Angle dependence of the orbital magnetoresistance in bismuth

7 MB·English

by  Aurelie Collaudin

Angle dependence of the orbital magnetoresistance in bismuth
Aur´elie Collaudin1, Benoˆıt Fauqu´e1, Yuki Fuseya2 Woun Kang3 and Kamran Behnia1
(1) Laboratoire de Physique Et d’Etude des Mat´eriaux (UPMC-CNRS-ESPCI) 10 Rue Vauquelin, 75005 Paris, France
(2) Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
3 Department of Physics, Ewha Womans University, Seoul 120-750, Korea
(Dated: March 18, 2015)
Wepresentanextensivestudyofangle-dependenttransversemagnetoresistanceinbismuth,with
amagneticfieldperpendiculartotheappliedelectriccurrentandrotatinginthreedistinctcrystallo-
graphicplanes. Theobservedangularoscillationsareconfrontedwiththeexpectationsofsemi-classic
transporttheoryforamulti-valleysystemwithanisotropicmobilityandtheagreementallowsusto
quantify the components of the mobility tensor for both electrons and holes. A quadratic temper-
5 ature dependence is resolved. As Hartman argued long ago, this indicates that inelastic resistivity
1 in bismuth is dominated by carrier-carrier scattering. At low temperature and high magnetic field,
0 thethreefoldsymmetryofthelatticeissuddenlylost. Specifically,a2π/3rotationofmagneticfield
2 around the trigonal axis modifies the amplitude of the magneto-resistance below a field-dependent
temperature. Byfollowingtheevolutionofthisanomalyasafunctionoftemperatureandmagnetic
n
field, we mapped the boundary in the (field, temperature) plane separating two electronic states.
u
In the less-symmetric state, confined to low temperature and high magnetic field, the three Dirac
J
valleysceasetoberotationallyinvariant. Wediscussthepossibleoriginsofthisspontaneousvalley
0
polarization, including a valley-nematic scenario.
1

]
l I. INTRODUCTION threefold symmetry is yet to be understood.
e
- Metals hosting a small concentration of high-mobility
r
t Electric conduction in solids is affected by the appli- carriers and displaying a large magnetoresistance have
s cation of magnetic field in a variety of ways. The most attracted much recent attention. In dilute metals such
.
at prominent is orbital magnetoresistance, which is the en- asWTe2[16]orCd3As2[15],resistivityenhancesbymany
m hancementofresistivityduetotheLorentzforcesuffered orders of magnitude upon the application of a magnetic
bychargedcarriersinpresenceofmagneticfield. Asearly field of 10 T. This is also the case of bismuth[6, 7, 17]
-
d as1928,Kapitzadiscoveredthattheelectricresistivityof and graphite[17, 18], two well-known semi-metals. Both
n bismuthincreasesbymanyordersofmagnitudeuponthe the amplitude of magnetoresistance and its field depen-
o applicationofalargemagneticfield[1]. Thelargeorbital dencehavebeenputunderscrutinyandareexploredand
c
magnetoresistance is one manifestation of the extreme discussed by experimentalists and theorists.
[
mobility of carriers in bismuth, itself a consequence of In this paper, we present an extensive study of trans-
2 thelightnessofelectrons,theultimatereasonbehindthe verseangle-dependentmagnetoresistanceinbismuthand
v
singular role played by this elemental semi-metal in the establisha detailedmapof transversemagnetoresistance
4
history of scientific exploration of electrons in metals[2]. for all possible orientations of magnetic field from room
8
5 During the last few years, bismuth has attracted new temperature down to 2K and up to a magnetic field of
1 attention (A recent review can be found in ref.[3]). 14T.Thestudyaimstoaddresstwodistinctissues. The
0 A number of intriguing observations on bismuth crys- first concerns the amplitude of magnetoresistance in a
1. tals exposed to strong magnetic fields have have been compensated semi-metal such as bismuth. Our results
0 reported[4–8]. The angle-resolved Landau spectrum has showthatinanyrealmaterial,theknowledgeofallcom-
5 been found to become exceptionally complex in high ponents of the mobility tensor is required to compute
1 magnetic field. In spite of this complexity (and in con- the magnitude of magnetoresistance. The emergence of
:
v trast to what was initially thought[4, 5]), the spectrum a valley-polarized state is the second issue addressed in
i resolved by experiment[5, 8–10, 14] is in agreement with this study.
X
theoretical expectations[9–12] based on the band struc- Numerous studies of angular oscillations of magne-
r
a ture of the system and its fine details. toresistance in strongly-correlated electron systems have
Open questions remain however. The three small been reported in the past. For three case studies, see
pockets of Fermi surface residing at the L-point of the Ref.[19–21]. These investigations used angular magne-
Brillouin zone host Dirac fermions with an extremely toresistance as a probe, knowing that it shows an ex-
anisotropic mass becoming as small as one-thousandth tremum when the peculiar topology of the Fermi sur-
of the free electron mass along the bisectrix axis. These face modifies the dimensionality of the cyclotron orbit
three Dirac valleys are interchangeable upon a 2π/3 ro- at a ’magic’ angle (dubbed either Lebed or Yamaji, af-
tation around the trigonal axis. But, according to two ter those who conceived these commensurability effects).
