both TRS andSRS of the Hamiltonians are preserved [8]. However, when a homogeneous magnetic
field is applied, electrons are extended, not localized due to broken TRS. In this situation, the Hamiltonians are classified as unitary class. It is a stimulating question
whether electrons localize or not in which TRS is locally and randomly broken by arandom magnetic field (RMF). An RMF means that localized magnetic moments takerandom positions in space. The universality class of this system is non-trivial since
TRS is locally broken but globally not. It has been suggested that the quantum states
in RMF systems are relevant to the physics of strongly correlated electron systems
such as fractional quantum Hall systems [9–11] or high Tc superconductors [12,13].
Nevertheless, the conduction of electrons in an RMF has been studied only by theoretical methods [14–16]. Our purpose is to clarify the conduction of electrons in RMF
systems experimentally.
In this letter, we measured the resistance of single crystals with 2D electronic systems with an RMF and considered the universality class of their systems. We formed
an RMF in 2H -TaSe2 crystals by doping magnetic impurities. The paramagnetism
possessed by 2H -Fex TaSe2 crystals indicates that the magnetic moments is random
in space, and so the crystals are ideal RMF systems. We observed a metal-insulatortransition by increasing of x from the result of a resistance measurement. The resistance of insulator single crystals became almost independent of temperature below∼20 K. This behavior is consistent with the unitary class of Anderson localization.
However, the estimated critical sheet resistance (CSR) is accordant to that of the chiralunitary class rather than that of the unitary class. The same result is provided from theapproximation of the critical exponent of the localization length ν. We propose that
2H -Fex TaSe2 is classified in chiral unitary class.

2 Experiments
We chose 2H -TaSe2 as the base system. 2H -TaSe2 is a typical quasi-2D conductor. A
layer of 2H -TaSe2 is constructed from two triangular sheets of Se atoms separated by
one sheet of Ta atoms [17,18]. The in-plane conductivity of the crystal is thousands
times greater than the out-of-plane value [19]. The temperature dependent of resistanceof 2H -TaSe2 single crystals is metallic above the super-conductive transition temperature Tc = 0.2 K. This Tc is lower value among transition metal dichalcogenides.

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43

We tried to realize an RMF in 2H -TaSe2 crystals by doping them with magnetic
impurities. If doped crystals exhibit paramagnetism, the crystals have an RMF since
the magnetic moments are random. We doped the crystals with iron as magnetic
impurities. Doped Fe atoms substitute the Ta atoms or intercalate between layers. If
the locations and magnetic moments of Fe atoms have no order, 2H -Fex TaSe2 crystals
form an RMF system.
All the samples were grown using the chemical vapor transport method. We reacted
FeTa alloys and selenium shots. The FeTa alloys were made from iron wires (0.2 mm
in diameter, 99.5 %) and tantalum wires (1.0 mm in diameter, 99.95 %). The wires
were melted in an argon atmosphere with an electronic arc furnace. Every alloy andshot were sealed in evacuated (∼10−6 Torr) quartz tubes and reacted at about 780 ◦ C
for a week before being finally quenched in water. The Fe doping rates x of thegrown crystals were measured using energy dispersive X-ray spectroscopy (EDS). We
obtained higher doping rate samples than the previous study [20]. To investigate themagnetic property of the crystals, we measured the temperature dependence of themagnetic susceptibility and the magnetic field dependence of the magnetic moments.
The susceptibility and moments of the crystals were measured using a quantum design
superconducting quantum interference device (SQUID) fluxmeter MPMS-XL. Several
crystals were wrapped in kapton foil and mounted in plastic straws. Crystalline axisof measured crystals does not correspond to the direction of a magnetic field.
We measured the temperature dependence of the resistances of single crystals
using the four-terminal method. The current was parallel to the conduction surface of the crystal. We performed measurements from room temperature to 0.5 or
0.3 K with cryostats. Furthermore, we measured the magnetoresistance of the crystals. The applied magnetic field was perpendicular to the conduction surface of thecrystals.

3 Results and Discussion
We confirmed that an RMF system was realized in 2H -Fex TaSe2 crystals measuring
the magnetic susceptibility. The temperature dependence of the magnetic susceptibility
χ for the 2H -Fex TaSe2 crystals (x = 0, 0.050, and 0.120) at 1.0 T is shown Fig. 1a.
The susceptibility of the pure sample (x = 0) decreased at 120 K because the charge
density wave (CDW) transition reduced Pauli susceptibility. This is a typical behaviorof 2H -TaSe2 [21]. The susceptibility of the doped samples (x = 0.050 and 0.120)
was independent of temperature above 40 K and increased greatly below 40 K. The
susceptibility above 40 K indicates that Pauli paramagnetism is dominant at hightemperature.
Figure 1b shows the temperature dependence of the susceptibility χ −χ0 normalized
to zero at 40 K. χ0 includes Pauli paramagnetism and the molecular diamagnetismof the crystals. The solid lines in Fig. 1b are fitting lines χ − χ0 by 1/T . χ − χ0is proportional to 1/T between 40 and 4 K (or between 0.025 and 0.25 K−1 ). The
gradients of χ − χ0 increased as the Fe doping rate rises. χ − χ0 indicates that Curie
paramagnetism is dominant at low temperature and that the doped iron contributes tothe paramagnetism when x = 0.050 and 0.120. Since there may be a few unexpected123

J Low Temp Phys (2016) 183:41–49

x=0

0 [10

x=0.120−

TCDW=120K

emu/g]−1(b)

2H−Fe xTaSe 2
1 B=1.0 T
x=0.120x=0.050x=0−62H−Fe xTaSe2
B=1.0 T−6

[10 emu/g]

(a)

(c)

2H−Fe xTaSe 2
0.02 T=2.0 K

m−m0 [emu/g]

44x=0.120x=0.050
0 x=0

−0.02

x=0.050
0

−2
0

100
200
300
Temperature T [K]

0

0.2
0.4
−1
1/T [K ]−50
5Magnetic field B [ T ]Fig. 1 Temperature dependence of the magnetic susceptibility and moments of 2H -Fex TaSe2 crystals
where x = 0, 0.050, and 0.120. Error bars are within the range of the plot points. a Shows the susceptibility
χ against T and b shows the susceptibility χ − χ0 at 1.0 T, where χ0 includes the Pauli paramagnetismand molecular diamagnetism of the crystals. We fit the susceptibilities χ − χ0 with 1/T , and show themas solid lines. c Shows the magnetic field dependence of the magnetic moments m − m 0 of 2H -Fex TaSe2
crystals at 2.0 K, where m 0 includes constant terms (Color figure online)

magnetic impurities in the sample or the holder, the χ −χ0 of x = 0 is also proportional
to 1/T in spite of the absence of iron.
Figure 1c shows the magnetic field dependence of the magnetic moments m − m 0
at 2.0 K. m 0 includes the Pauli paramagnetism and the molecular diamagnetism of thecrystals. m − m 0 was linear below ±3.0 T, but decreased above ±3.0 T because ofmagnetic saturation. The m − m 0 gradients increased as the Fe doping rate rises. No
magnetic hysteresis was observed. On the other hand, results of Fig. 1 a, b indicate thatmagnetic moments of doped iron atoms exhibits Curie paramagnetism. Considering
the results of the susceptibility, the iron atoms in 2H -Fex TaSe2 have no magnetic
order, in other words, magnetic moments of iron locate at random. We conclude thatRMF systems are formed in the crystals.
Figure 2a shows the temperature dependence of sheet resistances Rsheet for 2H Fex TaSe2 single crystals. This result has been reported recently [22]. The x = 0,0.009, 0.050, 0.054, 0.082, and 0.120 values were measured. The Rsheet values were
calculated from the resistances of the samples, the sample size, and the lattice constantof 2H -TaSe2 ; c = 6.2 Å. The sample size measured with an optical microscope anda field emission scanning electron microscope (FE-SEM). The values of Rsheet rose
drastically by increasing x. The gradients of temperature dependence changed frompositive to negative between x = 0.050 and 0.054. This indicates that metal-insulatortransition is induced by doping iron. For metal samples (x = 0, 0.009, and 0.050),
inflection points which indicate the CDW transition were observed at 118, 70, and64.3 K, respectively. This is typical behavior of 2H -TaSe2 [23]. There was no anomaly
such as the Kondo effect in the metal samples. In contrast, the Rsheet of x = 0.054,
0.082, and 0.120 increased with cooling. This temperature dependence is consistent
with an insulator. However, the values of insulator Rsheet were almost constant at
low temperature as if they were metal. We suggest that this behavior represents weak
localization with magnetic scattering classified in unitary class [20].
We measured the Hall coefficients RH to investigate effect of doped iron on carrier
density. Figure 2 (b) shows the correlation between estimated values of |RH | and Rsheet .

