Low temperature universality in disordered solids
Moshe Schechter
In collaboration with:
Philip Stamp (UBC)
Alejandro Gaita-Arino (UBC)
MS and Stamp, arXiv:0910.1283
Gaita-Arino and MS, in preparation

Low temperature universality in disordered solids
Moshe Schechter
In collaboration with:
Philip Stamp (UBC)
Alejandro Gaita-Arino (UBC)
Below
T

T
U

3 K
C v

T



T

 
1
 
2
Q

1  
/ 2
 l

10
 3
Zeller and Pohl, PRB 4, 2029 (1971)
Pohl, Liu, Thompson, RMP 74, 991 (2002)

Low temperature universality in disordered solids
Moshe Schechter
In collaboration with:
Philip Stamp (UBC)
Alejandro Gaita-Arino (UBC)
Below
T

T
U

3 K
C v

T



T

 
1
 
1
Q

1  
/ 2
 l

10
 3
Freeman and Anderson, PRB 34, 5684 (1971)

Standard tunneling model
2-level systems
Below T

T
U

3 K
1
2


0

0


P (

,

0
)
 p
0

0
C v

T



0
T
2
Q
 const
C v

T



T

 
1
 
2
Q

2
 l /
 
10
3
C
0
 p

0 c

2
2
 0 .
1 n

 c
2
2
Q

1  
C
0
/ 2

0

1 / C
0
Anderson, Halperin, Varma, Phil. Mag. 25, 1 (1972)
Philips, J. Low Temp. Phys. 7, 351 (1972)

Standard tunneling model
2-level systems
Below T

T
U

3 K
1
2


0

0


P (

,

0
)
 p
0

0
C v

T



0
T
2
Q
 const
C v

T



T

 
1
 
2
Q

2
 l /
 
10
3
C
0
 p

0 c

2
2
 0 .
1 n

 c
2
2
Q

1  
C
0
/ 2

0

1 / C
0
TLS in aging, 1/f noise, qubit decoherence
Anderson, Halperin, Varma, Phil. Mag. 25, 1 (1972)
Philips, J. Low Temp. Phys. 7, 351 (1972)

Standard tunneling model
2-level systems
Below T

T
U

3 K
1
2


0

0


P (

,

0
)
 p
0

0
C v

T



0
T
2
Q

1  const
C v

T



T

 
1
 
2
Q

1  
/ 2
 l

10

3
C
0
 p

0 c

2
2
 0 .
1 n

 c
2
2
Q

1  
C
0
/ 2

0

1 / C
0
1. What is tunneling?
C 10

3
T
U

3 K
4. Magnitude of specific heat, non-integer exponents

Theoretical models





Soft phonons
Large scale behavior of renormalized interactions
Renormalized dipolar TLS-TLS interactions
Frozen domains at the glass transition
Ad-hoc models for specific systems (KBr:CN)
Parshin, Phys. Re. B 49, 9400 (1994)
Leggett, Physica B: Cond. Matt. 169, 332 (1991)
Burin, J. Low. Temp. Phys. 100, 309 (1995)
Lubchenko and Wolynes, Phys. Rev. Lett. 87, 195901 (2001)
Sethna and Chow, Phase Tans. 5, 317 (1985); Solf and Klein, PRB 49, 12703 (1994)

Disordered lattices – KBr:CN
20% < x < 70% : Universal characteristics
70% CN – ferroelectric phase – glassiness not important
De Yoreo, Knaak, Meissner, Pohl, PRB 34, 8828 (1986)

CN impurities in KBr:KCl mixed crystals – strain vs. interactions
C
0
 p

0 c

2
2
Universal characteristics down to low x.
Tunneling strength linear in x
Strain, and not TLS-TLS interactions
Watson, PRL 75, 1965 (1995)
Topp and Pohl, PRB 66, 064204 (2002)

Amorphous vs. Disordered
Ion implanted crystalline Silicon – amorphisity not important
Liu et al., PRL 81, 3171 (1998)

