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Magnetic-Field Dependence of the Anomalous Noise Behavior

in a Two-Dimensional Electron System in Silicon

J. Jaroszyn´ski,1,* Dragana Popovic´,1and T. M. Klapwijk2

1National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310 2Department of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands

(Received 25 February 2003; published 3 June 2004)

Studies of low-frequency resistance noise show that the dramatic change in the dynamics of the two-dimensional electron system (2DES) in Si that occurs near the metal-insulator transition (MIT) persists in high parallel magnetic fields B such that the 2DES is fully spin polarized. This strongly suggests that charge, as opposed to spin, degrees of freedom are responsible for this effect. In the metallic phase, however, noise is suppressed by a parallel B, pointing to the role of spins. At low B, the temperature dependence of conductivity in the metallic phase provides evidence for a MIT.

DOI: 10.1103/PhysRevLett.92.226403 PACS numbers: 71.30.+h, 64.70.Pf, 71.27.+a, 73.40.Qv

The fascinating strong correlation physics exhibited by low-density two-dimensional (2D) electron and hole sys-tems [1] has been the subject of intensive research. In particular, the precise role of the spin degrees of freedom remains the central unresolved issue in this field [2]. Recent resistance (R) noise measurements on a 2D elec-tron system in Si [3,4] have revealed a striking change in the dynamics of the two-dimensional electron system (2DES) in the vicinity of the apparent metal-insulator transition (MIT), posing a question of whether charge or spin degrees of freedom are responsible for this effect. Since it is known [5] that a relatively weak magnetic field

B is required to fully spin polarize the 2DES in the relevant range of densities ns, experimental studies in parallel B [6] should be able to distinguish between the two possibilities. Here we present such a study, which shows that the dramatic change in the electron dynamics persists even when the 2DES is spin polarized, strongly pointing to charge, as opposed to spin, degrees of free-dom as the origin of this effect. At high ns, on the other hand, noise measurements in B suggest that electrons’ spins may play a relevant role.

The B  0 studies employed a combination of trans-port and low-frequency resistance noise measurements [3,4] to probe the system dynamics. By reducing ns, it

was found that, at some well-defined density ng, (i) the

dynamics suddenly and dramatically slowed down, and (ii) there was an abrupt change to the sort of correlated statistics characteristic of complicated multistate systems. These features were attributed to the glassy freezing of the 2DES at ng, with the data being consistent with the hierarchical picture of glassy dynamics. Hence, for brev-ity, this change in the noise behavior will be called the ‘‘glass transition.’’ The ‘‘glass transition’’ occurs [3] in the metallic phase, i.e., at ng > nc, where nc is the critical density for the MIT determined from the vanishing of activation energy in the insulating regime [7,8]. The in-termediate metallic phase with slow electron dynamics (MSED, or ‘‘metallic glass’’) is considerably wide in

strongly disordered samples [ng nc=nc  0:5 [3]], whereas in devices with low disorder ng is at most a few percent higher than nc [4]. In this work, we inves-tigate these same high peak mobility (), i.e., low dis-order devices using noise spectroscopy in a parallel B. Our results demonstrate the following: (a) the abrupt and dramatic changes in the noise behavior [features (i) and (ii) above] are qualitatively the same as in B  0, even when the electrons are spin polarized at high B; (b) the phase diagram constructed in the ns; Bplane shows that

the MSED phase is broadened by a parallel B; (c) the temperature (T) dependence of the conductivity in the MSED at low B provides evidence for a quantum phase transition (QPT) at ncB; (d) for ns> ngB > ncB

(metallic phase with fast electron dynamics — MFED), the noise is suppressed by a parallel B.

The experiment was performed on n-channel Si metal-oxide-semiconductor field-effect transistors (MOSFETs) with  2:5 m2=Vs at 4.2 K, fabricated in a Hall bar geometry with Al gates, and oxide thickness dox 147 nm [9]. R was measured using a standard four-probe ac technique (typically 2:7 Hz) in the Ohmic regime. A precision dc voltage standard (EDC MV116J) was used to apply the gate voltage, which controls ns. nswas always

varied at a relatively high T  2 K. Contact resistances and their influence on noise measurements were mini-mized by using a split-gate geometry, which allows one to maintain high ns ( 1012cm2) in the contact region

while allowing an independent control of ns of the 2D system under investigation in the central part of the sample (120  50 m2). The samples and the measure-ment technique have been described in more detail else-where [4].

