DOI: 10.1038/NPHYS3321 Atomically Thin Nb SiTe Enhanced ...atomically thin crystal. The overall...

$: DOI: 10.1038/NPHYS3321 Atomically Thin Nb SiTe Enhanced ...atomically thin crystal. The overall rectangle arrangement of diffraction spots is consistent with the orthorhombic structure$
Supplementary Information for “Enhanced Electron Coherence in

Atomically Thin Nb3SiTe6”

Contents

1. Structure characteristic of Nb3SiTe6

2. Characterization of Nb3SiTe6 thin flakes

3. Electronic structure of Nb3SiTe6

4. Resistivity upturn induced by EEI

5. Fitting of MR to the 2D WAL model

5.1 Fitting of MR using Hikami-Larkin-Nagaoka (HLN) model

5.2 Fitting of MR using or Iordanskii - Lyanda-Geller – Pikus (ILP) model

6 Electron diffusion constant

7 Temperature dependence of dephasing rate τϕ-1

8 Phonon dimensionality

8.1 Debye temperature determined by specific heat measurements

8.2 Phonon wavelength estimated from ab initio calculations

Enhanced electron coherence in atomically thin Nb3SiTe6

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http://dx.doi.org/10.1038/nphys3321


Figure S1 | a and c, Crystal structures of (a) Nb3SiTe6 and (c) MoS2. b and d, The top view of a single

layer of (c) Nb3SiTe6 and (d) MoS2.


Figure S2 | a, Raman spectra of Nb3SiTe6 crystals with various thicknesses (displaced vertically). The

spectrum for the 4.5 nm flake is multiplied by a factor of 8, in which the hump around 150 nm-1 is caused

by laser damage of the Nb3SiTe6 crystals. b, [010] zone (perpendicular to layers) selected area electron

diffraction (SAED) pattern of a Nb3SiTe6 atomically thin crystal. The overall rectangle arrangement of

diffraction spots is consistent with the orthorhombic structure of Nb3SiTe6, while the hexagonal pattern of

brighter spots reflects the hexagonal Te lattice plane. c, High resolution transmission electron

microscopy image of the atomically thin crystal shown in b.


Figure S3 | a-c, Projected electronic band

structure and density of states of Nb3SiTe6 for

(a) bulk, (b) bi-layer, and (c) mono-layer.

The gray and cyan curves in the left panels

represent energy bands derived from Nb-4d

and Te-5p orbitals respectively, while the

black curves in the right panels represent

total density of states. The Fermi energy is

marked by the horizontal red lines.

We have performed electronic band structure calculation for Nb3SiTe6, using density functional

theory in the framework of generalized gradient approximation (GGA) in Perdew-Burke-Ernzerhof 1

parameterization with periodic boundary conditions using Vienna Ab-initio Simulation Package 2-4.

Projector-augmented wave method along with a plane wave basis set with energy cutoff of 220 eV was

used. To calculate equilibrium atomic structures, the Brillouin zone was sampled according to the

Monkhorst–Pack 5 scheme with a k-points mesh 3×6×4. To avoid spurious interactions between

neighboring structures in a tetragonal supercell, a minimum of vacuum layer of 20 Å in all non-periodic

direction was included. Structural relaxation was performed until the forces acting on each atom were less

than 0.05 eV/Å.

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SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3321




Figure S1 | a and c, Crystal structures of (a) Nb3SiTe6 and (c) MoS2. b and d, The top view of a single

layer of (c) Nb3SiTe6 and (d) MoS2.


Figure S2 | a, Raman spectra of Nb3SiTe6 crystals with various thicknesses (displaced vertically). The

spectrum for the 4.5 nm flake is multiplied by a factor of 8, in which the hump around 150 nm-1 is caused

by laser damage of the Nb3SiTe6 crystals. b, [010] zone (perpendicular to layers) selected area electron

diffraction (SAED) pattern of a Nb3SiTe6 atomically thin crystal. The overall rectangle arrangement of

diffraction spots is consistent with the orthorhombic structure of Nb3SiTe6, while the hexagonal pattern of

brighter spots reflects the hexagonal Te lattice plane. c, High resolution transmission electron

microscopy image of the atomically thin crystal shown in b.


Figure S3 | a-c, Projected electronic band

structure and density of states of Nb3SiTe6 for

(a) bulk, (b) bi-layer, and (c) mono-layer.

The gray and cyan curves in the left panels

represent energy bands derived from Nb-4d

and Te-5p orbitals respectively, while the

black curves in the right panels represent

total density of states. The Fermi energy is

marked by the horizontal red lines.