sets of experimental studies[13, 14], the three Dirac val- Our experimental configuration is different. The mag-
leys become inequivalent at low temperature and high netic field is rotated in a way to keep the magnetic field,
magnetic field. The origin of this spontaneous loss of B(cid:126),andchargecurrent,(cid:126)j,perpendiculartoeachotherand
2
the transverse magnetoresistance is measured. In this average mobility of electrons and holes is as large as
configuration, the macroscopic Lorentz force between B(cid:126) < µe > + < µh >= 2×107 cm2V−1s−1. As we will
and(cid:126)j,isconstant. Ifthemobilitywereascalar,noangu- see below, the order of magnitude is confirmed by our
lar variation would arise. Large angular oscillations are magnetoresistance data, but the mobility is very differ-
visible even at room temperature and in fields as small ent for electrons and holes and along different orienta-
as 0.7T in bismuth[13], because carriers have drastically tions. Samplesofvariouscrosssections(circular,square,
anisotropic mobilities. triangular) were studied in order to check the effect of
Our results provide an opportunity to test the semi- sample geometry on the loss of threefold symmetry. No
classictransporttheoryinaparticularlyconstrainingfor- significant difference between crystals of different origin
mat. Since the structure of the three electron ellipsoids (commercial vs. home-grown) was observed.
and the single hole ellipsoid in bismuth are well-known,
one can attempt to fit the experimental data by assum-
ing reasonable values for the components of the mobility III. ANGULAR OSCILLATIONS OF
tensorsofelectronsandholesandestablishtheirtemper- TRANSVERSE MAGNETORESISTANCE IN
atureandfield-dependence. Wefindthat,saveforanum- THREE PERPENDICULAR PLANES
ber of important details, our experimental results are in
reasonable agreement with the expectations of the semi-
The Fermi surface of bismuth consists of a hole ellip-
classic theory. This paves the way to a quantitative un-
soid and three electron ellipsoids[24]. The hole ellipsoid
derstanding of transverse magnetoresistance in bismuth
hasalongeraxis,whichisthreetimeslongerthanthetwo
with obvious implications for other semi-metals and di-
other shorter axes and lies along the trigonal axis. The
lute metals.
longer axis of each electron ellipsoid is about 14 times
We also present a detailed study of the configuration longer than the two shorter axes and lies in a (bisectrix,
in which the current is applied along the trigonal axis trigonal) plane, slightly tilted off the bisectrix axis. In
and the magnetic field is rotating in the (binary, bisec- theabsenceofmagneticfield, chargeconductivityinbis-
trix)plane. Weconfirmthelossofthreefoldsymmetryat muthisalmostisotropic. But,behindthisquasi-isotropy
low temperature and high magnetic field previously re- hides an intricate structure of opposite and compensat-
ported by transport[13] and thermodynamic studies [14] ing anisotropies, which reveals itself by the application
andfindaboundaryinthe(field,temperature)planesep- of a rotating magnetic field.
arating two electronic states. The underlying threefold
Fig. 1 presents polar plots of transverse magnetoresis-
symmetry of the zero-field crystal lattice is lost in the
tance as the electric current is applied along one crystal
low-temperature-high-field state, but is kept in the high-
axis and the magnetic field is rotated in the plane per-
temperature-low-field state. A phase transition between
pendicular to the applied current. The figure compares
these two states is clearly detectable at low magnetic
the data obtained for three perpendicular planes at a
field. Withincreasingmagneticfield,thetransitionshifts
temperature of 30 K and a magnetic field of 0.5 T. The
to higher temperature and becomes broader. We discuss
threepolarplotsillustratehowthestructureoftheFermi
possible origins of this phase transition and consider the
surface leads to very different patterns in each case.
availabletheoreticalscenariosinvokingCoulombinterac-
When the current is applied along the trigonal axis
tionamongelectrons[22]orlatticedistortioninducedby
(Fig. 1a) , magnetoresistance shows six fold angular os-
magneticfieldreminiscentofJahn-Tellereffect[23]. None
cillations. The system has a C3 symmetry and remains
of the currently available pictures provide an adequate
invariant when the magnetic field rotates by 2π/3. In
description of the whole range of experimental facts.
this configuration, the hole conductivity does not de-
pend on the orientation of magnetic field. On the other
hand, electrons show a large variation in their magne-
II. EXPERIMENTAL toresistance. It becomes largest when the field is aligned
along a bisectrix orientation, since for this orientation of
Most measurements were performed using a Quantum magneticfield,Fermivelocityofelectronsandthemicro-
Design PPMS apparatus. These studies were comple- scopic Lorentz force they feel are maximum. When the
mented with measurements in Seoul using a two-axis current is applied along the bisectrix axis and the field
home-made set-up in particular to check for any arti- rotates in the perpendicular plane (Fig. 1b), magnetore-
fact resulting from misalignment. Three bismuth single sistance respects two angular symmetries. It is mirror
crystals were cut in a cuboid shape with typical dimen- symmetric(i.e. ρ(θ)=ρ(−θ))anditkeepstheinversion
sionsof(3×4×5mm3)andwereusedformappingthe symmetry(i.e. ρ(θ) = ρ(π +θ) ). When the current is
magnetoresistance in three perpendicular planes. The applied along the binary axis and the field rotates in the
typicalresidualresistivityofthesecrystalswas1µΩcm. third plane(Fig. 1c), only the last [inversion] symmetry
Taking a carrier concentration of n = 3×1017cm−3 for is kept. For the last two configuration, one expects a
both hole-like and electron-like carriers and using the maximal magnetoresistance for holes when the magnetic
simple expression σ = ne(< µ > + < µ >), the field is along the trigonal axis. Added to the sum of the
0 e h
zero-field conductivity would imply that the sum of the contributions of the three electron pockets, this leads to
3
Trig. q B Trig. q B
Bisect.
e2 e1
e2 e1
e2,3 e1
Bin.
e3
Bin.
