Glassy dynamics near the two-dimensional

metal-insulator transition

Acknowledgments:NSF grants DMR-0071668, DMR-0403491; IBM, NHMFL; V. Dobrosavljević, I. Raičević

J. Jaroszyński and Dragana Popović

National High Magnetic Field Laboratory Florida State University, Tallahassee, FL

• metal-insulator transition (MIT) in 2D electron and hole systems in semiconductor heterostructures (Si, GaAs/AlGaAs, …)

Background

Si

Si MOSFET

• critical resistivity ~h/e2

role of disorder?

• rs U/EF ns-1/2 10

role of Coulomb interactions?

• competition between disorder and Coulomb interactions: glassy ordering???

[Davies, Lee, Rice, PRL 49, 758 (1982); 2D: Chakravarty et al., Philos. Mag. B 79, 859 (1999); Thakur et al.,

PRB 54, 7674 (1996) and 59, R5280 (1999); Pastor, Dobrosavljević,

PRL 83, 4642 (1999)]

Our earlier work: transport and resistance noise measurements to probe the electron dynamics in a 2D system in Si MOSFETs signatures of glassy dynamics in noise

2D MIT in Si: melting of the Coulomb glass

T=0 phase diagram

ns* – separatrix (from transport)

ng – onset of slow dynamics (from noise)nc – critical density for the MIT from (T) on both insulating and metallic sides

-Insulator: (T=0)=0-Slow, correlated dynamics (1/f noise; 1.8)

-Metal: (T=0)0; d/dT<0-Fast, uncorrelated dynamics (1/f noise; =1)

High disorder (low-mobility devices): nc < ng < ns*

Low disorder (high-mobility devices): nc ns* n≲ g for B=0,

nc < ns* n≲ g for B≠0

(Coulomb glass)

-Metal: (T=0)0;

-Slow, correlated dynamics (1/f noise; 1.8)

(ns,T)=(ns,T=0)+b(ns)T3/2

Theory: Dobrosavljević et al.

nc ng ns* density

[Bogdanovich, Popović, PRL 88, 236401 (2002);Jaroszyński, Popović, Klapwijk, PRL 89, 276401 (2002);Jaroszyński, Popović, Klapwijk, PRL 92, 226403 (2004)]

• slow relaxations and history dependence of (ns,T) also observed for ns < ng

Samples: • low-mobility (high disorder) Si MOSFETs with LxW of 2x50 and 1x90 m2

[from the same wafer as those used for noise measurements in Bogdanovich et al., PRL 88, 236401 (2002); all samples very similar]• data presented for 2x50 m2 sample• Note: critical density nc(1011cm-2) 4.5 obtained from (ns,T=0) in

(ns,T)=(ns,T=0) + b(ns)T3/2,

which holds slightly above nc (up to n 0.2); below nc, is insulating

(decreases exponentially with decreasing T) - similar to published data• noise measurements in this sample give ng(1011cm-2) 7.5, the same as

published results

This work: a systematic study of relaxations as a function of ns and T

2468

10

-20000 0 20000 40000 60000 80000

t/s

T (

K)

789

101112

Vg

(V)

0.001

0.01

0.1

1

10

0

(e

2 /h)

Sample annealed @ Vg=11V(ns=20.26 x 1011cm-2) @ T=10K;

then cooled down to different T (here to 3.5 K);

then @ t = 0, Vg switched (here) to Vg=7.4 V (ns=4.74 x 1011cm-2) and relaxation measured.

After change of Vg, decreases fast, goes through a minimum and then relaxes up towards 0 , which is when sample is cooled down at Vg=7.4 V (i.e. equilibrium ).

To measure 0, after some time (here approx. 55000 s), T is increased up to 10 K to rejuvenate the sample and then lowered back to 3.5 K.

Example 1:

Note: large perturbation

0.01

0.1

1

10

0

(e2 /h

)

02468

10

-10000 0 10000 20000 30000

t/s

T (

K)

789

101112

Vg

(V)

Sample annealed @ Vg=11V @ T=10K;

then cooled down to different T (here to 1 K);

then @ t = 0, Vg switched (here) to Vg=7.4 V and relaxation measured.

After change of Vg, initially decreases fast to below 0, and then continues to decrease slowly.

In both cases, the systemfirst moves away fromequilibrium.

