Anomalous thermal conductivity in multiwalled carbon nanotubes with impurities and short-range order V E Egorushkin, N V Melnikova, A N Ponomarev, A A Reshetnyak1 Institute of Strength Physics and Material Science, Siberian Branch of the Russian Academy of Sciences, 2/4, prospect Academicheskii, Tomsk, 634021, Russia 1
E-mail:
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Abstract. Low-temperature thermal conductivity and thermopower of multiwalled carbon nanotubes considered within a bundle of nanotubes are calculated taking into account multiple scattering of electrons on the atoms of impurities (like single carbon atoms) and short-range order regions arising due to a some distribution of the impurities. The calculations are realized on a base of the temperature diagrammatic Feynman techniques and the results of our research are in a good quantitative and qualitative agreement with the corresponding experimental data for multiwalled carbon nanotubes with diameter less or equal 10 nm at T= cSU 0δ ( p − p′) G0 ( p ) . 2
(8)
Note, that in the Eq.(8) U = U (0) = U0 without using the approximation U(x) = U0δ (x) . In the latter case, U ( p − p′) = U 0 . Expression (8) coincides for small c with the analogous quantity < G (1) > for the case of “dirty” metals [12]. Let us turn to the next term in the Eq.(4)
G ( 2 ) ( r , r ′) =
R’
R C
r C
r’
or, equivalently, in analytical form,
Gi(2) ,j =
G0 ( r − R )c( Ri )U ( R − Ri )G0 ( R − R′)c( R j )U ( R′ − R j )G0 ( R′ − r ′)dRdR′ .
In p – representation after the change p ′′ ↔ p ′ we have
′ ′ ′ −i ( p − p1 ) Ri e Gi(2) , j ( p, p ) = c ( Ri )c ( R j )G0 ( p )G0 ( p ) U ( p − p1 )G0 ( p1 )U ( p1 − p )e
i ( p′− p1 ) R j
= G0 ( p )G0 ( p′) U ( p − p1 )G0 ( p1 )U ( p1 − p′)c( Ri )e − i ( p − p1 ) Ri c ( R j )e
i ( p′− p1 ) R j
dp1
(9)
dp1
Eq. (9) describes “double” scattering by one site and “single” scattering by two sites (the quotes assume the possibility of including t –matrix that means the repeated multiple scattering by one site and by two sites, correspondently). It is useful estimate those terms in (9): 1) for i = j c 2 ( Ri ) = c( Ri ) and therefore
− i ( p − p′ ) Ri ′ ′ Gi(2) U ( p1 − p′)dp1 ,i ( p, p ) = G0 ( p )G0 ( p ) U ( p − p1 )G0 ( p1 )c ( Ri )e
= c( Ri )e − i ( p − p′) Ri G0 ( p)G0 ( p′) U ( p1 − p′)G0 ( p1 )U ( p − p1 )dp1
.
(10)
′ After averaging of the propagator of GF Gi(2) ,i ( p, p ) in (10) with respect to configurations of impurities and then summation over index i [ c( Ri ) = c ;
e −i ( p − p′) Ri = δ ( p − p ′) ] we have
i
< G ( p, p′) >ii = cU δ ( p − p′)G02 ( p) G0 ( p1 )dp1 . (2)
2) for i ≠ j
2 0
− i ( p − p1 ) Ri ′ ′ Gi(2) c ( R j )e ≠ j = G0 ( p )G0 ( p ) U ( p − p1 )G0 ( p1 )U ( p1 − p )c ( Ri )e
Averaging and summation over i,j lead to the representation
(11) − i ( p1 − p′ ) R j
dp1 .
