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6. LOW-TEMPERATURE PROPERTIESOF NON-CRYSTALLINE SOLIDS
T > 1 K: other low-frequency excitations, “soft modes”, and the Soft-Potential Model
T > 1 K ?
U. Buchenau et al., Phys.Rev. B34, 5665 (1986)
2
Basic assumptions of the SOFT-POTENTIAL MODEL:
The SOFT-POTENTIAL MODEL
V(x) = W (D1 x + D2 x2 + x4 )
(i) The soft modes can on average be characterized by a single energy W (ii) D1 and D2 are randomly distributed around the origin of the D1 - D2 plane:
P(D1 , D2) = P(0, 0) Ps (iii) The interaction between the soft modes and the sound waves isbilinear in the displacement of the soft mode and in the strain fieldof the sound wave:
Vl,t = l,t x l,t
D2 < 0
V(x) = W x4
D2 > 0D2=D1=0
V(x) = W (D1 x + D2 x2 + x4 )
TUNNELING MODEL SOFT POTENTIAL MODEL
2
0
2,
DW
PPP s 2/2Djj
W
PP js
j
22
0
2/3
20 3
2exp DW 21 2DWD
SPM universal constants:
SOFT-POTENTIAL MODEL: Specific heat
)(ln
91
6)( exp3/1
3/12
, T
t
WTk
kPTCmin
BBsTLSp
56
, 212
)(
WTk
kPTC BBssmp
minTW 28.1
0.1 0.5 1 5 10
0.5
1
5
10
50
W /1.85
T5T
Phillips Zeller&Pohl
Debye
SiO2
(Cp-C
Deb
)/T
3 (J
/g·K
)
T (K)
53)( TCTCTCTC smDTLSp
)(ln
91
6)( exp3/1
3/12
, T
t
WTk
kPTCmin
BBsTLSp
56
, 212
)(
WTk
kPTC BBssmp
]·28.1[ minTW
SPM fit:
{
1 10
2
4
6
8
10
20
T3
Debye
2020.5 5
(B2O3)84 (Na2O)16
Cp/T
3 (J
/g∙K
)
T (K)
,
M.A.R. , Philos. Mag. (2004)
CALCULATION OF THE THERMAL CONDUCTIVITY
(
( )
( )
( )
max·6.1 TW
CBTkW min85.1
Tc
max6.1 TkW B
2.1/1
QcBTkW
SPM
M.A.R. and U. Buchenau, Phys. Rev. B 55, 5749 (1997)
L. Gil et al., Phys. Rev. Lett. 70, 182 (1993)
2121 exp),( ADPDDP s 2/3)/(169.0 gBTkWA
4/14/3minmax 07.1 gTTT
Ioffe-Regel limit:
2/IR
SPM WCE IRIR
3/122
phonon localization when itsmean free path decreasesdown to half the wavelength:
for Brillouin scattering:
