Weyl semimetals: the
next (next) graphene?
Leon Balents, KITP, UCSB
Anton Burkov, U. Waterloo
Gabor Halasz, U. Cambridge
Outline
• Weyl fermions
• Where to find them
• TR-breaking and Hall effects
• I-breaking
• Graphene-like physics
Weyl Fermion
• Massless Dirac fermion with fixed
handedness
• described by a 2-component spinor unlike
4-component (spin+particle/hole) Dirac
spinor
H = v�σ · �k
Level repulsion
•
von Neumann and
Wigner, 1929
•
In QMs, 3 parameters
must be tuned to
make 2 levels cross
•
led to a whole field of
statistics of energy
levels, quantum
chaos,...
or broken by any infinie potential U which preis an integer, and according to whether this
symmetry. It is vanishingly
curves to lie in planes of integer is odd or even the number of circuits of
Z; however a contact curve the second type along which contact between the
bands i and occurs must be odd or even. Since
symmetry axis at a point
or contact of any crystal with an inversion center can be made
generacy
in the form of U into
occurs.
an infinitesimal
In 3d band by
structures
withchange
non-degenerate
one whose space group is merely its translation
crystal with an inversion
- lacking
either inversion or TR - this
manifolds
f inequivalent bands
group plus the inversion, this implies certain
at a point k happens
on a sym- at restrictions
on the numbers of contact curves
isolated points
which may occur for crystals of higher symose that m'(k) and m'(k)
nal. Then if the the
existence of curves of
vectornon-degeracy
of the requires
g metry. Prediction
of course
Hartree case to (P„', iVPq,
contact of equivalent manifolds may therefore
breaking spin-rotation symmetry must pass be possible from a knowledge merely of the
urve of contact DEGENERACY
contact by
SOCof the different M'(k, ) at the eight
energies
may be a curve oftypically
s of the type just described,
points k„.
It
of contact of inequivalent
For a crystal without an inversion center, the
E.
of symmetry. Naturally if energy separation 8E(k+x) in the neigborhood
metry plane in the space of a point k where contact of equivalent maniernative must hold.
folds occurs may be expected to be of the order
+0, for all directions of x.
space group consists only of ~ as ~—
For a crystal with an inversion center, the
p plus an inversion, three
Weyl points in band
j
theory
•
•
))'
ACCIDENTAL
s vectors for a real representation
of the
e group of the crystal, and that the normal
es belonging to a representation
which is
ucible in the field of real numbers, even
gh reducible in the complex field, must all
the same frequency. 7 Thus mathematically
theory of normal modes and their frequencies
Cf.
&
E. Wigner, Gott.
GUST
Nachr. (1930), p. 133.
15, 1937
365
is just like the theory of electronic wave functions
and their energies: frequency can be plotted as a
function of wave vector, and sticking together of
two or more of these frequency bands will occur
at wave vectors k where G' has multidimensional
representations or where case (b) or case (c), as
defined above, occurs.
is a pleasure for me to express my thanks to
Professor
Wigner, who suggested this problem.
PHYSICAL REVIEW
VOLUM E 52
Accidental Degeneracy in the Energy Bands of Crystals
CONYERS HERRING
Princeton
Princeton, Net Jersey
(Received June 16, 1937)
University,
The circumstances are investigated under which two wave functions occurring in the Hartree
or I'ock solution for a crystal can have the same reduced wave vector and the same energy, It
is found that coincidence of the energies of wave functions with the same symmetry properties,
as well as those with different symmetries, is often to be expected. Some qualitative features
are derived of the way in which energy varies with wave vector near wave vectors for which
degeneracy occurs. All these results, like those of the preceding paper, should be applicable
also to the frequency spectrum of the normal modes of vibration of a crystal.