Bisect.
q
B
e3
a) b) c)
j // trigonal j // bisectrix j // binary
0 0 0
2 4
330 30 4 330 30 330 30
1 300 60 2 300 60 2 300 60
 )mmc( 001 270240 12090 )mm(c  002 270240 12090 )mm(c  002 270240 12090
4
2 210 150 210 150 4 210 150
180 180 180
T =30K B= 0.5T
FIG.1: PolarplotsoftransversemagnetoresistanceasafunctionoftheorientationofthemagneticfieldatT=30KandB=0.5
K. The three plots show the data for three perpendicular planes, Left: field rotates in the (binary, bisectrix) plane; Middle:
field rotates in the (binary, trigonal) plane and Right: field rotates in the (biscetrix, trigonal) plane. The electric current is
always applied along the crystal axis perpendicular to the rotating plane. Upper panels show the projection of the Brillouin
zone, as well as the hole and electron pockets of the Fermi surface in each rotating plane.
complex patterns. As seen in Fig. 1b and 1c, the total is the visibility of such angular oscillations at room tem-
magnetoresistance peaks at intermediate angles off the perature and in magnetic fields lower than 1 T[13]. This
high-symmetry axes. is a consequence of the large mobility of electrons, which
These polar plots resemble those reported by Mase, exceeds 104 cm2V−1s−1 at room temperature, and their
von Molnar and Lawson half a century ago[25]. These very anisotropic mass. No solid other than bismuth is
authors presented a study of the galvanometric tensor of currently known to present such properties.
bismuthforanarbitraryorientationofthemagneticfield The evolution of the angle-dependent magnetoresis-
for a single temperature (20.4 K) and a single magnetic tance in bismuth when the field rotates in the (binary,
field (0.576 T). The angular variation of our data is in bisectrix) plane was previous reported and the data wa
good agreement with their data. However, the absolute found to be described by an empirical formula for multi-
amplitude of the magnetoresistance in our data at20 K valleyconductivity[13]. Inthispaper,extendingthemea-
and 0.5 T, is roughly two times lower. surements to the two other planes, our aim is to see
The extreme sensitivity of magnetoresistance in bis- if transverse magnetoresistance in bismuth can be ex-
muth upon the orientation of magnetic field permits one plainedforthewholesolidanglewithasingletheoretical
to use a rotating magnetic field to tune the contribu- model based on a Boltzmann semi-classical picture. We
tion of each electron pocket (i.e. each of three valleys) will show that this is indeed the case and the empirical
to the total conductivity in the configuration presented formula used in ref.[13] is an approximation of a more
in Fig. 1a[13]. This provides an interesting opportu- general formula for multi-valley systems.
nity for “valleytronics”[26, 27], an emerging field of re- Weperformedanextendedstudyonseveralcrystalsin
search focused on manipulation of the valley degree of a wide range of temperature (2K<T <300 K) and mag-
freedom. Such angle-dependent magnetoresistance is ex- netic field(B <12 T). A selection of data for B =0.5 T
pected to be observable in any multi-valley system with and T < 75 K are presented in Fig. 2-4. Fig. 2 shows
anisotropic valleys, as recently demonstrated in the case the thermal evolution of the angular magnetoresistance
of SrMnBi [28]. What distinguishes bismuth, however, when the current is applied along the trigonal axis and
2
4
0 40K 0 75K
330 30 30K 330 30 50K
1.5 27.5K 40K
25K 8 30K
22.5K 27.5K
1.0 300 60 20K 300 60 25K
22.5K
17.5K 4 20K
0.5 15K 17.5K
m 12.5K m 15K
c 00..00 270 90 170KK mc 00 270 90 1120.K5K
m 7K
0.5 4 4.2K
2K
240 120
1.0 240 120
8
1.5 210 150
210 150
180
180 B= 0.5 T; J//Bisectrix
B=0.5 T; J// trigonal
1E-3 330 0 30 7500KK 0.01 330 0 30 754500KKK
40K 30K
30K 27.5K
27.5K 300 60 25K
1E-4 300 60 25K 22.5K
22.5K 1E-3 20K
20K 17.5K
17.5K m 15K
m 1E-5 15K c 270 90 12.5K
c 270 90 12.5K  10K
 1E-5 170KK 1E-3 74.K2K
2K
240 120
1E-4
240 120
0.01
210 150
1E-3
210 150 180
180
FIG. 2: Transverse angle-dependent magnetoresistance for a FIG. 3: Transverse angle-dependent magnetoresistance for a
currentappliedalongthetrigonalaxisandafieldrotatingin currentappliedalongthebisectrixaxisandafieldrotatingin
the (binary, bisectrix) plane for different temperatures. The the (binary, trigonal) plane for different temperatures. The
twopanelspresentthesamedataplottedinalinear(top)and twopanelspresentthesamedataplottedinalinear(top)and
logarithmic (bottom) scale. logarithmic (bottom) scale.
servable and new extrema emerge in ρ(θ) curves as the
themagneticfieldrotatesinthe(binary,bisectrix)plane.
system is cooled down. Note also the gradual saturation
This configuration is identical to the one studied in ref.