For a given nsand B, R was measured as a function of time t at T  0:24 K, although measurements at higher T were also performed at several selected B. At B  0, the temperature coefficient of the time-averaged resistivity

dhi=dT  0 at n s  9:7  1010cm2, similar to what

was obtained on the previous cooldown of the same

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sample [4]. At a fixed T and in the range of ns under investigation, hi exhibits a dramatic increase with B, followed by a much weaker dependence (‘‘saturation’’) at higher fields (B > 2–4 T). This large positive magne-toresistance at low B has been observed and studied extensively in many 2D systems [1], including other samples from the same source [9] as ours. In the satura-tion region, it has been shown [5] that the 2DES is spin polarized.

Figures 1(a) and 1(b) show the time series of the relative changes in resistance Rt=hRi and the corre-sponding power spectra SRf, respectively, for a fixed ns

and several B [10]. It is obvious that B has a strong effect on both the amplitude and the character of the noise, as discussed in detail below. In order to compare the noise magnitudes under different conditions, the power SRf  1 mHz is taken as the measure of noise. It is determined from the fits of the octave-averaged spectra to the form 1=ffor 104< f < 0:07 Hz [solid lines in Fig. 1(b)]. In addition, we have also analyzed the so-called second spectrum S2f2; f, which is the power spectrum of the fluctuations of SRf with time [11]. S2f2; f provides a direct probe of correlations between fluctuators: it is white (independent of f2) for uncorrelated, and S2/ 1=f12 for interacting fluctuators [11]. At B  0, the glass transition in Si MOSFETs was manifested by [crite-rion (i) above] a sudden and dramatic increase of SR, and a rapid rise of  from  1 to  1:8 [3,4], and [criterion (ii) above] a change of the exponent 1   from a white (zero) to a nonwhite (nonzero) value [4]. We adopt similar criteria for the glass transition in B.

Figures 2(a) – 2(c) present the B-field dependences of

SR, , and 1  , respectively, for several ns. For each

nsand B, S2f2; fwas calculated [12] for three different octaves f: 2–4, 4–8, and 8–16 mHz. In order to reduce the uncertainty in 1  , the exponent shown in Fig. 2(c)

represents the average 1   obtained from S2 in those three octaves.

At the highest ns, in the metallic phase (MFED), ‘‘pure’’ 1=f noise ( ’ 1) resulting from uncorrelated [1    0] fluctuators is observed in B  0 [3,4].

dSR=dB < 0 in Fig. 2(a) shows that parallel B suppresses

such noise, as seen already in the raw data [e.g., the bottom two traces in Fig. 1(a)]. In addition,  is reduced to 0:5 [Fig. 2(b)]. For B * 3–4 T, where the 2DES is fully spin polarized, the dependence of both SRand  on

Bbecomes weak. These data suggest that 1=f noise in the MFED is probably related to the electrons’ spins. As in

B  0 [3,4], here the noise does not depend on ns. Pure

1=f noise in B  0 was observed also in a 2D hole system in GaAs on the metallic side of and slightly below the apparent MIT, defined as dhi=dT  0 [13]. The increase of the noise power with decreasing hole density was attributed to approaching the percolation transition, but more work is needed to understand the origin of the differences between the two systems.

0 0.5 1.0 1.5 2.0 0 2000 4000 6000 -9 -8 -7 -6 -4 -3 -2 t (s) log [ fS R ] ∆ R /< R> [%] log [ f (Hz) ] B = 0 B = 0 B = 1.9 T B = 1.9 T B = 1.6 T B = 1.6 T a)T = 0.24 K b) B = 8.2 T B = 8.2 T <ρ> = 0.26 h/e2 <ρ> = 0.68 h/e2 <ρ> = 1.1 h/e2 <ρ> = 6.8 h/e2

FIG. 1. (a) Rt=hRi  R  hRi=hRi, and (b) the

corre-sponding power spectra SRf, at several B for ns 11:9 

1010cm2. In (a) traces are shifted for clarity, and the

corre-sponding hi are shown. In (b) SRf are averaged over octaves

and multiplied by f, so that 1=f spectrum is horizontal on this scale. Solid lines are linear least-squares fits with the slopes