Introduction
Ln2Ir2O7
Resistivity (polycrystalline samples)
pyrochlore oxides
107
106
Ln3+: (4f)n Localized moment
Magnetic frustration
Ln2Ir2O7 Pyrochlores
Ho
Conduction electrons
Ir[t2g]+O[2p]
105
Dy
104
Tb
105
103
conduction band
! (m" cm)
Ir4+: 5d5
106
Itinerant electron system
on the pyrochlore lattice
104
Dy
B
80 100
300
Ho
103
Tb
Gd
Eu
IrO6
60
Sm
102
Nd
101
Ln2Ir2O7
Pr
100
Ln
A
0
50
100
150
200
250
300
T(K)
O!
Metal Insulator Transition
(Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)
• Series of materials shows systematic MITs
D.Yanagashima,Y. Maeno, 2001
• Ir has λ≈0.5eV
K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) 043706.
(Ln=Nd, Sm, Eu)
1
4+
FULL PAPERS
is known to be attributed to
imensional antiferromagnets;
e lattice to induce a T -linear
ossible origin for T -linear
insulating state, Anderson
d.35) Further investigation is
f the T -linear contribution.
n dependence of the entropy
. To estimate !S, a smooth
ata outside the region of the
oken line) for Ln = Nd, Sm,
, 6(a), and 6(b), respectively.
was subtracted from the raw
of the C=T (!C=T ) for
in the inset. By integrating
7, 2.0, and 1.4 J/(K$mole) for
K. MATSUHIRA et al.
K. Matsuhira et al, 2011
Fig. 7.
(Color online) Phase diagram of Ln Ir O based on Ln3þ ionic
Exotic Possibilities
• Topological Mott Insulator?
D. Pesin+LB, 2010
Exotic Possibilities
• Topological Mott Insulator?
D. Pesin+LB, 2010
Probably not: commensurate
magnetic order seen in μSR
S. Zhao et al, 2011
WANATH, AND SAVRASOV
Weyl semimetal?
•
X. Wan et al, 2011
TOPOLOGICAL SEMIMETAL AND FERMI-ARC SURFACE . . .
LDA+U calculations
find Weyl state!
PHYSICAL REVIEW B 83, 205101 (2011)
insulating ground states evolve from a
allic phase via a magnetic transition.9,10
hown to arise from the Ir sites, since it
Lu, where the A sites are nonmagnetic.
ture remains unknown, ferromagnetic
d unlikely, since magnetic hysteresis is
ronic structure calculations can naturally
ution and point to a novel ground state.
agnetic moments order on the Ir sites
ttern with moment on a tetrahedron
ut from the center. This structure retains
a fact that greatly aids the electronic
hile the magnetic pattern remains fixed,
es evolve with correlation strength. For
in the absence of magnetic order, a
contrast to the interesting topological
Ref. 8. With strong correlations we find
all-in/all-out magnetic order. However,
ediate correlations, relevant to Y2 Ir2 O7 ,
state is found to be a Weyl semimetal,
ng Dirac nodes at the chemical potential
escribed above.
the possibility of an exotic insulating
n the Weyl points annihilate in pairs
e reduced; we call it the θ = π axion
our LSDA + U + SO calculations find
e intervenes before this possibility is
at local-density approximation (LDA)
stimates gaps, so this scenario could well
lly, we mention that modest magnetic
reorientation of the magnetic moments,
hase. Previous studies include Ref. 18, an
considered ferromagnetism. In Ref. 19,
del of Ref. 8 was extended to include
FIG. 1. (Color online) Sketch of the predicted phase diagram
for pyrochlore iridiates. The horizontal axis corresponds to the
increasing interaction among Ir 5d electrons while the vertical axis
corresponds to external magnetic field, which can trigger a transition
out of the noncollinear “all-in/all-out” ground state, which has several
electronic phases.
• They also pointed out very unusual surface
states
full Hamiltonian to linear order. No assumptions are needed
beyond the requirement that the two eigenvalues become
degenerate at k0 . The velocity vectors vi are generically
nonvanishing and linearly independent.
