Glassy dynamics near the two-dimensional metal-insulator transition Acknowledgments: NSF grants DMR , DMR ; 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/e 2 role of disorder? r s  U/E F  n s -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 n s * – separatrix (from transport) n g – onset of slow dynamics (from noise) n c – 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): n c < n g < n s * Low disorder (high-mobility devices): n c  n s * ≲ n g for B=0, n c < n s * ≲ n g for B≠0 (Coulomb glass) -Metal:  (T=0)  0; -Slow, correlated dynamics (1/f  noise;   1.8)  (n s,T)=  (n s,T=0)+b(n s )T 3/2 Theory: Dobrosavljević et al. n c n g n s * density [Bogdanovich, Popović, PRL 88, (2002); Jaroszyński, Popović, Klapwijk, PRL 89, (2002); Jaroszyński, Popović, Klapwijk, PRL 92, (2004)]

slow relaxations and history dependence of  (n s,T) also observed for n s < n g Samples: low-mobility (high disorder) Si MOSFETs with LxW of 2x50 and 1x90  m 2 [from the same wafer as those used for noise measurements in Bogdanovich et al., PRL 88, (2002); all samples very similar] data presented for 2x50  m 2 sample Note: critical density n c (10 11 cm -2 )  4.5 obtained from  (n s,T=0) in  (n s,T)=  (n s,T=0) + b(n s )T 3/2, which holds slightly above n c (up to  n  0.2); below n c,  is insulating (decreases exponentially with decreasing T) - similar to published data noise measurements in this sample give n g (10 11 cm -2 )  7.5, the same as published results This work: a systematic study of relaxations as a function of n s and T

Sample Vg=11V (n s =20.26 x cm -2 T=10K; then cooled down to different T (here to 3.5 K); t = 0, Vg switched (here) to Vg=7.4 V (n s =4.74 x cm -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 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

Sample T=10K; then cooled down to different T (here to 1 K); 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 system first moves away from equilibrium. Example 2:

Relaxations at different temperatures for a fixed final V g =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)) V g =7.4 V Scaling function: linear on a ln [  /(  0 a(T))] vs. (t/  low )  (  =0.3 for V g =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)). 4.4 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. (dashed lines are linear least squares fits with slopes at 0.4 K and at 0.3 K )

At lowest T (< 1.2 K), stretched exponential crosses over to a power law dependence with an exponent 0.07 but scaling in the power law region is not unambiguous. Scaling: /  low (T) [ a(T)] V g =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, (2003) in Au 0.86 Fe 0.14 ) V g =7.4 V 0.24 K 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 V g, i.e. n s : relaxations measured after a rapid change of Vg from 11 V to a given V g at many different T V g =7.4 V 1.2 K black dashed line – fit to Ogielski form A blowup of the region where curves collapse well:

  individual fits Ogielski formula individual fits Ogielski formula n s (10 11 cm -2 ) exponent  - power law exponent;  - stretched exponential exponent dashed lines are guides to the eye n c (10 11 cm -2 )  4.5  →0 at n s (10 11 cm -2 )   n g, where n g was obtained from noise measurements!!!  grows with n s – relaxations faster

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) = k 0 exp(-E a /T), with E a  19 K and k 0  6.25 s -1 for V g =7.4 V similar results are obtained for other V g in the glassy region (e.g. E a  20.8 K for V g =7.2 V, and E a  22 K for V g =8.0 V)  E a  20 K, independent of V g in this range (3.99 ≤ n s (10 11 cm -2 ) ≤ 7.43; 29 ≤ E F (K) ≤ 54) but k 0 =k 0 (V g ), i.e. k 0 =k 0 (n s )

1/  low (n s,T) = k 0 (n s ) exp(-E a /T) T=3 K a decrease of  low with decreasing n s does not imply that the system is faster at low n s ; since the dominant effect is the decrease of , the system is actually slower at low n s  low (s) n s (10 11 cm -2 ) individual fits Ogielski formula  dashed lines guide the eye

 low (T)  exp(an s 1/2 ) exp(E a /T), Coulomb energy U  n s 1/2 ; 1/r s =E F /U ~ n s 1/2 ln  low (s) n s (10 11 cm -2 ) ln  low (s) [n s (10 11 cm -2 )] 1/2 Blue line – fit to n s 1/2 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)) V g =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). V g =7.4 V (  0 -  )/  0  exp(-k 1 t/  high ) (  0 -  )/  0  exp(-k 2 t/  high )

Scaling parameter  high vs. T  high  exp[E A /(T-T 0 )], T 0  0, E A  57 K (Arrhenius) V g =7.4 V

Characteristic times  high /k 1 and  high /k 2 do not depend on V g in the range shown; they also do not depend on the direction of V g change (see below). The data shown were obtained by changing V g between the values given on the plot. The fits on this plot were made to all points. Final V g s (7.2 to 11) correspond to a density range from 3.99 to in units of cm -2 (E F from 29 K to 149 K).  high  exp[E A /(T-T 0 )], T 0 =0, E A  57 K

Examples of time scales: T=5 K,  high /k 1  34 s; T= 1 K,  high /k 1  years! (age of the Universe  years) the system appears glassy for short enough t < (  high /k 1 ) : relaxations have the Ogielski form  t -  exp[-(t/  low )  ], with  low  exp (an s 1/2 ) exp (E a /T), E a  20 K the system reaches equilibrium at (  high /k 1 )<(  high /k 2 ) << t: relaxations exponential (  high  exp (E A /T), E A  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, (2005)]  high →  as T→0, i.e. T g = 0 [see Grempel, Europhys. Lett. 66, 854 (2004)] consistent with noise measurements