 1.58, 1.24, 0.96, 0.58 (from top). 0.5 1.0 1.5 2.0 0 0.5 1.0 0 1 2 3 4 5 6 7 8 9 10 -7 -6 -5 -4 -3 -2 a) b) c) B (T) log [ SR (H z -1)] α 1 -β 9.85 9.56 9.71 10.4 11.2 11.9 12.6 14.0 14.8 16.2 19.1 9.85 9.56 9.71 10.4 11.9 14.8 16.2 11.2 14.0 19.1 12.6 9.85 9.56 9.71 14.0 16.2 11.2 14.8 11.9

FIG. 2. (a) SRf  1 mHz, (b) , and (c) 1   vs B for

ns1010cm2 shown on the plots; T  0:24 K. Other data have

been omitted for clarity. SRf has been corrected for the

white background noise. The arrow in (a) shows Bg for

ns1010cm2  11:9. The error bars on the right show maxi-mum standard deviations of the data, which, for clarity, were plotted after performing a simple three-point average. The

origin of the fluctuations of SRB for some ns at B * Bg

[e.g., for ns1010cm2  11:2] is not understood at this time.

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At lower ns, the noise at low B behaves as described above. However, for each ns, there is now a well-defined field Bg [Fig. 2(a)] where, after the initial decrease with

B, SR begins to increase dramatically, accompanied by a rapid rise of  and an increase of 1   to nonwhite values, indicative of the onset of strong correlations. This striking change of both the magnitude and character of noise within a narrow range of B is obvious even from the raw data [e.g., the middle two traces in Fig. 1(a)]. In analogy with the B  0 case, we identify Bg as the field

where glass transition occurs at a given ns. At even higher

B * 4 T, SR, , and 1   no longer depend on B,

consistent with the fact that the electrons are spin polar-ized. Nevertheless, Fig. 2 shows that the strong depen-dence of noise on ns is still present for a given B in this regime. The dependence of SR, , and 1   on ns is plotted explicitly in Fig. 3 for several B. The density where SR begins to increase dramatically, accompanied by changes in  and 1  , is identified as the glass transition density ngfor a given B. At even lower ns, such

that ns< ncB  0, dSR=dB > 0. Here, however, R

be-comes too large to measure with our ac technique. The values of Bgns and ngB determined in this way

have been used to construct a phase diagram shown in Fig. 4. The square symbols designate the boundary of the phase with a correlated, slow electron dynamics (SED), i.e., the onset of abrupt and dramatic changes in the noise

behavior, and clearly show an increase of ng with B, followed by a saturation at higher B. It is interesting to compare the behavior of ngB with ncB. In Ref. [8],

where samples almost identical to ours were used, ncB was determined based on both a vanishing activation energy and a vanishing nonlinearity of current-voltage characteristics when extrapolated from the insulating phase. We have used the activation energy method but the range of accessible T and ns in our ac measurements

was smaller compared to the dc technique and dilution refrigerator T (down to 30 mK) used in Ref. [8]. The data that we have available in the insulating regime are best described by h i / exp T0=Tn with n  1=2, which corresponds to variable-range hopping with a Coulomb gap [15]. The extrapolation of T0ns to zero, where only

nswith T0* 0:5 K were used, was used to determine nc shown in Fig. 4 (for nc at 1, 2, 3 T, see below). Strictly speaking, the rather limited range of data does not allow one to make an accurate distinction between different forms of activated h Ti. This experimental uncertainty is reflected in rather large error bars for ncshown in Fig. 4.

Nevertheless, the agreement between our results for the form of ncB dependence and that obtained in Ref. [8] is

remarkably good (Fig. 4).