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45

(b) R

sheet[

10(a)Rsheet [

10

6

]

6

]

T = 4.2 K

x=0.120x=0.082

10

4

10

2

T = 1.5 K
RH < 0T = 4.2 K

10

4

x=0.054

10

−2

10

0

10

|R H| [10x=0.050

(c) R

TCDW

sheet [

−10

2

3

m /C]

]

6

10

10

2

x=0.009

TCDW
TCDW
2H−FexTaSe2

x=0
10

0

1

unitary: 51.6 k
4

10

2

10
10
Temperature T [K]210

T =100 K= 90 K= 80 K= 70 K= 60 K= 50 K= 40 K= 30 K= 20 K
−2

−1

0

[

1

/K ]

Fig. 2 a The temperature dependence of sheet resistances Rsheet for 2H -Fex TaSe2 single crystals. Inflection points of x = 0, 0.009, and 0.050 are defined as the TCDW . Lines indicate gradients of Rsheet at 50
K. b The correlation between estimated |RH | and Rsheet . All values of RH are negative. An orange open
dot indicates a literature data of M. Naito et al. [24]. c The correlation of Rsheet and α between 100 and 20
K. Solid lines are fitted with α ∝ log Rsheet for each temperature and a dashed line indicates α = 0 (Color
figure online)

All values of the RH are negative. A blue rectangle indicates a boundary between metal
and insulator phase, corresponding to minimum Rsheet of x = 0.050 and maximum
Rsheet of x = 0.054. The values of the |RH | are same order over the metal-insulatortransition (inside of a green rectangle in Fig. 2b), except for the largest Rsheet . This
indicates that change of carrier density does not induce metal-insulator transition in
2H -Fex TaSe2 crystals. From the results of Fig. 2a, b, it is considered that doped
iron brings decline of mobility of electrons rather than carrier doping. Therefore, we
propose that the disorder (random magnetic potential) formed by doped iron induces
Anderson localization and the metal-insulator transition in the crystals.
The temperature dependence of the insulator samples was consistent with that ofunitary class; however, the CSR of this metal-insulator transition was not. We discuss
the universality class of 2H -Fex TaSe2 based on predicted critical resistances andcritical exponents in 2D systems (Table 1). To estimate the CSR, we focused on thegradients α of the temperature dependence after the CDW transition. α is given byα=

d log Rsheet.
d log T

(1)

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Table 1 Predicted critical resistances and critical exponents ν for six universality classes
Universality class

TRS

SRS

ChS

Predicted critical resistance [k]

Predicted ν

Orthogonal

Yes

Yes

No

None [8]None [8]UnitaryNo

Yes / NoNo2h/e2 = 51.6 [25]2.616 ± 0.014 [26]h/0.60e2 = 43.0 [27]Symplectic

Yes

No

No

h/1.42e2 = 18.2 [28]2.80 ± 0.04 [28]Chiral orthogonal

Yes

Yes

Yes

π h/4e2 = 20.3∗ [29]Non-predicted
0.35 ± 0.03 [30]Non-predicted

Chiral unitaryNo

Yes / No

Yes

Less than h/1.49e2 = 17.3∗ [30]

