An anisotropic model for the minimum thermal conductivity
Zhen Chen1#* and Chris Dames1,2*
1
Department of Mechanical Engineering, University of California, Berkeley, California
94720
2
Division of Materials Sciences, Lawrence Berkeley National Laboratory, Berkeley,
California 94720
(#Currently with the Department of Electrical Engineering, Stanford University)
(*Corresponding: zchen23@stanford.edu; cdames@berkeley.edu)
Supplemental Material
A. Expressions of k min-c and k min-ab for layered and chain-like materials.
It is helpful to convert Eq. (2) to an integral over frequency,
,
(S1)
where we introduce a new quantity, Gs, which can be understood as the density of states
weighted by the square of the ŝ direction projected velocity:
Gs (w ) = 8p 3 òò
1
( v × ŝ) dS
2
v
w
,
(S2)
where dS is an elemental area on an isoenergy surface in k space (Fig. 1a of Ref. S1).
Thus, Gs represents the product of the square of the phonon velocity component
along the direction of heat transfer (here, ŝ ) and the number of phonon modes per unit
frequency in the shell between  and +d, integrated over the entire FBZ. We refer to
Gs as the v2DOS (v2 indicating square-of-velocity-weighted), and its role in the thermal
conductivity (Eq. S1) is analogous to the role of the vDOS in the irradiation [Eq. (13) of
Ref. S1].
Substituting the anisotropic Debye dispersion and ellipsoidal FBZ models from
Ref. S1 into Eq. S2, we obtain expressions for v2DOS:
ì vc
2
ï 2 2 w , 0 £ w £ w D,c
6
p
v
ab
ïï
3
Gc,Layered (w ) = í
2
2
(w D,ab
- w 2 ) ù2
ï vc 1 é w D,c
ê
ú ,
ï 2 2
2
2
6p vab w êë w D,ab
- w D,c
úû
îï
(S3)
w D,c £ w £ w D,ab
ì vc
2
ï 2 2 w , 0 £ w £ w D,ab
ï 6p vab
ï
3
ì
ü
Gc,Chain-like (w ) = í
2
2
2
2
ï vc 1 ïw 3 - é w D,c (w - w D,ab ) ù ï ,
ê
ú ý
2
2
ï 6p 2 v 2 w í
êë w D,c - w D,ab úû ï
ab
ï
ï
î
þ
î
(S4)
w D,ab £ w £ w D,c
ì 1
2
ï 6p 2v w , 0 £ w £ w D,c
c
ïï
1
Gab,Layered (w ) = í
2
2
3
2
é w D,ab
w D,c
-w2 ù
ï w D,cw é w D,ab - w ù 2
,
ê 2
ï 2 ê 2
2 ú
2
2 ú
ïî 4p vc êë w D,ab - w D,c úû 12p vcw êë w D,ab - w D,c úû
ì 1
2
ï 6p 2 v w , 0 £ w £ w D,ab
c
ïï
Gab,Chain-like (w ) = í
2
2
2
ï 1
w é w D,c w - w D,ab
2
ï 2 w - 2 ê
2
2
6p vc
4p vc ê w D,c - w D,ab
ë
îï
(
) ùú
1
2
1
+
2
úû 12p vcw
(
(S5)
w D,c £ w £ w D,ab
2
2
é w D,c
w 2 - w D,ab
ê
2
2
êë w D,c - w D,ab
) ùú
úû
(S6)
3
2
,
w D,ab £ w £ w D,c
Substituting Eqs. S3-S6 into Eq. S1, we obtain expressions for the minimum
thermal conductivity:
,
,
(S7)
(S8)
,
(S9)
,
(S10)
where
. These results hold for all temperatures and anisotropy ratios, and
were evaluated numercially to generate the points in Fig. 1 of the main text.
B. Quantify the contributions of phonon focusing effect and FBZ effect to the ratio of
k min-iso k min-aniso .
We can now quantify how the two major effects depicted in Fig. 1, phonon
focusing and FBZ truncation, cause the breakdown of the apparently conservative k min-iso
treatment of a layered material. For WSe2 at room temperature, all three branches of the
k min-iso model saturate to their high T limit (isotropized D = 130 K and 173 K for TA and
LA branches, respectively). Thus applying the conservative approximation of Ref. S2 to
2
Eq. (1), we simply add k min-iso = kB kiso,m
vc (12p ) for each branch. For the k min-aniso model,
however, we underline that it is nontrivial to interpret the behavior of each of the three
branches. As we can see from Table II, for WSe2 at room temperature, the TA and TL1
branches approximate their intermediate T limit, while the TL2 branch is close to its high
T limit. Fortuitously, in Table I both of these two limiting cases collapse into the same
(
)
2
2
vc3 4p vab
expression. Thus comparing k min-aniso = k B kc,m
to Eq. (1), we deduce Eq. (3) in
the main text, in which we utilize the fact hatom =
1
6p 2
3
2
kiso,m
= 6p1 2 kab,m
kc,m . Equation (3) can
be interpreted in two groups: ( vab vc ) comes from the phonon focusing effect, and
2
( kab,m kc,m ) 3 comes from the FBZ truncation effect. Note that the prefactor 13 originates
4
1
3
from the projection of group velocity along the heat transfer direction integrated over the
full 4 solid angle for isotropic materials. We lose this 13 factor for highly anisotropic
materials because the FBZ truncation effect limits the integration to a much smaller solid
angle.