The energy dispersion
"#
3
2
is conelike, $E = v0 · q ±
i=1 (vi · q) . One can assign a
chirality (or chiral charge) c = ±1 to the fermions defined as
c = sgn(v1 · v2 × v3 ). Note that, since the 2 × 2 Pauli matrices
appear, our Weyl particles are two-component fermions. In
contrast to regular four component Dirac fermions, it is not
possible to introduce a mass gap. The only way for these modes
along high-s
insulating, a
an extension
using the pa
At the L po
with opposi
be argued th
U where th
Appendix).
connected to
the smaller
borne out by
analysis pert
subsection) a
a Weyl semi
Indeed, i
1.5 eV, we
within the "in Fig. 4 and
There also a
D FERMI-ARC SURFACE . . .
Fermi Arcs
• On most surfaces, metallic Fermi surfaces
which are not closed - “arcs” - terminate at
the projections of the Weyl points
PHYSICAL REVIEW B 83, 205101 (2011)
(b)]. Hence, this surface state
on the surface Brillouin zone
d for every curve enclosing
ergy, there is a Fermi line in
terminates at the Weyl point
c beginning on a Weyl point
a Weyl point of the opposite
ty of the Weyl points within
n determining the number of
posite chirality line up along
cancellation and no surface
O7 , at U = 1.5 eV, a Dirac
occur at the momentum
ordinate system aligned with
and equivalent points (see
s occurring on the edges of a
opposite chirality occupying
points (0.52,0.52,0.30)2π/a
the case of U = 1.5 eV, the
0.52(4π/a). Thus, the (111)
surface states connecting the
6 for the (110) surface states
or the (111) surface]. If, on
urface orthogonal to the (001)
e chirality are projected to the
the edges of the cube. Thus,
for this surface.
onsiderations, we have con-
FIG. 6. (Color online) Surface states. The calculated surface
energy bands correspond to the (110) surface of the pyrochlore
iridate Y2 Ir2 O7 . A tight-binding approximation has been used to
simulate the bulk band structure with three-dimensional Weyl points
as found by our LSDA + U + SO calculation.
2 2The plot
7 corresponds
to diagonalizing 128 atoms slab with two surfaces. The upper inset
shows a sketch of the deduced Fermi arcs connecting projected
bulk Weyl points of opposite chirality. The inset below sketches the
24 Weyl points
predicted in Y Ir O
Heterostructuring
Can we engineer Weyl points
in a heterostructure?
• A: yes! And you can do it with topological
insulators
TI
normal I
TI to NI transition
TI
NI
strong
tunneling
across the
NI “heals” TI
Tunneling
across TI
slabs kills the
3d TI
TI to NI transition
TI
NI
in between is a
(quantum) phase
transition
strong
tunneling
across the
NI “heals” TI
Tunneling
across TI
slabs kills the
3d TI
TI to NI transition
TI
NI
in between is a
(quantum) phase
transition
strong
tunneling
across the
NI “heals” TI
We can turn this
critical point into the
Weyl semimetal by
breaking I or TR
Tunneling
across TI
slabs kills the
3d TI
10
S. Murakami, 2007
Figure 5. Phase diagram for the QSH and ordinary insulating (I) ph
TR breaking
Bandstructure HgTe
• Dope with magnetic impurities (already
E
achieved in Bi-based TIs)
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
4.0nm
7.0 nm
6.2 nm
k
E1
H1
normal
Bandstructure
HgTe
gap
E1
H1
Δs
inverted
gap
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
TI
E
4.0nm
7.0 nm
6.2 nm
k
TI
E1
H1
normal
gap
Δd
H1
model just in terms
of surface states
m = exchange energy
E1
inverted
gap
TR breaking
Bandstructure HgTe
• Dope with magnetic impurities (already
E
achieved in Bi-based TIs)
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
4.0nm
7.0 nm
6.2 nm
k
E1
H1
normal
Bandstructure
HgTe
gap
E1
H1
Δs
inverted
gap
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
TI
E
4.0nm
7.0 nm
6.2 nm
k
TI
E1
H1
normal
gap
Δd
E1
H1