Chiral symplectic

Yes

No

Yes

Non-predicted

Six classes of single-particle Hamiltonians are classified in terms of the presence or absence of time-reversal
symmetry (TRS), spin-rotational symmetry (SRS), and chiral symmetry (ChS). The critical values of theorthogonal class do not exist because all the electrons localize [8]. The critical resistance of the unitaryclass is at the zero-temperature transition between quantum Hall liquids or between a quantum Hall liquidand an insulator [25]∗ Critical resistances of the chiral orthogonal class and the chiral unitary class are not critical values of thephase transition, but the conductance of the Dirac fermion. In the chiral unitary class, the critical resistancedepends on the strength of the magnetic field. The critical resistance is maximum; h/1.49e2 = 17.3 k forthe strongest field [30]Figure 2c shows the correlation between the Rsheet and the α calculated from Fig. 2a.
We obtained Rsheet and α values between 100 and 20 K, below the TCDW and above
the temperature which Rsheet becomes independent of T . When the α equals zero, theRsheet is independent of temperature, hence it is the critical value. We estimated that the
CSR is 11.7 ± 5.4 k. From Table 1, the CSR of the unitary class is 2h/e2 = 51.6 k,
namely larger than the estimated value. Furthermore, we estimated a critical exponentof the localization length ν. The ν is expressed in β function given byβ(g) = −2 d log g,
p d log T(2)where g is dimensionless conductance and p is a constant determined by the scattering mechanism of electrons. The α (Eq. 1) and the β function (Eq. 2) are regarded
equivalent each other by(3)
g = h/e2 (Rsheet )−1 ,except for the constant terms [31,32]. We calculated the value of ν using ν =(gc β  (gc ))−1 , where gc is the critical g of the metal-insulator transition, and obtained
it as 0.31 ± 0.18. From Table 1, this is also inconsistent with that of unitary class(2.616 ± 0.014). The estimated values both of the CSR and the ν correspond to themof chiral unitary class (less than h/1.49e2 = 17.3 k and 0.35 ± 0.03, respectively)
rather than standard unitary class.
We suggest that symmetric property of an RMF differs from that of unitary class,namely random scalar potentials with a homogeneous magnetic field. To provide an
intuitive understanding, we propose a model focused on the spacial variation of phaseof electrons. This model is created based on a work by Miller and Wang [33]. Figure 3
shows schematics of (a) random scalar potentials with a homogeneous magnetic field,

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47

Fig. 3 Schematics of a the random scalar potential with a homogeneous magnetic field and b the RMF,
and phase φm against the location of an electron x. Times symbol indicates scalar potentials, circled dot
and circled times indicate magnetic fluxes, and arrows indicate vector potentials A. Circles below graphs
indicate the phase φm on each location. The helicity of the phase is finite in (a), but zero in (b) for a long
distance (Color figure online)(b) an RMF, and correlation between the phase and the location of an electron x.
When electrons move in the magnetic field, the field adds
 phase φm to wave functionsof electrons. The φm value is given by φm = −ı h̄ −1 e A(r) · dr, r is the electron
location and A(r) is the vector potential. We regard the spatial variation of the phaseas the fluid of the phase.
In Fig. 3a, when an electron moves in the positive direction, φm turns counterclockwise on a complex plane as blue arrows on circles. The chirality for a direction neverchanges for a long distance. That is to say, the helicity of phase is finite in Fig. 3a.
In this case, the symmetry of chirality of the phase is broken because the chirality is
distinguishable. On the other hand, in Fig. 3b, when an electron moves in the positive
direction, the φm turns clockwise first as red arrows on circles and then switches tocounterclockwise as the blue arrows. If the direction of the magnet fluxes are random,
this switching must exist. This means that the helicity of the phase is zero for a long
distance in Fig. 3b. In this case, the clockwise chirality and the counterclockwise chirality cannot be identified from each other, namely symmetry of the chirality is valid.
Our model implies that the RMF is different essentially from random scalar potentials
with a homogeneous magnetic field. The TRS is absent from both of them, but thesymmetric property of the chirality is different from each other. Essence of Anderson
localization is interference of wave functions, hence we suggest that the difference ofthe symmetry affects the localization and the conduction of electron.