in amplitude of magnetoresistance below 7 K, indicat-
[13] and the results are quite similar. The evolution of
ing that mobility has attained its maximum amplitude
the angle-dependent transverse magnetoresistance with
in this temperature range.
decreasing temperature for the two other planes of rota-
For a quantitative treatment of the data, we need to
tion isshownin Fig. 3and Fig. 4. Inall three cases, the
revise the semi-classic transport theory applied to the
evolutionissmoothwithincreasingmagnetoresistanceas
case of multi-valley system with anisotropic valleys.
temperature decreases. This is a consequence of the fact
that all components of the mobility tensor enhance with
decreasing temperature. At zero magnetic field, the en-
IV. THE SEMI-CLASSIC TRANSPORT
hancementinmobilitywithdecreasingtemperatureleads
THEORY
toanenhancementofconductivity. Inpresenceofafield
as small as 0.5 T, on the other hand, it leads to an in-
FollowinganearlierworkbyAbelesandMeiboom[29],
crease inresistivitywithdecreasingtemperature. Thisis
Aubrey[30] wrote the following equation for the charge
because the amplitude of the orbital magnetoresistance
conduction by a Fermi pocket with a carrier concentra-
is set by the mobility and it exceeds by far the zero-field
tion n in presence of electric and magnetic fields:
resistivity of the system in this “strong-field” limit.
The thermal evolution of the morphology of the three
curvesisinstructive. Forthefirstconfiguration(I (cid:107)trig- (cid:126)j =σˆ.E(cid:126) =neµˆE(cid:126) +µˆ(cid:126)j×B(cid:126) (1)
onal),thereisnovisiblechangeinthestructureofangle-
dependent magnetoresistance with cooling. In the case Here, e is the electric charge and σˆ and µˆ are the con-
of the other configurations, a qualitative evolution is ob- ductivity and the mobility tensors. In the absence of
5
Here mˆ−1 is the inverse of the mass tensor. In the
0 75K
20 330 30 50K simplest case derivation τ is a scalar. However, this is
40K
30K notnecessaryandonecaneasilygeneralizetoacasewith
27.5K different relaxation times along x, y, z.
10 300 60 25K
22.5K Aubrey’s important contribution was to find that by
2170.K5K defining a tensor Bˆ whose components are projections of
m
c 15K magneticfieldalongthethreeperpendicularorientations,
 00 270 90 12.5K
m 10K onecanwriteageneralsolutiontoEq. 1inthefollowing
7K
4.2K manner[30]:
2K
10
240 120
σˆ =ne (µˆ−1+Bˆ)−1 (5)
20 210 150
180 The components of the matrix Bˆ are:
B=0.5 T; J//Binary
0 75K  
330 30 50K 0 −B3 B2
0.01 4300KK Bˆ = B3 0 −B1
27.5K −B B 0
2 1
300 60 25K
22.5K
1E-3 20K Here, B , B and B are the projections of the mag-
1 2 3
17.5K
m 15K netic field along the three principal axes:
c 270 90 12.5K
 10K B 
7K 1
1E-3 4.2K B(cid:126) =B2
2K
B
240 120 3
It is worthy to underline that no assumption has been
0.01
210 150 made on the magnitude of the magnetic field. This is
180 to be contrasted with treatments based on the Jones-
Zener expansion, which are only valid in the weak-field
FIG. 4: Transverse angle-dependent magnetoresistance for a limit (µB < 1)[32]. The Aubrey approach[30], on the
current applied along the binary axis and a field rotating in other hand, is most appropriate in the strong-field limit
the(bisectrix,trigonal)planefordifferenttemperatures. The (µB >1).
twopanelspresentthesamedataplottedinalinear(top)and Now, to see the physics behind this picture, consider
logarithmic (bottom) scale. a spherical Fermi surface with an isotropic (i.e. scalar)
mobility of µ. In this case, the orientation of the mag-
netic field has no importance. Let us assume it oriented
magnetic field, Eq. 1 becomes the familiar expression:
along the z-axis (that is, B = B =0 and B = B). In
1 2 3
σˆ(B =0)=neµˆ.