H=
�
k⊥ ,ij
m = exchange energy
inverted
gap
[vF τ z (ẑ × σ) · k⊥ δi,j + mσ z δi,j + ∆S τ x δi,j
+
�
1
1
∆d τ + δj,i+1 + ∆d τ − δj,i−1 c†k⊥ i ck⊥ j
2
2
TR breaking
Bandstructure HgTe
• Dope with magnetic impurities (already
E
achieved in Bi-based TIs)
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
4.0nm
7.0 nm
6.2 nm
k
E1
H1
normal
Bandstructure
HgTe
gap
E1
H1
Δs
inverted
gap
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
TI
E
4.0nm
7.0 nm
6.2 nm
k
TI
E1
H1
normal
gap
Δd
E1
H1
�2k±
m = exchange energy
=
∆(kz ) =
2
vF2 k⊥
�
+ [m ± ∆(kz )]
2
∆2s + ∆2d + 2∆s ∆d cos(kz d)
inverted
gap
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
2
TI
ΔD
TI
TI
TI
ΔD
Ins
TI
Δd
Weyl
semimetal
m
Ins
d
hematic drawing of the proposed multilayer strucshed layers are the TI layers, while hashed layers
dinary insulator spacers. Arrow in each TI layer
magnetization direction. Only three periods of the
e are shown in the figure, 20-30 unit cells can perown realistically.
m = exchange energy
rface of each layer. k⊥ is the momentum in
0
0
ΔS
(a) m=0
0
QAH
0
m
Ins
ΔS
(b) m≠0
FIG. 2. (Color online) Phase diagrams for (a) m = 0 and
(b) m �= 0. In (a), the red line represents the phase boundary between topological insulator (TI) and ordinary insulator
(Ins). In (b), due to TR symmetry breaking, the distinction
between topological and ordinary insulators is moot, so the TI
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
TI
kz
+
Δd
k0
-
m = exchange energy
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
TI
kz
Bµ (k) =
1
�µνλ d̂ · ∂ν d̂ × ∂λ d̂
8π
+
Δd
k0
∂µ Bµ (k) =
�
i
m = exchange energy
qi δ(k − ki )
“monopoles” of Berry curvature
Quantum Hall effect
c.f. Haldane 1988
kz
m(kz)
H = vkx σ x + vky σ y + m(kz )σ z
Quantum Hall effect
c.f. Haldane 1988
kz
m(kz)
H = vkx σ x + vky σ y + m(kz )σ z
σxy = 0
Quantum Hall effect
c.f. Haldane 1988
kz
m(kz)
H = vkx σ x + vky σ y + m(kz )σ z
σxy
e2
=
h
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
TI
c.f.Volovik, 2005
kz
+
Δd
k0
-
m = exchange energy
σxy
e2 k0
=
h 2π
semi-quantum AHE
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
TI
kz
+
Δd
k0
-
m = exchange energy
in
general
e2
σµν =
�µνλ Qλ
2πh
�
�ki qi + Q
� =
� RLV
Q
i
TR breaking
• Dope with magnetic impurities (already
achieved in Bi-based TIs)
Δs
TI
Δd
Σxy
1.0
0.8
0.6
0.4
0.2
m
0.5
m = exchange energy
1.0
1.5
�S
QAHE in finite multilayer
I breaking
• Asymmetric heterostructure, or intrinsic I
breaking
Δs
TI
+V
electrostatic potential asymmetry
-V
�2k± = vF2 (|k⊥ | ± V )2 + |∆(kz )|2
Δd
m = exchange energy
I breaking
• Asymmetric heterostructure, or intrinsic I
breaking
Δs
TI
+V
electrostatic potential asymmetry
-V
�2k± = vF2 (|k⊥ | ± V )2 + |∆(kz )|2
Δd
Naively gives nodal ring at
critical point with Δs=Δd
m = exchange energy
I breaking
• Asymmetric heterostructure, or intrinsic I
breaking
Δs
TI
+V
electrostatic potential asymmetry
-V
�2k± = vF2 (|k⊥ | ± V )2 + |∆(kz )|2
Δd
Need to include kdependence of Δs,Δd
m = exchange energy
I breaking
• Asymmetric heterostructure, or intrinsic I
breaking
Δs
TI
ky
- +
Δd
kx
+ -
no AHE
m = exchange energy
I breaking
• Asymmetric heterostructure, or intrinsic I
breaking
Δs
ky
-+
TI
kx
Δd
σxx
m = exchange energy
TI
+-
σyy
NI Δs-Δd
Hg1-xCdxTe structures
•
Checked this with semirealistic 10-orbital tight
binding model for
(Hg,Cd)Te superlattices
with asymmetry
•
Advantage: can be grown
with very high quality
•
Disadvantage: strain
must be controlled
HgTe
CdTe
HD = τ [βx ky� σ + βy kx� σ + βz kz σ ] ,
(14)
tion of the layer thicknesses N1,2 , the strain �0 ≡ �(HgTe) in
the multilayer structure, the amplitude U0 of the superlattice
potential, and the asymmetric displacement δ0 .
