TaIrTe
4”
F. Le Mardelé,1,∗ D. Santos-Cottin,1 E. Martino,1, 2 K. Semeniuk,2 S. Ben David,1 F. Orbanić,3 M. Novak,3 Z. Rukelj,1, 3 C. C. Homes,4 and Ana Akrap1,†
1
Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland 2IPHYS, EPFL, CH-1015 Lausanne, Switzerland
3
Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, HR-10000 Zagreb, Croatia 4
Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, New York 11973, USA
(Dated: June 8, 2020)
In the Supplementary Materials we show the sample characterization through quantum oscillations, and derive theo-retical expressions for the optical conductivity, Hall coefficient and Seebeck coefficient.
A. Sample characterization through quantum oscillations
FigureS1shows the results and analysis of the quantum oscillations of the resistivity. Two different directions of the magnetic field were investigated. Results for B k b are shown in Fig. S1a–c, the top tree panels, where the maximum field was 14 T. Results for B k c are in shown Fig. S1d–f, the bottom three panels, with a field up to 34 T.
Fig. S1a and d show the resistivity as a function of the inverse magnetic field after background subtraction. Fig. S1b and e show the the fast Fourier transformation of these oscillations, in order to extract the different frequencies for each pocket. Fig. S1c and d show how we can extract the effective mass for the same frequencies, thanks to the Lifshitz-Kosevitch formula.
In both directions we detect three frequencies. The results are summarized in in TableI.
Table I. Effective masses of the frequencies obtain from the quantum oscillation for two magnetic field direction. Effective masses m∗
Magnetic field direction :
B k b B k c
F1 0.34 ± 0.05 me 0.51 ± 0.06 me
F2 0.32 ± 0.04 me 0.81 ± 0.02 me
F3 0.28 ± 0.02 me 0.67 ± 0.03 me
B. Real part of dielectric response
In the main text we mostly focus on the real part of optical conductivity, σ1(ω). For completeness, in Fig.S2we show here the real component of the complex dielectric function, 1(ω).
C. Magnetotransport data
The Hall coefficient and Hall mobility in the main text are calculated from the experimentally determined ρxx and
ρxy, which we show in Fig.S3
∗florian.lemardele@unifr.ch †ana.akrap@unifr.ch
0.1 0.2 1/B(Tesla) −2 −1 0 1 2 3 ρx x (µ Ω .c m ) 1.8K 2K 3K 4K 6K (a)
B k b
100 200 F(T) 0 5 10 A m pl it ud e (a rb -u ni ts ) F3=109T F2=72T F1=40T (b) 0 2 4 6 8 10 Temperature (K) 0 2 4 6 8 10 12 A m pl it ud e (a rb -u ni ts ) F3 : m∗= 0.28 ± 0.02m e F2 : m∗= 0.32 ± 0.04m e F1 : m∗= 0.34 ± 0.05m e (c) 0.03 0.05 0.07 0.09 1/B(Tesla) −1.5 −0.5 0.5 1.5 ρx x (µ Ω .c m ) 1.2K 3.5K 4.5K 6K 8K (d)B k c
100 200 300 F(T) 0 5 10 15 20 25 A m pl it ud e (a rb -u ni ts ) F3=223T F2=168T F1=81T (e) 0 2 4 6 8 10 Temperature (K) 0 10 20 30 40 50 60 70 A m pl it ud e (a rb -u ni ts ) F3 : m∗= 0.67 ± 0.03me F2 : m∗= 0.81 ± 0.02m e F1 : m∗= 0.51 ± 0.06m e (f)Figure S1. The top 3 panels represent the analysis of quantum oscillations with B k b. (a) Resistivity oscillations as a function of the inverse magnetic field after background subtraction, at different temperatures. (b) FFT of these oscillations. (c) Mass calculation of the frequencies with the Lifshitz-Kosevich formula.
The bottom 3 panels represent the analysis of quantum oscillation with B k c. (d) Resistivity oscillations as a function of the inverse magnetic field after background subtraction, at different temperatures. (e) FFT of these oscillations. (f) Mass calculation of the frequencies with the Lifshitz-Kosevich formula.