We have performed electronic band structure calculation for Nb3SiTe6, using density functional

theory in the framework of generalized gradient approximation (GGA) in Perdew-Burke-Ernzerhof 1

parameterization with periodic boundary conditions using Vienna Ab-initio Simulation Package 2-4.

Projector-augmented wave method along with a plane wave basis set with energy cutoff of 220 eV was

used. To calculate equilibrium atomic structures, the Brillouin zone was sampled according to the

Monkhorst–Pack 5 scheme with a k-points mesh 3×6×4. To avoid spurious interactions between

neighboring structures in a tetragonal supercell, a minimum of vacuum layer of 20 Å in all non-periodic

direction was included. Structural relaxation was performed until the forces acting on each atom were less

than 0.05 eV/Å.





Our DFT-PBE band structure calculations revealed that the metallicity of Nb3SiTe6 originates

from the specific bonding state of Nb ions, as detailed in supplementary information. From the projected

band structure and density of state shown in Fig. S3a, it can be seen that the valance bands crossing the

Fermi level are derived from Nb-4d orbitals, indicating that the transport properties of Nb3SiTe6 are

dominated by Nb-4d electrons. Moreover, the decrease of dimensionality was found to lead to an

unambiguous reconstruction of electronic band structure (Figs. S3b and S3c). When the thickness

approaches single sandwich layer, the valance bands become much narrower and the gap between

conduction and valence bands is doubled (~0.8 eV), while the Fermi level still crosses valence bands.

Figure S4 | a, Projections of one slab of the Nb3SiTe6 structures. b, The isosurface (0.04e/Å3) of the

spatial electron distribution of the Nb/Si plane. The unit cell is represented by solid rectangle.

The origin of metallicity in Nb3SiTe6 compound is further revealed by our DFT calculations. Like

Nb3GeTe6 6, Nb3SiTe6 has two inequivalent Nb sites, as shown in Fig. S4a. Nb1 is surrounded by 6 Te

neighbors, while Nb2 and Nb3 not only are bonded with Te but also forms additional metal-metal

bonding with each other (Fig. S4b). This gives rise to two different electronic configurations of Nb atoms:

4d1 for Nb1 (Nb+4) and 4d2 for Nb2 and Nb3 (Nb+3). Therefore the Nb3SiTe6 formula can be represented

as 2

6

24

2

3

TeSiNb)Nb( . Such electronic configuration is characterized by unoccupied states of Nb1 in the

valence band, thus resulting in metallic character of conductivity.


Figure S5 | logarithmic temperature dependence of low temperature resistivity upturns for (a) 6nm, (b)

8nm, (c) 10nm and (d) 15nm flakes. The solid lines are the guide for the eyes.



The 2D WAL can be described according to the Hikami-Larkin-Nagaoka (HLN) model 7-9:

2

2

4 221 3 1 1 13 3[ ( ) ( ) ( )]

2 2 2 2 2 2

SO s ie SO s s i

WAL

B B BB B B B BeGB B B

(1)

where is digamma function; Bx = ħ/(4eDx) with x = SO, i, s and e referring to the characteristic fields

for spin-orbit, inelastic, magnetic spin-flip and elastic scattering; 1/x is the corresponding scattering rate

for each process and D is the diffusion constant. The inelastic and spin-flip scattering contribute to

quantum decoherence, leading to the dephasing rate -1 = i

-1 +2s

-1 and the corresponding characteristic

dephasing field B = ħ/(4eD) = Bi + 2Bs.

To perform the fit, we presented the measured MR data in the form of conductance change

ΔG = G(B) - G(B=0). Given that Nb3SiTe6 is non-magnetic, our initial trail are performed with fixed Bs=0.

The classical orbital MR (B2) is also considered during fitting. As shown in Figs. 3a, the HLN model

agrees well with the T=2K magnetoconductance ΔG(B) of Nb3SiTe6 thin flakes. Even for the 6 nm sample

which shows relatively stronger noise in MR (see Fig. 3a), the HLN model can still describe its ΔG(B)

well with relatively larger uncertainty in fitting parameters (Figs. 3a and 3b).





We have also performed the fitting for the MR data at various temperatures (2-30K) for Nb3SiTe6

flakes of each thickness. The signature of classical MR (B2 dependence) becomes more remarkable at

higher fields and at higher temperatures owing to the rapid suppression of WAL, as shown in Figs. 4a-c.