We first consider the dependence on �0 . If U0 �= 0 and
δ0 �= 0, there is a range in �0 close to zero where band touching is observed. This band touching is robust because it remains intact for an infinitesimal change in any of the external
parameters �0 , U0 , and δ0 . The upper and lower limits of the
range are functions of U0 and δ0 as illustrated in Fig. 2, and
we verify the expectation from Section II C that the range increases with both U0 and δ0 . For the reasonable values of
U0 ∼ 0.1 eV and δ0 ∼ a/2, this range is ∆�0 ∼ 0.002.
(l)
where the coefficients βx,y,z are again to be determined from
a comparison with the full model. The reduced Hamiltonian
finally reads H = H0 + HS + HD . It is a considerable simplification with respect to H, and it only contains seven parameters that need to be extracted from the full model.
Hg1-xCdxTe structures
C.
Conditions for robust band touching
Checked this with semirealistic 10-orbital tight
binding model for
(Hg,Cd)Te superlattices
with asymmetry
•
Advantage: can be grown
with very high quality
•
Disadvantage: strain
must be controlled
asymmetry
strain
•
Ε0
Ε0
TI
0.02
TI
W
qualitative level. Nevertheless, it provides useful guidelines
0.02
0.015 phase in this multifor the realization of the Weyl semimetal
NI
layer0.01
structure. The strain �0 has to be positiveNI
to avoid bandW
0.01
overlap but not too large because
∆ 0 !a
U0 !eV that would be hard to achieve
0.00
experimentally.
This gives a restriction on
the
thickness
of
the
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
HgTe layers: the ideal dimensionless thickness of 4 ≤ N1 ≤ 5
corresponds
to an actual thickness of 2 nm < d < 3 nm,
FIG. 2. Critical strains �0 against U0 at constant δ01 = a/2 (left) and
which
is onδ0the
reasonability.
against
at border
constantofUexperimental
0 = 0.1 eV (right). The phase boundaries
0.03
Σ
1.0
strain
Band touching between the HOB and the LUB occurs in
the full model when the two middle eigenvalues are equal in
the simplified model. It can be shown that for a 4 × 4 matrix of the form H, this is only possible if the direction of the
� = (βx ky� , βy kx� , βz kz ) lies halfway between the divector B
� (1,2) = (αx(1,2) kx� , αy(1,2) ky� , 0). The
rections of the vectors A
two bands then cross each other as |�k| is increased without
changing the direction of �k, whereas anti-crossing happens
otherwise. Since the above condition requires the three vec� has to
tors to lie in the same plane, the third component of B
vanish. Due to βz �= 0 in general, we find that robust band
touching can only occur in the kz = 0 plane.
Restricting our attention to this plane simplifies the problem
� (1,2) and B
� become 2D vectors. If we change the
because A
ratio ky� /kx� gradually from 0 to ∞, the ratios of the corre� (1,2) change in the same direction,
sponding components in A
� change in the opposite direction between 0
while those in B
and ±∞. This means that whether band touching happens at
any �k is determined entirely by the signs of the different pa(1,2)
rameters. Since we always choose αx,y > 0, the condition
becomes straightforward: band touching occurs if and only if
βx and βy have the same sign.
