doi: 10.1038/nphys3321 atomically thin nb site enhanced ...atomically thin crystal. the overall...
TRANSCRIPT
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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1. Structure characteristic of Nb3SiTe6
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.
2. Characterization of Nb3SiTe6 thin flakes
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.
3. Electronic structure of Nb3SiTe6
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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1. Structure characteristic of Nb3SiTe6
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.
2. Characterization of Nb3SiTe6 thin flakes
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.
3. Electronic structure of Nb3SiTe6
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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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.
4. Resistivity upturn induced by EEI
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.
5. Fitting of MR to the 2D WAL model
5.1 Fitting of MR using Hikami-Larkin-Nagaoka (HLN) model
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).
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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.
4. Resistivity upturn induced by EEI
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.
5. Fitting of MR to the 2D WAL model
5.1 Fitting of MR using Hikami-Larkin-Nagaoka (HLN) model
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).
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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.).
5.2 Fitting of MR using or Iordanskii - Lyanda-Geller – Pikus (ILP) model
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
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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.).
5.2 Fitting of MR using or Iordanskii - Lyanda-Geller – Pikus (ILP) model
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
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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.
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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.
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8. Phonon dimensionality
8.1 Debye temperature determined by specific heat measurements
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 Phonon wavelength estimated from ab initio calculations
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).
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8. Phonon dimensionality
8.1 Debye temperature determined by specific heat measurements
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 Phonon wavelength estimated from ab initio calculations
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).
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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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