this case, the solution implied by Eq. 5 becomes:
Abeles and Meiboom[29] argued that one can derive
Eq. 1 starting from the linearized Boltzmann equation,  σ σ 0
⊥ H
which can be stated in the following manner [31]: σˆ =−σH σ⊥ 0
0 0 σ
(cid:107)
e∂fk0→−v .E(cid:126) + e(→−v ×B(cid:126)).−∇−−→g =−gk (2) Here, the transverse (σ ), Hall (σ ) and longitudinal
∂(cid:15) k (cid:126) k k k τ ⊥ H
k (σ ) components of the conductivity tensor take the fol-
(cid:107)
Here,f0istheFermi-Diracdistributioninequilibrium, lowing familiar expressions:
k
g represents the deviation from this equilibrium distri-
k neµ neµ
bution,(cid:15)kistheenergeyand→−vk thevelocityofanelectron σ⊥ = 1+µ2B2;σH = 1+µ2B2µB;σ(cid:107) =neµ (6)
withawave-vectork andτ istherelaxationtime. Inthis
picture, the charge current is defined as: Inotherwords, magneticfieldmodifiesonlythetrans-
verse conductivity and leaves the longitudinal conduc-
(cid:126)j =e(cid:90) →−vkgk−d→k (3) twiveiigtyhsutnhcehtarnagnesdv.ersMemoraegovneert,ortehseistHanalclecionmthpeonheignht-fioeultd-
(µB > 1) limit. Now, consider an ellipsoidal Fermi sur-
Eq. 1 condenses all material-dependent parameters in face, with an anisotropic mobility:
the mobility tensor defined as:
 
µ 0 0
1
µˆ =0 µ2 0
µˆ =τmˆ−1 (4) 0 0 µ
3
6
Rotating the magnetic field in the (1, 2) plane and Usingthisformalism,onecancomputethecomponents
measuring electric conductivity along the third axis, ac- of total conductivity tensor for a given set of seven pa-
cording to Eq. 2, one would expect to find: rameters (µ , ν and n). In order to compare with
1−4 1,3
theexperiment,oneneedstoinvertthecalculatedtensor
and obtain the relevant component of the resistivity ten-
neµ
σ = 3 (7) sor(ρˆ=σˆ−1). Inourparticularcase, wehavemeasured
33 1+B2µ (µ cos2θ+µ sin2θ)
3 2 1 resistivity, ρ , ρ , ρ , along three principal axes. The
11 22 33
link between ρ and the components of the conductivity
ii
Here θ is the angle between the magnetic field and the
tensor can be written as:
binary (i.e. 2nd) axis. When µ (cid:29) µ , this equation
1 2
becomes identical to the empirical formula used to fit
the data obtained for the field rotating in the (binary, 1
ρ = (9)
bisectrix) plane[13]. ii σ +δσ
ii ii
Let us now turn to a multi-valley system where the
Where:
totalconductivityisthesumofthecontributionsbyeach
valley. In the specific case of bismuth, one can write: σ σ σ +σ σ σ −σ σ σ −σ σ σ
δσ = 12 23 31 13 21 32 22 13 31 33 12 21
11 σ σ −σ σ
22 33 23 32
(10)
(cid:88)
σˆ = σˆe+σˆh. (8)
tot i
i=1−3 σ σ σ +σ σ σ −σ σ σ −σ σ σ
δσ = 12 23 31 13 21 32 11 23 32 33 12 21
22 σ σ −σ σ
Here σˆh is the conductivity tensor of the hole pocket 11 33 13 31
(11)
and σˆe is the conductivity tensor of the electron pocket
i
indexed i.
The structure of electron ellipsoids in bismuth is com- δσ = σ12σ23σ31+σ13σ21σ32−σ11σ23σ32−σ22σ13σ31
plex. They have no circular cross sections and do not 33 σ σ −σ σ
11 22 12 21
lie along any symmetry axis. Their mobility tensor, like (12)
theireffectivemasstensor[33],hasfourdistinctandfinite
components. In the case of the electron ellipsoid which One may wonder if there is any room for harmless ap-
has its longer axis in the (x, z) plane, this tensor can be proximations here. These expressions contain numerous
expressed as: off-diagonal components. Onsager relations imply that
σ (B) = −σ (−B). Agkoz and Saunders have detailed
ij ji
  therestrictionsimposedbyOnsagerreciprocityrelations
µ 0 0
1
µˆe1 =0 µ2 µ4 fsotrraatleld32thcarytsitnaltlohgeracapsheicopfoAin7tsgtrrouucptus[r3e4]o.fTbhisemyudtehmaonnd-
0 µ µ
4 3
antimony, the off-diagonal components of the conductiv-
ity tensor can have both even and odd parts. In par-
Thethreeelectronpocketsareequivalenttoeachother
ticular, when the field is along the binary axis, there
through a 2π/3 rotation. Therefore the mobility tensor
is a so-called ’Umkehr’ effect, leading to the inequality
for the two other electron ellipsoids would be:
σ (B )(cid:54)=−σ (B ). Thismeansthatthefirsttwoterms
23 1 32 1
µˆe =Rˆ−1 . µˆe . Rˆ ; µˆe =Rˆ−1 . µˆe . Rˆ ofthenominatorsinequations10to12donotcancelout.
2 2π/3 1 2π/3 3 4π/3 1 4π/3 It is true that in a compensated semi-metal with an
equal concentration of electrons and holes, the opposite
Here, Rˆ is the rotation matrix for a rotation angle of
θ Hallresponseofelectronsandholescancanceleachother.
θ around the trigonal axis :
In case of negligible off-diagonal components, one would
simplyfindρ (cid:39)σ−1. Whenthemagneticfieldisaligned
cosθ −sinθ 0 ii ii
along each of the three high-symmetry axes, one can ex-
Rˆθ =sinθ cosθ 0 perimentally verify that ρ (cid:28) ρ and thus σ (cid:28) σ .
ij ii ij ii
0 0 1
However, forarbitraryorientationsofthemagneticfield,
the magnitude of the off-diagonal components is large
The hole pocket is an ellipsoid with a circular cross enoughtogeneratenon-negligibleeffects. Wefoundthat
sectionperpendiculartothetrigonalaxis. Ithasidentical in order to find a satisfactory fit, the full expressions of
projectionsinthebinaryandbisectrixplanes. Therefore, equation9-12aretobeused. Inparticular,inthecaseof
thereareonlytwodistinctandfinitecomponentsandits the third configuration, i.e. the field rotating in the (bi-
mobility tensor νˆ can be written as: sectrix, trigonal)plan], withouttakingδσ intoaccount,
ii
even a qualitative agreement with the experimental data
 
ν 0 0 is impossible to obtain.