Let us now consider the special case of the symmetric potential with δ0 = 0. By repeating the symmetry considerations
in Section II B and taking into account the additional four-fold
roto-reflection symmetry around the z axis, we find that the
seven parameters from the full model are no longer indepen(1,2)
(1,2)
dent because αx
= αy
and βx = −βy . This shows that
band touching can only occur in this scenario if at least one of
0.0
separate three phases: the normal insulator (NI), the topological insulator (TI),
Ε 0 and the Weyl semimetal (W). The layer thicknesses are
N10.02
= N2 = 4 in both subfigures.
FIG. 4. (C
(red solid l
TI
Now we turn our attention to the layer thicknesses. Keepingthe norma
0.01
the HgTe thickness N1 =W
4 constant and increasing the CdTethrough th
thickness N2 between 4 and 8 reveals that such an increaseµ = vF2 (∆
decreases
∆�0 . This is understandable because δ0 becomesmeasured
0.00
smaller with respect
NI to the superlattice periodicity. Keeping
the CdTe thickness N2 = 4 constant and increasing the HgTe
M
thickness
N1 between 3 and 7 shifts the range in �0 towards0 during
!0.01
1
more negative values. By arguing on physical N
grounds
thateach Wey
(x, y, z) b
the system
is
in
the
NI
phase
when
the
HgTe
layers
are
thin
3
4
5
6
7
and in the TI phase when the HgTe layers are thick, we can
FIG.distinguish
3. Phase diagram
of the
the strain
�0 and
between
thesystem
NI andagainst
TI phases
around
thethephase
HgTe
thickness
other parameters
are constant: U0 = 0.2
1 . The
with
robustNband
touching
in between.
σl =
6
eV, δ0 To
= conclude
a/2, and Nthat
4. phase
The phase
boundaries
separate
four
2 =this
is indeed
a Weyl
semimetal,
it
phases:
theto
normal
insulator
(NI),condition:
the topological
insulator
(TI), overlap.
the
needs
satisfy
one more
the lack
of band
band overlap metal (M), and the Weyl semimetal (W).
Even if there is robust band touching between the HOB andwhere we
HgTe thickness
Graphene-like Physics
• 2d graphene physics can already be
achieved in HgTe quantum wells
Dirac peak at B=0
Bandstructure HgTe
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
E
4.0nm
7.0 nm
6.2 nm
k
E1
H1
normal
gap
H1
E1
inverted
gap
Peak width and mobilities comparable with/better than free standing graphene
Scattering mechanisms: probably mass fluctuations + Coulomb (fit is Kubo model)
L. Molenkamp: HgTe QWs are “better graphene”
“3d graphene”
• The transport behavior of 3d Dirac/Weyl
fermions is subtle and interesting!
• Naive argument (no disorder or
interactions):
Re σ(ω, T = 0) ∝ ω
• insulating?
With impurities
• Usually impurities induce elastic scattering
that dominates at low T
• Here, Born approximation is valid
(disorder is irrelevant in RG sense)
1/τ ∼ uimp ω 2
• Contrast graphene: higher order
corrections induce non-zero scattering
−
1/τ
∼
e
rate at zero frequency (SCBA)
c
uimp
Bandstructure HgT
With impurities
B.A Bernevig, T
• Neutral impurities w/o interactions leads
E
Re σ(ω, T ) ∝ σ0 f (ω/T 2 )
he optical conductivity, we assume a
range impurity scattering potential of
�
a
δ(r − ra ),
(8)
σ Σxx
σ1.0
0
7.
