Glassy dynamics near the twodimensional

1
Glassy dynamics near the
two-dimensional
metal-insulator transition
J. Jaroszynski and Dragana Popovic
National High Magnetic Field Laboratory
Florida State University, Tallahassee, FL
Acknowledgments NSF grants DMR-0071668,
DMR-0403491 IBM, NHMFL V. Dobrosavljevic, I.
Raicevic
2
Background

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

• critical resistivity h/e2
• role of disorder?
• rs ? U/EF ? ns-1/2 ? 10
• role of Coulomb interactions?

3

• 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, Dobrosavljevic,
• 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

4
Bogdanovich, Popovic, PRL 88, 236401
(2002) Jaroszynski, Popovic, Klapwijk, PRL 89,
276401 (2002) Jaroszynski, Popovic, Klapwijk,
PRL 92, 226403 (2004)
T0 phase diagram

• -Metal ?(T0)?0
• d?/dTlt0
• Fast, uncorrelated
• dynamics
• (1/f? noise ?1)

-Insulator ?(T0)0 -Slow, correlated dynamics
(1/f ? noise ? ? 1.8)
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
High disorder (low-mobility devices) nc lt ng lt
ns Low disorder (high-mobility devices) nc ?
ns ? ng for B0,
nc lt ns ?
ng for B?0
(Coulomb glass)
Theory Dobrosavljevic et al.
5

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

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

• 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,T0) in
• 
  ?(ns,T)?(ns,T0) 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

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Example 1
Sample annealed _at_ Vg11V (ns20.26 x 1011cm-2) _at_
T10K then cooled down to different T (here to
3.5 K) then _at_ t 0, Vg switched (here) to
Vg7.4 V (ns4.74 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 Vg7.4 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.
Note large perturbation
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Example 2
Sample annealed _at_ Vg11V _at_ T10K then cooled
down to different T (here to 1 K) then _at_ t
0, Vg switched (here) to Vg7.4 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.
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Relaxations at different temperatures for a fixed
final Vg7.4 V
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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 ?/?0a(T)g(t/?low(T))
a(T) ? (?low)-?
Scaling function linear on a ln ?/(?0a(T))
vs. (t/?low)? (?0.3 for Vg7.4V) scale for over
4 orders of magnitude in t/?low, i.e. a
stretched exponential dependence for intermediate
times (just below minimum in ?(T)).
a(T))
Vg7.4 V
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At even shorter times (best observable at lowest
T) power-law dependence ?/?0 ?
t-?
(dashed lines are linear least squares fits
with slopes 0.068 at 0.4 K and 0.071 at 0.3 K )
In this region, scaling may be achieved by a
nonunique combination of horizontal and vertical
shifts.
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Scaling
a(T)
Vg7.4 V
/?low(T)
At lowest T (lt 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.
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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)
Yes!
black dashed line fit to Ogielski form

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A blowup of the region where curves collapse well
black dashed line fit to Ogielski form
Vg7.4 V

• 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

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individual fits Ogielski formula
exponent
?
individual fits Ogielski formula
?
ns (1011 cm-2)

• ? - 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

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Scaling parameters vs. T
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

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• 1/?low(T) k0 exp(-Ea/T), with Ea?19 K and k0
  ? 6.25 s-1
• for
  Vg7.4 V
• similar results are obtained for other Vg in the
  glassy region
• (e.g. Ea?20.8 K for Vg7.2 V, and Ea?22 K for
  Vg8.0 V)
• ? Ea ? 20 K, independent of Vg in this
  range
• (3.99
  ns(1011cm-2) 7.43 29 EF (K) 54)
• but k0k0(Vg), i.e. k0k0(ns)

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

• dashed lines
• guide the eye

T3 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

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ln ?low (s)
Blue line fit to ns1/2
?low(T) ? exp(ans1/2) exp(Ea/T),
Coulomb energy U ? ns1/2 1/rsEF/U ns1/2

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

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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 T5 K curve.
This means scaling ?/?0 f(t/?high(T))
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Scaling function describes relaxation of ? to
?0 from below. There are two simple exponential
regions (the slower one is not always seen).
21
Scaling parameter ?high vs. T
?high ? expEA/(T-T0), T0 ? 0, EA ? 57 K
(Arrhenius)
Vg7.4 V
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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).
?high?expEA/(T-T0), T00, EA ? 57 K
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Conclusions

• the system appears glassy for short enough t lt
  (?high/k1)
• relaxations have the Ogielski form ? t-?
  exp-(t/?low)?,
• with ?low ? exp (ans1/2) exp (Ea/T), Ea?20 K
• the system reaches equilibrium at
  (?high/k1)lt(?high/k2) ltlt t
• relaxations exponential (?high ? exp (EA/T),
  EA? 57 K)

?high ? ? as T?0, i.e. Tg 0
Examples of time scales T5 K, ?high/k1 ? 34
s T 1 K, ?high/k1 ? 1013 years! (age of the
Universe ? 1010 years)
see Grempel, Europhys. Lett. 66, 854 (2004)

• consistent with noise measurements

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)