For the thinnest sample we have successfully fabricated (i.e. 6 nm), the WAL signature at higher

temperatures (>2K) can hardly be resolved in MR due to strong noise, which may be associated with the

enhanced difficulty of fabricating good electric contact in thinner flakes, and/or stronger heating effects

during measurements due to its higher resistivity. Therefore, successful fits are only obtained for the 8, 10

and 15 nm samples which show clear WAL signature and relatively smooth MR data (Figs. 4a-c.).


The HLN model assumes that the Elliott-Yafet (EY) type spin-orbit coupling dominates spin

relaxation, which is generally the cause for metals. However, in material without bulk or structural

inversion symmetry, such as semiconductor quantum well and heterostructures, D’yakonov-Perel’ (DP)

type spin-orbit coupling dominates. In this case the model developed by Iordanskii, Lyanda-Geller, and

Pikus (ILP model) is more appropriate 10,11:

22 0

210

1 0 1 1

'2 1 3 2 1 2(2 1)1 3{ [ ]' '2 ( 2 ) ( ) 2 [(2 1) 1]

12ln ( ) 3 }2

SO SO SOn n

WALSO SO SO SOn

n n n n

tr

B B Ba a a ne B B BG B B B Ba na a a a a n aB B B B

BB CB B

(2)

12

SOn

B Ba nB B ,

Where C is the Euler’s constant, is digamma function. BSO(BSO’), Bϕ, Btr refers to the characteristic

fields for spin-orbit, dephasing, and elastic scattering, similar to the definition in the HLN model. The

essential difference between ILP and HLN models, i.e., mechanisms of SOC, is reflected in the definition

of BSO and BSO’ (See ref. 10,11 for details).

Figure S6 | a, The fits of the 2K magnetotransport data of 8, 10 and 15 nm Nb3SiTe6 thin flake crystals to

the ILP model. b, The coherence length l at T=2K obtained from HLN and ILP model. The error bars

represent uncertainties in the fits.

Though Nb3SiTe6 system seems fall in the frame of EY mechanism, we have also fitted our data

using ILP mode. As shown in Fig. S6a, the ILP model also fits the MR data well. More importantly, the

coherence length obtained from the ILP fits is almost the same as those from the HLN fits (Fig. S6b). In

fact, though ILP model considers DP type spin relaxation, the EY spin-relaxation can also be included

into ILP model without changing the form of the equation (Eq. 2) 11. When the spin relaxation is not

dominated by the DP mechanism, the characteristic field for DP SOC vanishes, and ILP model reduces to

HLN model 10,11. Therefore, the similar fitting results indicate that the EY mechanism dominates the spin-

relaxation in our Nb3SiTe6 thin flakes, and HLN model is sufficient to describe the WAL of Nb3SiTe6.

6. Electron diffusion constant 　　　





Figure S7 | The diffusion constant D at T=2K extracted from Einstein relation.

7. Temperature dependence of dephasing rate τϕ-1

At low temperatures, the total dephasing rate τ can be expressed by:

1,

1,

10

1

pheee (3)

where 10 is the finite zero temperature dephasing rate, 1

,

ee and 1,

phe represent the dephasing rate

caused by inelastic e-e and e-ph scattering, respectively. Given both of the inelastic scatterings from e-e

and e-ph interactions yield power law term in the temperature dependence of dephasing rate , the total

dephasing rate can be further written as:

1 10

phPPeAT BT (4)

where PeAT and phPBT are the contributions from e-e and e-ph inelastic scatterings respectively.

Normally, τ should vanish at zero temperature in the presence of only e–e and e–ph

scattering . Our observation of finite τ implies dephasing process even at zero temperature, which

leads us to suspect possible magnetic scattering from occasional magnetic Nb ions with different

oxidization state (e.g., ions at sample surface or near point defects). We further performed fitting using Eq.

1 with Bs being included as a free fitting parameter. Nevertheless, this process did not yield enhanced

fitting quality and the obtained Bs is extremely small or simply zero. Indeed, finite τ is widely observed

in various mesoscopic systems (see ref and references therein). Possibilities other than magnetic origin

have also been widely discussed Although we may not completely exclude magnetic scattering, it

should not play essential role in enhancing WAL signature with reducing thickness.

Although the origin of the finite τ0-1 at zero temperature is still unclear , the dephasing

mechanisms of e-e and e-ph inelastic scatterings have been well studied. It was found that both of these

two processes are strongly depend on dimensionality . The e-ph interaction is found to dominate the

dephasing in 3D but contributes much less at lower dimensions, with the exponent Pph varying from 2 to 4

for all dimensions . In contrast, the e-e interaction dominate the low temperature dephasing for 1D and

2D systems, with exponent Pe being 2/3 and 1 for 1D and 2D system respectively .