1
νˆ=0 ν1 0 . In 1974, Su¨mengen, Tu¨retken and Saunders[35] em-
0 0 ν ployed this formalism to quantify the components of the
3
7
a) 0 100K 0 50K 0 10 K
1.0 330 30 DFiatta 6 330 30 DFiatta 100 330 30 DFiatta
300 60 300 60 300 60
0.5 3 50
)m )m )m
c  00..00270 90 c  00270 90 c  00270 90
m m m
( 0.5 (  3 ( 50
240 120 240 120 240 120
1.0 210 150 6 210 150 100 210 150
180 180 180
b) 0.4 330 0 30 100 KDF iatta 2 330 0 30 50K DFiatta 6 330 0 30 20K DFiatta
300 60
0.2 300 60 1 300 60 3
)mc 00..00270 90 )mc 00270 90 )mmc00270 90
m( 0.2 240 120 m( 1 240 120 ( 3 240 120
6
210 150
0.4 210 150 2 210 150 180
180 180
c) 0.3 330 0 30 100 KD ata 1.5 330 0 30 50K Data 10 330 0 30 15K Data
0.2 300 60 Fit 1.0 300 60 Fit 5 300 60 Fit
)mmc( 0000....0101270 90 )mmc( 0000....0505 270 90 )mmc( 00 270 90
0.2 240 120 1.0 240 120 5 240 120
0.3 210 150 1.5 210 150 10 210 150
180 180 180
FIG. 5: Comparison between the data and the theoretical fit based on the optimal parameters at B=0.5 T and three different
temperatures for : a) Magnetic field rotating in the (binary, bisectrix) plane; b) magnetic field rotating in the (trigonal,
bisectrix) plane; c) magnetic field rotating in the (trigonal, binary) plane.
mobility tensor in bismuth using the experimental data areshowninFig. 5. Thisagreementestablishesthatthe
publishedadecadebeforebyMaseandco-workers[25]for large anisotropic magnetoresistance of bismuth can be,
asingletemperature(20.4K)andasinglemagneticfield mostly if not entirely, explained by the semi-classic the-
(0.576 T). They found that experimentally-observed an- ory. At lower temperatures and higher magnetic fields,
gular variation can be theoretically reproduced and the the fits become less satisfactory.
extractedcomponentsofthemobilitytensormatchthose There are several reasons for the gradual inadequacy
found by measuring components of resistivity, ρij in the ofthemodelasthetemperaturelowersandthemagnetic
low-field (µB (cid:28) 1)limit[36, 37]. In the next section, we fieldincreases. Firstofall,Landauquantizationbecomes
are going to use this procedure for our extensive set of sharp enough to introduce additional angular structure
data. notexpectedbythemodelusedhere. Thesecondreason
is the loss of the threefold symmetry, which will be dis-
cussed in detail in the following section. A third is the
V. THE MOBILITY TENSOR apparent effect of the applied magnetic field on the com-
ponentsofthemobilitytensor. Thislastfeatureappears
According to the theoretical frame described in the onlyafterputtingunderscrutinythedetailsofthefitting
previous section, the angle-dependent transverse magne- procedure.
toresistance can be fit using equations 9-12 with seven We found that in some cases, two different sets of fit-
adjustable parameters. These are the four components tingparameterscouldbothyieldsatisfactoryfitsofcom-
of the mobility tensor of electrons, µ , the two com- parablequality. Inordertominimizethisuncertainty,we
1−4
ponents of the mobility tensor of holes, ν and the attempted to reduce the number of adjustable parame-
1,3
carrier density of holes, n , which is equal to the sum tersbyexcludingthecarrierdensityandthetiltangleas
h
of the carrier densities of the three electron pockets variable parameters.
(n =n +n +n ). It has been known that carrier density in bismuth re-
h e1 e2 e3
We tried such fits and found that good fits with plau- mains constant up to 50 K and then begins to increase
sible parameters can be achieved. Polar plots comparing as thermally excited carriers are introduced across the
theexperimentaldataandtheexpectationsofthetheory small gap at the L-point[40]. We took a fixed value of
8
n =n =3×1017 cm−3 (the magnitude obtained from
e h
deHass-vanAlphenmeasurements[39])forT <50Kand
let it evolve as a free parameter for temperatures above.
Thetiltangle,θ,betweenthelongeraxisoftheelectron
ellipsoid and the crystalline bisectrix axis, generates the
fourth mobility component, µ . If the relaxation time
4
tensor happens to be aligned along the high-symmetry
crystals axes, the mobility tensor and the mass tensor
should have identical tilt angles. It was measured to
be 6.4 degrees by de-Haas-van-Alphen effect studies[39]
and its controversial sign was settled by refined com-
parison between the data on structural and electronic
properties[38]. The recent study of angle-resolved Lan-
dau spectrum[9] found a tilt angle(∼ 6.2◦ ) close to the
previous reports.
Inthecaseofthemobilitytensor,thetiltanglecanbe
expressed as [35, 36]: FIG. 6: Temperature dependence of the components of the
electron mobility tensor, µ , obtained by fitting our angle-
i
dependent magnetoresistance data at B =0.5T for a current
1 2µ
θ = arctan( 4 ) (13) appliedalongthetrigonalaxis. Alsoshownaretheresultsre-
2 µ2−µ3 portedforzeromagneticfieldbyHartmanbelow15K[36]and
Hartman found that the tilt angle of the mobility ten- by Michenaud and Issi above 77 K [37]. Solid lines represent
aT−2 temperaturedependence(Nodataonµ wasreported
sor is very close to this value and concluded that the 4
in the latter case).
relaxation time tensor has little or no tilt[36]. We found
that the best fits to our data point to a value close to
6.8◦ and in order to reduce the fitting uncertainty im-
posed this as a constraint to our fitting procedure.