D. DOS calculation
The tilted 3D Dirac system can be described by the following simple Hamiltonian: ˆ
H0= ~wkzI + ~vk · σ, (S1)
where σx,y,z are Pauli matrices, v Dirac velocity and w tilt velocity. The diagonalization of (S1) is straightforward and gives:
εc,vk = ~wkz± ~v|k|. (S2)
Density of state per unit volume is defined as
g(ε) = 1 V X kσ δ(ε − εk) = 2 (2π)2 1 (~v)3 Z ∞ 0 %d% Z ∞ −∞ dzδ(ε − γz +p%2+ z2). (S3)
In the above expression, we have changed the sum to integral and introduced variables v~kx = x, v~ky = y and
v~kz= z, after which we have transformed into cylindrical coordinate system by substituting % = p
−1500 −1000 −500 0 ε1 0 100 200 300 400 500 Wave number (cm−1) 295 K 250 K 200 K 150 K 100 K 50 K 5 K 0 10 20 30 40 50 60
Photon energy (meV)
Figure S2. The real component of the dielectric function, 1, shown for different temperatures in the far infrared range.
0 1 2 Magnetic field (T) 0 0.5 1 1.5 2 2.5 ρx y (µ Ω cm ) 2 K 5 K 10 K 25 K 50 K 100 K 200 K 300 K 0 2 4 6 8 10 Magnetic field (T) 0 10 20 30 40 50 ρx x (µ Ω cm ) 1.8 K
Figure S3. Resistivity tensor elements, ρxyand ρxx.
we have set γ = w/v. We perform the % integration first. The delta function is then simply:
δ(ε − γz +p%2+ z2) = δ(% − %0)
%0 q
%2
0+ z2, (S4)
where the %0is the zero of the argument of δ function ε−γz+p%20+ z2= 0. This explicitly gives %0=p(γz − ε)2− z2. This also imposes the boundary for the z integration. Since %0 is a real, the sub-root expression should be positive. This is true in the interval between z+= ε/(γ − 1) and z−= ε/(γ + 1), hence (S3) reduces to
g(ε) = 2 (2π)2 1 (~v)3 Z z+ z− dz(γz − ε) = 1 π2(~v)3 ε2 (γ − 1)2(γ + 1)2 = cε 2 (S5)
I. INTERBAND CURRENT VERTICIES
In the general form of the 2 × 2 Hamiltonian
H = bk ak a∗k dk
the interband L 6= L current vertices are JαkLL=X ``0 e ~ ∂H``0 k ∂kα Uk(`, L)Uk∗(`0, L) (S2)
where Uk(`, L) are the elements of unitary matrix defined as U ˆHU−1= E
Uk(`, L) = eiϕkcos(ϑ k/2) eiϕksin(ϑk/2) − sin(ϑk/2) cos(ϑk/2) ! (S3)
with the definitions
ak= |ak|eiϕk, tan ϕk= Im ak Re ak , tan ϑk= 2|ak| bk− dk . (S4)
Therefore in the general case of (S1), Eqn. (S2) gives ~ eJ vc αk= tan ϑk 2p1 + tan2ϑk ∂(bk− dk) ∂kα + i|ak| ∂ϕk ∂kα +p 1 1 + tan2ϑk ∂|ak| ∂kα . (S5)
Now we can determine the above derivations for the Hamiltonian (S1). We obtain
∂|ak| ∂kα = ~ v2kxδα,x+ v2kyδα,y p(vkx)2+ (vky)2 (S6) and ∂ϕk ∂kα =v 2(k xδα,y− kyδα,x) (vkx)2+ (vky)2 (S7) and trivially ∂(bk− dk) ∂kα = 2~vδ α,z. (S8) with tan ϑk= q k2 x+ ky2 kz (S9)
Then inserting (S6) and (S7) in (S5) for α = x we have
|Jvc xk| 2= e2v2 k 2 y k2 x+ k2y + k 2 ykz2 (k2 x+ k2y)(k2x+ ky2+ kz2) ! (S10)
Similarly for the z component
|Jvc zk| 2= e2v2 k 2 x+ k2y k2 x+ ky2+ kz2 (S11)
A. Optical conductivity in the limit of vanishing intraband relaxation constant
Optical conductivity is calculated in the case of a vanishing relaxation constant Γ. Introducing Ω = ~ω, and setting Γ → 0, the real part of the interband conductivity is
Re σααvc(Ω, T ) =~π V X kσ |Jαkvc| 2fkv− f c k εc k− εvk δ(Ω − εck+ εvk), (S12)
with components α ∈ {x, y, z} and the interband current vertices Jαkvc. We note that the expression (S12) is finite only for Ω = εc
k− εvk = 2
p
%2+ z2where we have immediately used the cylindrical coordinates like in the DOS calculation above. The energies can be for the same reason written as