We used the above Eq. 4 to fit the temperature dependence of l and summarized the fitting

parameters in Table 1. At low temperatures, e-e interactions dominate the dephasing ( PeAT >> phPBT )

and the temperature dependence of τ for all three samples is asymptotic to T -1 (Pe = 0.95, 0.97 and

1.05 for 8, 10 and 15 nm samples, respectively), consistent with the theoretical predictions . With

increasing temperature, phonon contribution grows rapidly due to larger exponent Pph. The Pph acquired

from our fitting is 3.63, 2.80 and 2.13 for 8, 10, and 15 nm samples respectively, in agreement with the

theoretical predictions and experimental observations of Pph = 2~4 for 2 or 3 dimensions Although

further theoretical works are needed to understand the non-integer Pph value and its thickness dependence,

the sharp changes of Pph clearly suggest the variation of e-ph interactions with the flake thickness.

More importantly, from the fitting parameters listed in Table 1 we can find that the contributions

from the zero temperature dephasing rate 10 and e-e dephasing PeAT are similar for all samples, while

the e-ph interactions differ a lot. The exponent Pph of the e-ph term phPBT increases noticeably from the 15

nm sample to the 8 nm sample, implying the variation of e-ph interactions with the flake thickness. In

addition, the pre-factor is reduced by nearly three orders of magnitude in the 8nm sample as compared to

the 15 nm sample, implying that the e-ph scattering contribute much less to dephasing in thinner samples.





8. Phonon dimensionality


Figure S8 | a, Temperature dependence of specific heat of an Nb3SiTe6 bulk single crystal. b, The fit of

C/T to γ + βT2 at low temperatures (T ≤ 5K).


8.2.1 ab initio calculations for isotropic medium

The phonon wavelength can be also estimated from the ab initio calculations. The Debye

temperature can be evaluated from the averaged sound velocity, mυ , by the equation

1/334

AD m

B

Nh nk M

(2)

where NA is Avogadro’s number, is the density, M is molecular weight and n is number of atoms in the

unit cell 13. Averaged wave velocity (for the isotropic medium) can be obtained from

3/1

3312

31

ltm

(3)

where t and l are transverse and longitudinal elastic wave velocity obtained by using the isotropic

shear modulus G, the bulk modulus B and the density ρ from Navier’s equation:

2/1

G

t and

2/1

343

GB

l (4)

The shear modulus G and bulk modulus B of the orthorhombic crystalline material can be approximated

by equations: G = ½ (GR + GV) and B = ½ (BR + BV), where GR and BR are Reuss shear and Reuss bulk

moduli, GV and BV are Voigt shear and Voigt bulk moduli which can be calculated from the Eqs. (12-15)

of Ref. 13 using elastic constants given in Table 1S which we obtained following this work.

Table 1S. Elastic constants of Nb3SiTe6, Cij, GPa.

C11 C22 C33 C44 C55 C66 C12 C13 C23

118.3 29.3 108.3 5.0 37.8 4.8 11.8 25.8 11.6

It was obtained that Debye temperature D equal to 174K, yielding the phonon wavelength at 2K

to be 87 nm which corresponds very well with experimental estimations and also suggests the phonon

confinement for the thinner samples (15 nm) showing enhancement of WAL signature.

8.2.2 ab initio calculations for single crystal

Also the Debye temperature can be evaluated from the averaged sound velocity obtained by

integrating the elastic wave velocities over various directions of the orthorhombic single crystal 13:

,aVn

khB

D31

021

31

49

(5)

where V is the volume of the unit cell and a0 is a function which represents the average velocity of elastic

waves in different directions of the single crystal. The last term can be evaluated from the following

equation 14:

.fffffffa GFEDCBA 825157513444801008455193780 0 (6)

Here fA, fB, fC, fD, fE, fF and fG are values of the function a0 in the [100], [001], [010], [101], [ 031 ], [021]

and [102] directions, respectively (see Ref. 14 for details) and depend only on the elastic constants (see

Table 1S).





This calculation gives a Debye temperature of 147K, which is lower than that determined by heat

capacity measurements. Using this Debye temperature the phonon wavelength at 2K was estimated to be

74 nm, lower than that estimated from heat capacity measurements (~100 nm) and calculations for

isotropic medium (~87 nm), but still suggests the high phonon confinement for the thinner samples

(15 nm) showing enhancement of WAL signature.

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DOI: 10.1038/NPHYS3321 Atomically Thin Nb SiTe Enhanced ...atomically thin crystal. The overall...

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