We can use Table II to quantify these various contributions to k min-aniso of WSe2 at
(
)
4
room temperature. The FBZ truncation effect [ 13 kab,m kc,m 3 ] contributes a factor of 2.6
to the ratio of k min-iso k min-aniso . The phonon focusing effect should be averaged over all
three branches: the velocity ratio [ ( vab vc ) ] is 0.6, 2.0, and 8.0 for TL2, TA, and TL1
2
branches, respectively. Weighting this by each branch’s contribution to the total thermal
conductivity (76%, 18%, and 6%, respectively; dashed lines in Fig. 3), finally the phonon
focusing effect contributes a factor of 1.3 to the ratio of k min-iso k min-aniso . Combining
these two effects, we arrive at an analytical prefactor of k min-iso k min-aniso = 3.4 . As a
comparison, the exact numerical prefactor is 4.0. This 15% discrepancy originates from
the fact that in order to obtain a compact expression for Eq. (3), we chose the simplified
high T limit for TL2 from Table I, instead of the exact but lengthy expression from Table
S1.
C. Comparison to a prior k c model.
We compare our expression of k min-c for highly layered materials in the high T
limit (Table 1 of the main text) to an earlier k c model [Ref. S3], which also exploits the
anisotropic Debye dispersion, though not the anisotropic First Brillouin zone (FBZ) or an
explicitly evaluated minimum mean free path. This model starts from a frequency
independent kinetic theory, and highlights the scaling relation between the thermal
conductivity and anisotropic velocities. It groups the relaxation time and specific heat
into one fitting parameter. For highly anisotropic layered materials, we can simplify Eq.
2
1 of Ref. S3, and obtain the scaling relation k c µ vc4 vab
. As a comparison, our model
2
gives k min-c µ vc3 vab
. This weaker vc dependence originates from the FBZ truncation
effect (Fig. 1b in the main text) and the more detailed spectral integration of Eq. S1.
(
)
Table S1. High temperature limits [ T ≫ min q D,c ,q D,ab ] of k min for materials with
arbitrary anisotropy ratios, where w D,c and w D,ab are Debye frequencies along c-axis and
ab-plane, respectively. These expressions simplify further for extreme anisotropy ratios,
which are given in Table I of the main text.
Layered
k min-c
k min-ab
é
2
2
3w 2 w 3
kB
v 3w w
å 2c êê w 2 D,ab- wD,c2 - 2 D,ab D,c
2
12p pol vab
D,ab
D,c
w
- w D,c
êë
D,ab
(
)
3
2
Chain-like
æ
öù
w D,c
p
ç - tan -1
÷ú
2
2 ÷ú
çè 2
w D,ab
- w D,c
ø úû
(
é
2
æ
ö w 2 3w - w
w D,c
kB
1 ê 3w D,abw D,c
p
D,ab
D,c
ç - tan -1
÷ - D,c
å
1
ê
2
2
2
24p pol vc w 2 - w 2
w D,ab + w D,c
çè 2
w D,ab - w D,c ÷ø
êë D,ab
D,c
(
)
é
æ w + w 2 - w 2 ö 3w 2 w 2 ù
3w 2 w 3
kB
v
å 2c êê 2 D,ab 2D,c 23 ln çç D,c w D,c D,ab ÷÷ - w 2 D,ab- w 2D,c úú
12p pol vab
D,ab
D,c
D,ab
w
w
è
ø
êë D,c
úû
D,ab
(
) ùú
ú
úû
)
é
2
2
4
æ w + w2 -w2 ö ù
3w D,ab
w D,c
kB
1 ê 3w D,abw D,c
D,c
D,ab ú
ç D,c
÷
å
3 ln
2
2
ê
2
24p pol vc w D,c - w D,ab w 2 - w 2
w D,ab
÷ø ú
èç
êë
úû
D,c
D,ab
(
)
References:
S1 Z. Chen, Z. Wei, Y. Chen, and C. Dames, Phys. Rev. B 87, 125426 (2013).
S2 C. Chiritescu, D.G. Cahill, N. Nguyen, D. Johnson, A. Bodapati, P. Keblinski, and P.
Zschack, Science 315, 351 (2007).
S3 W.P. Hsieh, B. Chen, J. Li, P. Keblinski, and D.G. Cahill, Phys. Rev. B 80, 1 (2009).