Tau and S TLSs
Change of axis – S excitations
180 flips – tau excitations



X
 x


S i z



X

 x

 x

 i z

Weak linear Tau coupling to phonons
H
  i

  s
S i z   w
  i z
 
X
 x



Weak linear Tau coupling to phonons
H
  i

  s
S i z   w
  i z
 
X
 x



Weak linear Tau coupling to phonons
H
  i

  s
S i z   w
  i z
 
X
 x


 s

E
C

5 eV
 w

E


0 .
1 eV g


 w s

E

E
C

0 .
01

0 .
03
~ deviations from inter-atomic distance

DFT calculation of weak and strong coupling constants
- Confirm theoretical prediction
 w in agreement with experiment: positive identification of TLSs, prediction for S-TLSs
A. Gaita-Arino and M.S., in preparation

Effective TLS interactions
H
  i

 s

S i z   w
  i z
 
X
 x


H
S

  ij

J ij
SS
S i z
S j z 
J ij
S

S i z
 j z 
J ij
  i z
 j z

J
0
SS 

 c
2 s
2
R
0
3

J
0

300 K
T int
 gT
U
 g 2 T
G
J
0
S
 

 c s
2
 w
R
0
3
 gJ
0

3

10 K J
0
 

 c
2 w
2
R
0
3
 g
2
J
0

100 mK
C
0
 p

0 c

2
2

0 .
1 n

 c
2
2
P (

,

0
)
 p
0

0
 p
0

0 .
1 n
C s

0 .
1 n s

 c
2 s
2

0 .
1 n
S

1
J
0
R
0
3

 c
 s
2
2
C 

0 .
1 n

 c
2
 w
2

0 .
1

 s w

0 .
1 g

10

3 n


1 gJ
0
R
0
3


 c s

2 w

Dipole gap – strength of the weak
H
S

  ij

J ij
SS
S i z
S j z 
J ij
S

S i z
 j z 
J ij
  i z
 j z
 n


1 gJ
0
R
0
3
E
 gJ
0 n s
0 
1
J
0
R
0
3
 gn

E

J
0
E s i  
J
SS ij
S j z  j
 j
J ij
S
  j z
E
 j   i
J
S
 ij
S i z   i
J ij
  i z
E
S i
  j

E
S i 
E
 j 
2 J
S
 ij

0
Efros and Shklovskii, J Phys C 8, L49 (1975)

J ij
S
  c ij
S

R
3 ij
J
0
S

 a
3
0
DOS of S-TLS n s
( E )
 n s
( E )
 j

( E s

E
 j

2 U j
)
 n s
( E ) P ( E s
)
C
S

C

 n s
 s
2  n

 w
2 
P ( E )

0 .
1 1
 a
0

6

E

0 .
2 E

 gJ
0

3 K

Summary
- At low energy tau TLSs dictate physics
- Universality and smallness of tunneling strength
- Tunneling states: inversion pairs. Intrinsically 2-level systems
- Accounts for energy scale of ~3K
- Below 3K – effectively noninteracting TLS!
- Above 3K – crossover to l /
 
1
- Strain important, not glassiness or amorphous structure
- Agreement with experiments: n


1 gJ
0
R
0
3

T
G

1
, mixed crystals

Amorphous Solids
Local order – small deviations from lattice, ~3% in 1 st n.n. distance
Disorder contribution to  w

1 / R
4 and random
Utmost experimental / numerical test: finding that low T TLSs are inversion pairs easier experimental test: Existence of S TLSs, with strong phonon interaction and gapped DOS (phonon echo)

Conclusion



Existence of inversion pairs give rise to the universality and smallness of the tunneling strength
Explains well the various experimental results
Future work:






Experimental and numerical verification in disordered solids
Calculation of the specific heat and thermal conductivity
Extension to amorphous solids
TLS in 1/f noise and qubit decoherence
Relation to glass transition
Molecular resonances