Figure 6 and Figure 7 show the the temperature-
dependence of the mobility components of electrons and
holes extracted from our fits to the data at B = 0.5
T.Our dataiscomparedwithzero-field valuesextracted
fromthegalvanometriccoefficientsofbismuthinthelow-
magnetic field limit. Several remarks are in order.
First of all, the sheer magnitude of the mobility of
Dirac electrons in bismuth is remarkably large. At 10
K, µ in bismuth becomes as large 1000 T−1 or 107
1
cm2V−1s−1. This is slightly lower than what Hartman
foundatthesametemperature. Accordingtohisresults,
µ becomes as large as 108 cm2V−1s−1. Therefore, and
1
this may be worth being recalled, electrons in bismuth
are by far more mobile than carriers seen in any other
FIG. 7: Temperature dependence of the components of
three-dimensional solid[15].
the hole mobility tensor, ν , obtained by fitting our angle-
Second, the mobility tensor is anisotropic. Qualita- i
dependent magnetoresistance data at B = 0.5T for current
tively, this reflects the anisotropy of the band mass.
applied along the binary or bisectrix axis. Also shown are
The anisotropy is larger for electron, which have a more the results reported for zero magnetic field by Hartman by
anisotropic mass, than holes. The mobility of elec- Hartmanbelow15K[36]andbyMichenaudandIssiabove77
trons is lowest along the bisectrix, which is the direc- K[37] (No ν was reported in the latter case).
3
tion for which the electrons are heaviest. As for holes,
they are lighter and more mobile perpendicular to the
trigonal axis than parallel to it. However, the corre- As seen in Fig. 6 in the case of electrons, a reasonable
spondence between the two tensors remains qualitative. match is observed between the different sets of results.
The mass anisotropy of electrons, extracted from studies In the case of holes, on the other hand, as seen in Fig.
of quantum oscillations[9, 39] is as large as 200. The 7, a difference is visible at low temperature: the mobil-
anisotropy of mobility does not exceed 40. This im- ities extracted from data saturate to values significantly
plies an anisotropic relaxation time tempering the mass lower than what was found in the zero-field limit. This
anisotropy. may be due to a difference in the ultimate carrier mean-
It is instructive to compare our mobilities at B = 0.5 free-path in different samples. The residual resistivity in
T for various orientations of magnetic field to those re- our samples is an order of magnitude larger than those
portedby previousauthorsin thezero-fieldlimit[36, 37]. studied by Hartman. It may also partially arise from a
9
100K 90 50K 90
1.0 120 60 0.5T 1.0 120 60
1T
1-dezilamroN0000....0505 180150 300 23468111TTTTT024TTT 0000....0505 180150 300
210 330 210 330
1.0 240 300 1.0 240 300
270 270
30K 90 0.5T 20K 90
1.0 120 60 12TT 1.0 120 60
3T
1- dezilam 000...050 180150 300 468111TTT024TTT 000...050 180150 300
ro
N
0.5 0.5
210 330 210 330
1.0 240 300 1.0 240 300
FIG. 8: Temperature dependence of the components of the 270 270
electron mobility tensor, µi, obtained by fitting our angle- 1.100K 120 90 60 012.TT5T 1.50K 120 90 60
dependentmagnetoresistancedataatB =0.5Tforthethree 3T
ptelamnpeesroafturoret.ation. Thediscrepancyincreaseswithdecreasing 1- dezilam000...050 180150 300 468111TTT024TTT 000...050 180150 300
ro
N
0.5 0.5
210 330 210 330
field-induced decrease in the magnitude of ν1 and ν3. 1.0 240 270 300 1.0 240 270 300
SAMPLE S1
Theeffectofmagneticfieldonthevariouscomponents
of mobility tensor shows itself in another fashion. Fig. 8
FIG. 9: Evolution of the angle-dependent magneto-
compares the magnitude of the components of the mo-
conductivitywithdecreasingtemperature. Eachpanelshows
bility tensor of electrons for the three rotating planes.
a polar plot of the inverse of magnetoresistance normalized
As seen in the figure, the discrepancy remains within
to its maximum value at a given magnetic field. At high-
the error margin at high temperatures, but exceeds it in temperature,thethreefoldsymmetryoftheunderlyinglattice
the low-temperature regime. Moreover, the discrepancy ispreserved. Asthetemperaturedecreases,thissymmetryis
has a clear pattern. All µ components become lower lostaboveathresholdmagneticfield,whichdecreasesinam-
1−4
in the case of the third configuration, with the current plitude with cooling.
applied along the bisectrix axis and the magnetic field
rotating in the (binary, trigonal) plane. This indicates
that the components of the mobility tensor are affected
bythemagnitudeandorientationofmagneticfieldinthe 3neν
σh = 3 (14)
low-temperature limit. 33 1+ν ν B2
1 3
Hartman[36] noticed that the components of mobil-
Therefore, the fit to the experiment at a fixed mag-
itytensorpresentaquadraticdecreasewithtemperature
netic field B yields only a single parameter proportional
over a wide temperature window. This T−2 tempera-
to the right hand side of this equation and not an inde-
ture dependence of mobility (and consequently zero-field
pendent estimation of ν and ν . The mystery is that
conductivity) is a hallmark of electron-electron scatter- 1 3
the ν and the ν ν found fromthe data obtainedin this
ing. Long ago, Baber[41] argued that in a metal with 1 1 3
configuration are five times larger than what was found
multiple electron reservoirs coupled to the lattice ther-
forthetwootherconfigurations. Thisdiscrepancyiswell
mal bath, electron-electron scattering should give rise to
beyondouruncertaintymargin. Itremainsthemainfail-
a T2 inelastic resistivity. In the case of metals hosting
ureofthesemi-classicalmodelemployedhereandpoints
correlated electrons, this T2 resistivity has been exten-
toanadditionaltransportprocessnottakenintoaccount
sivelydocumentedandisoneofthetwomainingredients
in the picture employed here.
of the much-discussed Kadowaki-Woods ratio[42].