−i ( p − p′)R < G(2) ( p, p′) >i≠ j = G0 ( p)G0 ( p′) U( p − p1)G0( p1)U( p1 − p′) i≠ j e−i( p− p1)Ri e 1 j c(Ri )c(Rj ) .(12) + c2} dp1 =U02G0 ( p) c2G02 ( p)δ ( p − p′) +G0( p′) c( p − p1) c( p1− p′) G0 ( p1)dp1
}
{
{
{
}
One should be noted that the first term in the Eq.(12), U c G ( p )δ ( p − p′) , coincides with the corresponding term in the GF for the “dirty” metal [13]. In its turn, from the analysis of the second term, 2 2 0
3 0
U 02G0 ( p )G0 ( p′) < c( p − p1 ) c( p 1 − p′) >G0 ( p1 ) dp1 ,
(13)
under the suggestion that the electron scattering on the defects in disordered system to be elastic it follows p = p′ and the Eq. (13) takes the form
U 02G02 ( p ) < c ( p − p1 ) >G0 ( p1 ) dp1 . 2
(14)
Collecting the resulting terms (8), (11), (12) together with use of the Eqs. (13), (14) we get
G ( p, p′) = G0 + G (1) + G (2) + G (2) = G0 + cU 0δ ( p − p′) G0 ( p ) ii i≠ j , (15) 2 2 2 2 + U 0 G0 ( p )δ ( p − p′) c G0 ( p1 ) dp1 + c G0 ( p ) + c( p − p1 ) G0 ( p1 ) dp1. 2
The representation (15) permits one to find self-energy part Σ , electron relaxation time τ =−2ImΣ and properly the GF averaged over disorder. To this end, we start from the Dyson equation (16) G ( p ) = G0 ( p ) + G0 ( p ) G ( p ) , −1
or, in the first order with respect to Σ ,
G ( p ) = G0 ( p ) + G02 ( p )
From the Eqs. (15), (17) we may to derive the expression for
= cU 0 + c U G0 ( p ) + cU 2
2 0
2 0
G0 ( p1 )dp1 +U
2 0
,
(17)
as follows,
c( p − p1 )
2
G0 ( p1 ) dp1.
(18)
, Im Σ . However, they
The first two summands do not give contribution to the imaginary part of
determine the contribution to a change of chemical potential (i.e. of the Fermi level), µ ∗ = µ − cU 0 , and contributions of the following higher order infinitesimal. Therefore, to calculate the electron relaxation time τ we have to consider two last terms in the Eq. (18). As the result for Im Σ = Im Σ 1(2) + Im Σ 2(2) , we have, firstly,
Σ1(2) = cU 02 G0 ( p1 ) dp1 . Pass to integrating with respect to scalar ξ , ξ = p 2 2m − µ , and omitting the factor (2π ) −3 we get
G0 ( p1 )dp1 =
ν0 4π
dn d ξ G0 (ξ ) = ν 0
n =1
dξ = −iπν 0 signε , ε − ξ + i 0 signε
with angle vector n . Thus, we obtain for Σ1(2) the expression,
Σ1(2) = −icU 02πν 0 sign ε = −
i
sign ε ,
2τ imp
(19)
where ν 0 is the density of electron states on the Fermi level, and
(τ ) imp
−1
= 2π cU 02ν 0 .
(20)
The last expression coincides with the corresponding expression for electron relaxation time calculated within the theory of “dirty” metals with impurities [13, 14]. Secondly, for the quantity of the formula
c( k )
2
=
(2) 2
from (18),
(2) 2
= U 02
∞
c( p − p1 )
2
G0 ( p1 )dp1. and validity
c(1 − c) N α i cos kRi (see formula before Eq.(1)) [12] we derive, N i=0 N ∞ cos( p − p1 ) Ri dp1 (2) 2 c (1 − c ) αi . 2 = U0 N i = 0 0 ε − ξ ( p1 ) + i 0 sign ε
, cos y = The standard Euler formula for cos y, y = ( p −p 1 )R
Therefore,
1 2
(e
iyi
)
+ e −iyi leads to the representation
∞ e ±ip1Ri Ri dp1 4π sin( p1 Ri ) p1 dp1 = . ε − ξ ( p ) + i 0 sign ε R ε − ξ ( p ) + i 0 sign ε 1 1 0 0
(2) 2
(21)
has the form
(2) 2
4π 2U 02 mc(1 − c) N cos( pRi ) ixRi signε αi =− e , N Ri i =0
(22)
where we have introduced the notation for variable x, x = 2m(ε + ε F + i 0 signε ) Substituting expressions (19) and (22) into Eq. (18) we obtain the self-energy part of GF,
i sin ε 4π 2U 02 mc(1 − c) N cos( pRi ) ixRi signε αi ( p, ε ) = − − e . 2τ N Ri i=0
(23)
( p, ε ) where the former term will correspond to the renormalization of the chemical potential µ → µ ∗ = µ − real and, the latter one for small It is easily to extract the real and imaginary parts of
momentum of electrons: p → 0 takes the form
Im ( p, ε ) p →0 = Im (ε ) = −i
1 2τ
4π 2U 02 mc (1 − c ) N sin xRi αi + signε . N Ri i =0
Therefore, for small electron energy ε we have:
Im (ε ) ≈ − 2i ( s ign ε) τ −1 ,
(24)
where,
1 1 x = + 4π 2U 02 mc(1 − c) 2τ 2τ imp N
N
i =1
αi .