6.2 nm
to non-zero DC conductivity
al to the z-axis. For more details on
eader to Ref. [9].
his section we will focus on diagonal
eristics of the Weyl semimetal, namely
tivity. Some of the results, presented
in [9], but not derived in detail. As we
the frequency dependence of the optif the Weyl semimetal is very unusual,
or experimental characterization of this
(r) = u0
4.0nm
3
E1
0.8
normal
gap
0.6
H1
0.4
∝ T2
0.2
Ω
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
ω T2
FIG. 1. (Color online). Optical conductivity of the Weyl
semimetal, in units of the DC conductivity σDC . The ω/T 2
ratio on the horizontal axis is in units of 32π 2 γ/h3 vF3 .
H1
excited
carriers
E1
1/τ ∼ uimp ω 2
With interactions
• Coulomb interactions are marginal characterized by dimensionless fine
structure constant α=e2/εvF
• Leads to strong scattering
1/τ ∼ α2 max(ω, T ) � uimp ω 2
• Then expect�
σdc ∼ e
2
2
�
vF3
�
vF2 τ
kB T
∼
α
power law
insulator
Experiment?
• Experiments on Eu Ir O
2 2
insulator
2
! (m"cm)
2000
06 GPa
88 GPa
49 GPa
61 GPa
06 GPa
88 GPa
0.01 GPa
1.34 GPa
2.15 GPa
0
find “weak”
7
! (m"cm)
1500
10.01 GPa
20
11.34 GPa
0
1000
7.88 GPa
40
0
50 100 150 200 250 300
T (K)
2.06 GPa
2.88 GPa
3.49 GPa
4.61 GPa
6.06 GPa
7.88 GPa
10.01 GPa
11.34 GPa
12.15 GPa
TMI
500
0
300
at P = 2.06 to
orresponding
indicated by
he inset is an
≥ 7.88 GPa).
nt differences
FIG. 2. The phase diagram for Eu2 Ir2 O7 constructed from
our resistivity data. At low pressures, P < 6 GPa, the finite temperature MIT is indicated by red squares. At high
pressures, P > 6 GPa, the transition between conventional
and diffusive metallic states is indicated by blue circles. The
T ∗ cross-over is shown by orange triangles. All the lines are
guides to the eye. The quantum critical point (QCP) lies on
the P axis at P = 6.06 ± 0.60 GPa. Notice the weak temper-
0
50
100
150
T (K)
200
250
300
FIG. 1. Resistivity as a function of temperature at P = 2.06 to
12.15 GPa. The approximate location of TM I , corresponding
to a peak in the second derivative of ρ(T ), is indicated by
arrows for the three lowest pressure curves. The inset is an
expanded view of the higher pressure curves (P ≥ 7.88 GPa).
Tafti et al, 2011 and private communications
ical pressure [9], however there are important differences
FIG. 2. The
our resistivit
nite tempera
pressures, P
and diffusive
T ∗ cross-over
guides to the
the P axis at
Donors
• This is likely related to combined effect of
small carrier density and Coulomb
scattering from donors - O vacancies
• Follow ideas of calculation for graphene but
for 3d
c.f. Nomura+MacDonald, 2007
Donors
• Screening
e2
V (q) ∼ 2
q + ξ −2
• Scattering
τ −1 ∼ n
�
ξ −2 ∼ αkF2
d3 q δ(�q − �F )|V (k + q)|2 v(k · q)
�
2
1−cos2 θ
∼ e kF α d cos θ [2(1+cos θ)+α]2
c.f. Nomura+MacDonald, 2007
Donors
• Conductivity
σ ∼ e2
�
kF2
vF
�
vF2 τ
2 1/3
∼ f (α)e n
1
f (α) ∼ 1 + 2
α ln α
• Mean free path
σ ∼ e2 kF · kF �
c.f. Nomura+MacDonald, 2007
kF � ∼ f (α)
Conclusions
• Weyl semimetals occur in the same sorts of
materials as topological insulators (and
others!), if inversion or time reversal are broken
• They can be designed as intermediate states
between certain TIs and NIs
• They have unique transport properties and
surface states, and in some respects are 3d
analogs of graphene, with interactions and
defects playing crucial roles