Example 2:

-2

-1

0

0 1 2 3 4

6

5

4.6

every 0.1 K

4.2

3.4

1.2

1.6

1.8

2

2.2

2.4

2.6

2.83.2

1 K

0.24 K

4.4

log t/s

log

(t

,T)/ 0(T

) 0(T

) =

an

ne

al V

g=7

.4 V

1

0 K

Relaxations at different temperatures for a fixed final Vg=7.4 V

I “Short” t (i.e. just before the minimum in ): data collapse as shown after a horizontal shift low(T) and a vertical shift a(T).

This means scaling:

/0=a(T)g(t/low(T))

Vg=7.4 V

Scaling function: linear on a ln [/(0a(T))] vs. (t/low) (=0.3 for Vg=7.4V) scale for over 4 orders of magnitude in t/low, i.e. a stretched exponential dependence

for intermediate times (just below minimum in (T)).

-4

-2

0

0 10 20 30

0 1 2 3 4 5

(t/low

)0.3

ln / 0

log t/low

4.4 K

3.2 K

1.2 K

2.4 K

3.7 K

[

a

(T))

]

a(T) (low)-

At even shorter times (best observable at lowest T):

power-law dependence /0 t-

In this region, scaling may be achieved by a nonunique combination of horizontal and vertical shifts.

-0.5

-0.4

-0.3

-0.2

-0.1

1 2 3 4

T = 1.0 KT = 0.8 KT = 0.6 KT = 0.4 KT = 0.3 K

Vg: 11 7.4 V

log t/s

log

/

0

(dashed lines are linearleast squares fits withslopes 0.068 at 0.4 Kand 0.071 at 0.3 K )

-2

-1

0

-2 -1 0 1 2 3 4 5 6

m=-0.06963L=5.459fit to stretched

log t

log

/

0

At lowest T (< 1.2 K), stretched exponential crosses over to a power lawdependence with an exponent 0.07 but scaling in the power law region is not unambiguous.

Scaling:

/low(T)

[

a(

T)]

Vg=7.4 V

Can we describe all the data with the following (Ogielski) scaling function? /0 t- exp[-(t/low)] = (low)- (t/low)- exp[-(t/low)]

f(t/low)(It works in spin glasses:C. Pappas et al., PRB 68,054431 (2003) in Au0.86Fe0.14)

Vg=7.4 V-2

-1

0

1

-25 -20 -15 -10 -5 0 5

log t/low

log

[/

(0( lo

w)-

)]

0.24 K0.5

1.0

2.4

3.2

4.4

Yes!

black dashed line – fit to Ogielski form

• curves collapse well down to 0.8 – 1.2 K; extract exponents and • experiment and analysis repeated for different Vg, i.e. ns: relaxations measured after a rapid change of Vg from 11 V to a given Vg at many different T

Vg=7.4 V-2

-1

0

-3 -2 -1 0 1 2 3 4 5

log t/low

log

[/

(0( lo

w)-

)]

1.2 K

2.4

4.4

3.2

black dashed line – fit to Ogielski form

A blowup of the region where curves collapse well:

0

0.10

0.20

0.30

0.40

0.50

2 3 4 5 6 7 8 9

individual fits

Ogielski formula

individual fits

Ogielski formula

ns (1011 cm-2)

expo

nent

- power law exponent; - stretched exponential exponent• dashed lines are guides to the eye• nc (1011cm-2) 4.5

• →0 at ns (1011cm-2) 7.5-8.0 ng, where ng was obtained from noise measurements!!!

• grows with ns – relaxations faster

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1.0

y = -8.229x1 +3.795, r2=0.9946

a=0.1321,

b=0.06501

Vg=7.4 V

1/T

horiz

onta

l shi

ft on

log

scal

e

log[low(2.4 K)/low(T)] vs. 1/T

• black line is an Arrhenius fit to the data in the regime where curves collapse well; Arrhenius fit works well over 7 orders of magnitude in 1/low

Scaling parameters vs. T

1/low(T) = k0 exp(-Ea/T), with Ea19 K and k0 6.25 s-1

for Vg=7.4 V

• similar results are obtained for other Vg in the glassy region (e.g. Ea20.8 K for Vg=7.2 V, and Ea22 K for Vg=8.0 V)