(25)
The parameter of short-range order α i in (25) is negative in the first coordinate sphere ( α1 < 0 ) and positive in the second one ( α 2 > 0 ). It is important that the coordinate sphere parameters α i may change a sign in dependence of atoms which generate the pairs. The first term in (25), τ imp −1 represents the inverse relaxation time of electrons scattered by impurities. The second one corresponds to the inverse relaxation time of electron scattered by structural inhomogeneities of shortrange order type. At last, the one-electron GF averaged with respect to configurations in “dirty” metallic CNT has the form (see, for details Refs.[12-14]:
(
G (ε , p ) = ε − ξ ( p ) + 2i τ −1 ( p ) sign ε
)
−1
,
(26)
where ξ ( p ) = p 2 m − µ , p and m – momentum and mass of electron, µ is the renormalized electron chemical potential. We shall use expressions (25) and (26) to calculate the contribution of multiple elastic electronic scattering by the short-range order regions. ∗
2
∗
3. Thermal conductivity As far as we consider model CNTs representing at low-temperature region the metallic systems with elastic electron scattering we may use the Wiedemann - Franz law to determine a thermal conductivity
χ (T ) = 13 ( e−1π kB ) Tσ (T ) . 2
(27)
where kB, e, n are respectively the Boltzman constant, charge of the electron and concentration of electrons with σ (T ) = m −1e 2τ n being by Drude conductivity. Because of the electron relaxation time is given by the Eq. (25), we have for the first coordination sphere, after the following substitution ε → ε l = π T (2l + 1) (for vanishing integer l)
σ (T ) =
ne 2 1 , 2 2 4π cU 0 mν 0 1 − BT 1 2
(28)
where B = (ν 0 N ) 2 2π (1 − c) m3 2 . Finally, we obtain −1
χ (T ) =
kB 2n T , 2 12cU 0 mν 0 1 − BT 1 2
(29)
It is useful to estimate the above contribution to thermal conductivity of multi-walled CNTs. Doing so, we have for the concentration of electrons in CNT, n=1028-1029m-3; e=1,6 10-19C; m= 9,1 10 -31 kg; v0=1eV-1=0,625 1019 J-1; =0,1; U 02 =(0,05eV)2=5 10-4 (1,6 10-19)2 J2. Hence, we get
( 4π
2
cU 02 mν 0 ) ne 2 = 7,1 ⋅10 6 S ⋅ m −1 and B = ( Np0 ) 4π 2 2π (1− c)m3 2kB1 2 =1,4K−1 2 (with restored −1
−1
dependence of Plank constant and momentum on the Fermi level p0 ). Therefore, the estimation holds
( 12cU
2 0
mν 0 ) kB 2 n ≈ 4,6 W ⋅ −1
−2
⋅ m−1 ,
(30)
that permits one to derive for T~50K the value for thermal conductivity, χ ( ) ∝ −26W ⋅ −1 ⋅ m −1 . The last result is to be in a good qualitative and quantitative agreement with the experimental data [6] for multi-walled CNTs with impurities with diameter less or equal 10 nm and we present on Figure 3 the low-temperature dependence of the calculated thermal conductivity.
Figure 3. Thermal conductivity of a model and single MCNT with diameter 200 (curve 1), 80 (curve 2) and 14 (curve 3). (Obtained from the experimental data in Ref.[6]). Theoretical temperature dependence of the input of the multiple elastic scattering of the electrons on the shortrange ordered regions in (T) (curve 4 as red type). 4. Thermopower In our recent paper [11] we have proposed a theory of low-temperature thermopower in CNTs. On a base of the Mott formula
S (T ) = 13 e k B2π 2T (τ ∂τ −1 / ∂ε ) −1
ε =ε F
,
we have calculated the corresponding one-particle contribution to thermopower in the form, −1 −1 S (T ) = − (18 e N ) k B2π m (1 − c ) α i Ri 2T , R0 = 2π ( k 0 ) , i
(31) (32)
where Ri , i are the radius of coordinate sphere and the parameters of short-range order, which includes the coordinate numbers [14] and k 0 is the wave-vector on Fermi level. From (33) we would like to make the following conclusions: • At low T the magnitude and curve slope of S(T) significantly depend on the size of inhomogeneity regions and do not determined by the parameters of electronic structure. • The change of thermopower sign for MCNTs in the oxygenated and deoxygenated states related to the geometry of a new structure of MCNTs, so that for i>0 we have S(T)