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J Low Temp Phys (2016) 183:41–49(a) R(B)/R
1.08

R(B)/Rminmin(b) 1.08

x=0.054, T=0.35 K
x=0.120, T=0.50 K
1.06

1.06

1.04

1.04

1.02

1.02

1

x=0.054, T=0.35 K

x=0.120, T=0.50 K

1
10

−2

−1

0

10
10
Magnetic field B [T]

1

10

0

2
B

1/2

4
1/2

[T ]

Fig. 4 Magnetoresistances of x = 0.054 and 0.120. We measured these values at minimum temperatureand the applied magnetic field was perpendicular to the conduction surface of the crystals. a Shows themagnetoresistance R(B)/Rmin versus log B. The lines are fitting lines obtained with R(B)/Rmin ∝ log B.
b Shows the magnetoresistance R(B)/Rmin plotted by B 1/2 . The lines are fitting lines obtained with
R(B)/Rmin ∝ B 1/2 (Color figure online)

We observed that the magnetoresistance of the insulator 2H -Fex TaSe2 is inconsistent with the theory of weak localization. Figure 4a shows the magnetic fielddependence of magnetoresistance R(B)/Rmin for x = 0.054 and 0.120 versus log B.
The applied magnetic field was perpendicular to the conduction surface of the crystals.
The lines are fitting lines obtained using R(B)/Rmin ∝ log B. Positive magnetoresistance is observed in both of insulator samples. This is a distinctive feature of symplecticclass in the weak localization, not unitary class. Although the R(B)/Rmin values were
not proportional to log B, which is expected in 2D symplectic class [5].
On the other hand, we found that the magnetoresistance of 2H -Fex TaSe2 can be
fitted by B 1/2 rather than log B. Figure 4b shows the magnetic field dependenceof magnetoresistance R(B)/Rmin for x = 0.054 and 0.120. Both R(B)/Rmin were
proportional to B 1/2 above 1.5 T1/2 . This behavior is similar to the magnetoresistanceof dirty graphite [34]. Y. Koike et al. observed that the resistivity of dirty graphite wasalso proportional to B 1/2 in a strong magnetic field. They explained this behavior withthe weak localization and a Coulomb interaction between electrons, but the explanation
was quantitatively insufficient.

4 SummaryIn this paper, we reported the resistance measurement of iron-doped 2H -TaSe2 . We
confirmed that 2H -Fex TaSe2 crystals exhibit paramagnetism by measuring the magnetic susceptibility and moments. Based on the results, we propose that 2H -Fex TaSe2
offers a good way to investigate a 2D electronic system with an RMF. We measured thetemperature dependence of the resistances of crystals and discovered a metal-insulatortransition caused by increasing x. This temperature dependence is inconsistent with123

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49

the Kondo effect and the weak localization of the orthogonal class, but consistent withthe localization of the unitary class. However, the observed CSR and critical exponent
ν are 11.7 ± 5.4 k and 0.31 ± 0.18, which are accordant to them of chiral unitaryclass (less than h/1.49e2 = 17.3 k and 0.35 ± 0.03, respectively) rather than standard unitary class. We conclude that 2H -Fex TaSe2 should be classified in the chiralunitary class. This paper is the first experimental report of an RMF system and thechiral unitary class.
Acknowledgments We thank Naoto Yasuda, Noriyuki Okinaka, and Tomohiro Akiyama for making the
FeTa alloys; Takaaki Minamidate, Noriaki Matsunaga, and Kazushige Nomura for measurements of theresistance and the magnetic susceptibility. We also thank Sandeep Kumar Kataria and Migaku Oda for theirearlier study, and Yasuhiro Asano and Pierre Monceau for useful discussions.

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