Ea 20 K, independent of Vg in this range

(3.99 ≤ ns(1011cm-2) ≤ 7.43; 29 ≤ EF (K) ≤ 54) but k0=k0(Vg), i.e. k0=k0(ns)

1/low(ns,T) = k0(ns) exp(-Ea/T)

T=3 K

• a decrease of low with decreasing ns does not imply that the system is faster at low ns; since the dominant effect is the decrease of , the system is actually slower at low ns

10-1

100

101

102

103

104

2 4 6 8

low (

s)

ns (1011 cm-2)

individual fitsOgielski formula

• dashed lines guide the eye

low(T) exp(ans1/2) exp(Ea/T),

Coulomb energy U ns

1/2 ; 1/rs=EF/U ~ ns1/2

ln

low (

s)

-5

0

5

10

2 4 6 8

ns (1011 cm-2)

-5

0

5

10

1.5 2.0 2.5

ln

low (

s)

[ns (1011 cm-2)]1/2

Blue line – fit to ns1/2

• strong evidence for the dominant role of Coulomb interactions between 2D electrons in the observed slow dynamics

II Long t (i.e. above minimum in (t), observable at highest T):all collapse onto one curve after horizontal shift (no vertical shift needed, as expected: all relax to 0 i.e. to 0 on this scale). Data

collapsed onto T=5 K curve.

This means scaling:

/0= f(t/high(T))

-2

-1

0

-2 -1 0 1 2 3 4

3.64.2

4.6

3.2 K

5K

log t/high

(T)

log

/

0

Vg=7.4 V

Scaling function – describes relaxation of to 0 from below. There are two simple exponential regions (the slower one is not always seen).

0.001

0.01

0.1

1

0 200 400 600 800 1000

t/high

(0-

)/ 0

Vg=7.4 V(0-)/0exp(-k1t/high)

(0-)/0exp(-k2t/high)

Scaling parameter high vs. T

high exp[EA/(T-T0)],

T0 0, EA 57 K (Arrhenius)

Vg=7.4 V

-10

-5

0

3 4 5 6

high

(from scaling)high

/k1 (from fits)

high

/k2 (from fits)

a-b/(x-c), r2=0.9999

a=11.04, b=53.58, c=0.1569

a=0.1992,

b=1.585,

c=0.05785

a-b/(x-c), r2=0.9992

a=5.935, b=56.83, c=0.02327

a=0.7938,

b=6.545,

c=0.2337

a-b/(x-c), r2=0.9996

a=7.900, b=56.67, c=0.05007

a=0.3324,

b=2.661,

c=0.09683

T/K

ln 1

/ high

(1

/s)

Characteristic times high/k1 and high/k2 do not depend on Vg in the range

shown; they also do not depend on the direction of Vg change (see below). The data shown were obtained by changing Vg between the values given on the plot.

The fits on this plot were made to all points. Final Vgs (7.2 to 11) correspond to a density range from 3.99 to 20.36 in units of 1011cm-2 (EF from 29 K to 149 K).

highexp[EA/(T-T0)],

T0=0, EA 57 K

-12

-10

-8

-6

-4

-2

3 4 5 6

a-b/x Arrheniusa-b/(x-c) VF6->7.4->1111->811->7.411->7.2

a-b/(x-c), r2=0.9976a=7.381, b=52.96, c=0.1545

a=0.5199,

b=4.254,

c=0.1603, prob=1.000

a-b/x, r2=0.9975a=7.870, b=57.11

a=0.1092,

b=0.4374, prob=1.000

T/K

ln(k

1/ high

)

Examples of time scales:

T=5 K, high/k1 34 s;

T= 1 K, high/k1 1013 years! (age of the Universe 1010 years)

• the system appears glassy for short enough t < (high/k1) :

relaxations have the Ogielski form t- exp[-(t/low)],

with low exp (ans1/2) exp (Ea/T), Ea20 K

• the system reaches equilibrium at (high/k1)<(high/k2) << t:

relaxations exponential (high exp (EA/T), EA 57 K)

Conclusions

Note: The system reaches equilibrium only after it first goes farther away from equilibrium![Also observed in orientational glasses and spin glasses; see also “roundabout”

relaxation: Morita and Kaneko, PRL 94, 087203 (2005)]

high → as T→0, i.e. Tg = 0

[see Grempel, Europhys. Lett. 66, 854 (2004)]

• consistent with noise measurements