One aspect of our data remains currently beyond our
understanding. In the first configuration, in which the VI. EMERGENCE OF A VALLEY-POLARIZED
current is applied along the trigonal axis and the field is STATE
rotatinginthe(binary,bisectrix)plane,thecontribution
of the hole pocket is expected to be independent of the Letusnowturnourattentiontothelossofvalleysym-
orientation of magnetic field. Indeed, for this configura- metry, a feature reported in two previous experimental
tion, the hole conductivity becomes simply: studies[13, 14]. In order to explore this feature, we have
10

performed extensive measurements with numerous sam-
ples performed in two different laboratories (Paris and B=0.1 T
Seoul) to document the emergence of spontaneous valley 1.10
polarization.
When an electric current is applied along the trigo-
1.05
nal axis, the three electron valleys remain degenerate.
Of course, this degeneracy is lifted by a magnetic field
SAMPLE P1
applied along an arbitrary orientation in the (binary, bi- )1.00
e
sectrix) plane. However, the C3 symmetry of the sys- g
a B=0.1 T
tem implies that a 2π/3 rotation of the magnetic field
r
around the trigonal axis should not affect the physical ev1.05
a
properties,sincethedegeneracyremainsliftedinexactly
(
the same way. Therefore, a magnetic field rotating in  SAMPLE S2
the (binary, bisectrix) plane, should generate an angle- /
)
1
dependent magnetoresistance keeping a six-fold symme- n1.00
try in a polar plot, resulting from the combination of ib
(
the C3 symmetry and the inversion symmetry. This is 1.10 B=0.1 T
indeed the case of our data at high temperature and/or
low magnetic field. At low temperature and high mag-
netic field, on the other hand, this symmetry was found 1.05

to be lost in all the samples studied.
SAMPLE S3
Fig. 9 presents a detailed study on a bismuth single 1.00
crystal with a square cross-section. Each panel of the
figure shows polar plots of ρ−1 at a given temperature 0 10 20 30
for different magnetic fields. In the following, ρ−1 would T(K)
be called “conductivity” keeping in mind that, because
0 2K
of the contribution of the off-diagonal elements σ (cid:54)=ρ−1. 2.5 330 30 4K
Ateachtemperature, thedataisnormalizedtothemax- 6K
2.0 8K
imum value of ρ−1 to allow an easy examination of the 10K
evolutionasthetemperaturedecreasesandthemagnetic 1.5 300 60 12K
14K
field increases. One can see that the sixfold symmetry )1.0 16K
clearly present at 100 K for all magnetic fields is lost m0.5
c
at 5K for all magnetic fields. In the intermediate tem- S00..00 270 90
perature range, the sixfold symmetry is present at low m
(0.5 Bin1
magnetic field but is lost at higher fields. 1
-1.0
Thelossofthreefoldsymmetrywithcooling,originally
1.5 240 120
reportedinref.[13], hasbeenconfirmedinmorethanten
2.0
different crystals of bismuth with different shapes and
cut from three different mother single crystals and us- 2.5 210 150
ing different experimental set-ups in Paris and in Seoul. 180
The low-temperature asymmetries found in various sam- SAMPLE P1
ples were different from one another. In particular, the
departure from threefold symmetry is more striking in
FIG. 10: Top: The ratio of ρ−1 to 1(ρ−1 +ρ−1 +ρ−1 )
sampleswithasquarecross-sectionthaninsampleswith bin1 3 bin1 bin2 bin3
as function of temperature in three different single crystals
a cylindrical or a triangular cross-section. On the other
of bismuth at B = 0.1 T. In all three cases, a sudden jump
hand, the temperature and magnetic field thresholds for occursatT ∼9K.Bottom: polarplotsofρ−1 forsampleP1.
the loss of symmetry were roughly similar.
Various possible experimental artifacts as the source
ofthelossofthreefoldsymmetrycanberuledout. Two-
axis rotation experiments performed have found a simi- have caused a detectable variation among the three elec-
lar loss of threefold symmetry[13]. Therefore, it cannot tron pockets[43]. Our bismuth crystals are twinned and
be a consequence of uncontrolled misalignment between therefore in addition to the dominant domain, there are
the magnetic field, the crystal axes and applied electric three minority domains. The presence of these minor-
current. The fact that the frequency of quantum os- ity domains generates a secondary set of Landau peaks
cillations remains identical for different orientations of visible by the Nernst measurements[10]. But, we cannot
magnetic field[14](see also below) rules out any possi- think of anyway their presence can distort the threefold
bleroleofinternaluncontrolledstrain,sinceexperiments symmetry of the dominant domain, the only source of
under strain have shown that presence of strain would magnetoresistance in our study.