εc,v= γz ±p%2+ z2= γz ± Ω/2 (S13)
This fact is to be used several times. First we change the |Jvc
αk|2 to cylindrical coordinates |Jvc xk| 2= e2v2 %2sin 2ϕ %2 + z2%2cos2ϕ %2(%2+ z2) (S14) we have Re σvcαα(Ω, T ) = ~e2v2π (2π)3 2 (~v)2 1 Ω Z ∞ 0 %d% Z ∞ −∞ dz Z 2π 0 dϕ sin2ϕ +z 2cos2ϕ (Ω/2)2 δ(Ω − 2p%2+ z2) [f (γz − Ω/2) − f (γz + Ω/2)]
The angular integration is trivial and only brings a π2 term. We can perform the z integration first. Similar to the DOS calculation, we have
δ(Ω − 2p%2+ z2) = δ(z − z0) 2|z0|
q
%2+ z2
0 (S15)
with two zeros, z0= ±p(Ω/2)2− %2. The realness of zeros further restricts the integration of % to (0, ∞) → (0, ω/2). The above integral then becomes:
Re σvcαα(Ω, T ) = e 2 4~2v Z Ω/2 0 %d% 1 +(p(Ω/2) 2− %2)2 (Ω/2)2 ! P s=± h f (sγp(Ω/2)2− %2− Ω/2) − f (sγp(Ω/2)2− %2+ Ω/2)i p(Ω/2)2− %2
Notice that by introducing the variable ν =p(Ω/2)2− %2 we have
dν = −%d%/p(Ω/2)2− %2 (S16)
and the above expression reduces substantially
Re σααvc(Ω, T ) = σ0 ~v Z Ω/2 0 dν(1 + 4ν2/Ω2) X s=±1 [f (sγν − Ω/2) − f (sγν + Ω/2)] (S17)
Here we use σ0= e2/(4~). In the low temperature limit (T → 0), the sum of the Fermi-Dirac distributions in above expression simplifies to a Heaviside step function, but since the argument contains both Ω and ν, it is not easily evaluated. Instead, (S17) is numerically evaluated.
B. Chemical potential, Hall constant and Seebeck coefficient
In the case of linear 3D Dirac dispersion chemical potential can be analytically determined from the Sommerfield expansion of the total electron concentration:
n =
Z
g(ε)f (ε)dε → ε3F = µ3+ π2k2BT2µ. (S18)
Then the cubic equation can be analytically solved for µ(T ). The Seebeck coefficient is defined within the constant relaxation time approximation as
Sα= − 1 eT P Lkσ(v L αk)2(εLk − µ)(∂fkL/∂εLk) P Lkσ(v L αk)2(∂f L k/∂ε L k) = − 1 eT P Lkσ(ε L k− µ)(∂fkL/∂εLk) P Lkσ(∂f L k/∂ε L k) (S19)
where we have used v2
αk→ v2/3. Since the total number of electrons is conserved, we have:
∂n ∂T = 0 = X Lkσ ∂fL k ∂T = X Lkσ εL k− µ T + ∂µ ∂T ∂fL k ∂εL k (S20)
Inserting (S20) in (S19) we finally get SD= 1 e ∂µ ∂T (S21)
Hall constant is defined as:
RH = −1/(nHe), (S22)
where we used the Hall concetration for the 3D Dirac
nH = 1 3π2 1 ~3v3 1 µ µ2+(πkBT ) 2 3 2 . (S23)
Also, we define the effective electron concentration
nα= − 1 V X kσ mevαk2 ∂fk ∂εk . (S24) Approximating vαk→ v2/3, we have: nα= mev2 3 1 V X kσ ∂fk ∂εk = mev 2 3 Z g(ε)∂f ∂εdε = mev2 3 c µ2+(πkBT ) 2 3 . (S25)
This defines the mobility as
σ(0) = eµn = e2τ nα/me→ µ =
eτ me
nα
n . (S26)
In the 3D Dirac T = 0 case the above relation gives
µ = eτ me
mev2
εF
. (S27)
This gives the Hall coefficient for a total of 4 Dirac points in the Brillouin zone, with a tilt γ = 0.37
RH≈ 50~
3v3µ
e (µ2+ (πkBT )2/3)2 (S28)
C. Infrared phonon optical response
The response function of the induced infrared phonon dipole mode of frequency ωT Oto the macroscopic electric field is given as
χ(ω) = A
ω2
T O− ω2− iγω
(S29)
The conductivity is then defined as σ(ω) = −iωχ(ω) and when compared to the experiment we can extract from the data: ωT O= 192 cm−1, γ = 0.5 meV = 4 cm−1= 1.3 × 10−12s, and A = 2.85 × e
2
ma.u.Vpc where ma.u.and Vpcare the atomic unit of mass and primitive cell volume.