Furthermore, this is the first time that it has been possible to determine nc in B by studying h Ti on the

metallic side of the MIT. In particular, in a relatively

narrow range of ns for B  1; 2; 3 T, the data are best described by the metallic (h T  0i > 0) power-law behavior h ns; B; Ti  h ns; B; T  0i  bns; BT1:5 [Fig. 5(a)], similar to what was observed in the MSED phase of highly disordered samples at B  0 [3]. The extrapolated T  0 conductivities go to zero at ncB in a power-law fashion h ns; B; T  0i / n with 1:5

[see Fig. 5(b); n ns=ncB  1], in agreement with

theoretical expectations near a QPT [16]. At B  0, there is some evidence [17] of similar behavior with 1–1:5,

0 0.5 1.0 8 10 12 14 16 18 20 1.0 1.5 2.0 -6 -5 -4 -3 -2 -1 ns (1010cm-2) log [ SR (H z -1)] 1 -β a) b) c) ×××× ⊗ ×××× ×××× ×××× ×××× ×××× ×××× ×××× α 1 T 0 2 T 3 T 4 T 9 T ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ×××× ××× ××× ×××× ×××× ×××× ××××

FIG. 3. (a) SRf  1 mHz, (b) , and (c) 1   vs ns for

several B shown on the plot. The same symbols are used in all three panels. The dotted lines are guides for the eye. The maximum error bars are shown in the upper right.

0 1 2 3 4 5 6 7 8 9 10 8 9 10 11 12 13 14 15 16 B (T) ns (10 10 cm –2) Metal (MFED) Metal (MSED) Insulator (ISED) ng nc nc from Ref. [8]

FIG. 4. T  0 phase diagram. The dashed lines are guides for

the eye. 䊏: boundary between phases with fast and slow

electron dynamics;䊉: boundary between metallic and

insulat-ing phases. The Ref. [8] data have been shifted up by 0:85 

1010cm2to make the n cB  0 values coincide [14]. P H Y S I C A L R E V I E W L E T T E R S week ending 4 JUNE 2004 VOLUME92, NUMBER 22 226403-3 226403-3

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obtained from T  0 in the metallic phase of different Si MOSFETs. Considering that ncB for B  1; 2; 3 T

shown in Fig. 4 have been obtained by approaching the MIT from the metallic side, the agreement between our results and those of Ref. [8] is even more remarkable.

The phase diagram in Fig. 4 makes it clear why the metallic T3=2 temperature dependence of [Fig. 5(a)] is observed in such a narrow range of ns: it is characteristic of the MSED phase, even in B of up to 3 T. At B  0, the MSED phase is very narrow and the T3=2 correction is, therefore, difficult to observe in these samples, in contrast to highly disordered ones [3]. A parallel B increases the width of the MSED phase (Fig. 4). In addition, the range of T where T3=2 holds increases with parallel B (not shown). The increase of ng and nc, and the broadening of the MSED phase with B can be understood to result from the suppression of screening by a parallel B [18], which increases the effective disorder. This, in turn, favors glassiness, consistent with theoretical expectations [19]. It is also interesting that the prefactor bns; Bof the

T3=2 correction [slopes in Fig. 5(a)] does not seem to depend on ns, in contrast to the strong ns dependence seen in highly disordered samples at B  0 [3]. Further careful investigation is required in order to determine whether this difference can be attributed to the effects of

B or to the effects of disorder. Such a study, however, along with a detailed analysis of h Ti in the MSED phase at high B, is beyond the scope of this Letter.

In summary, noise measurements in high-mobility Si MOSFETs in parallel B show that a dramatic change in the electron dynamics near the MIT persists even when the 2DES is spin polarized. The results are consis-tent with the theory [19 – 23] where charge, as opposed to spin, degrees of freedom are responsible for glassy ordering of the 2DES near the MIT. In the metallic phase (MFED), however, electrons’ spins may play a relevant role.

We are grateful to V. Dobrosavljevic´ for useful discus-sions. This work was supported by NSF Grant No. DMR-0071668, and NHMFL through NSF Cooperative Agreement No. DMR-0084173.

*Also at Institute of Physics, PAS, Warsaw, Poland. Electronic address: jaroszy@magnet.fsu.edu

[1] E. Abrahams, S.V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73, 251 (2000), and references therein. [2] S.V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. 67,

1 (2004), and references therein.

[3] S. Bogdanovich and D. Popovic´, Phys. Rev. Lett. 88, 236401 (2002).

[4] J. Jaroszyn´ski et al., Phys. Rev. Lett. 89, 276401 (2002). [5] T. Okamoto et al., Phys. Rev. Lett. 82, 3875 (1999); S. A. Vitkalov et al., Phys. Rev. Lett. 85, 2164 (2000); E. Tutuc

et al., Phys. Rev. Lett. 86, 2858 (2001).

[6] As usual, a parallel field is used in order to avoid complications due to the orbital motion of electrons. [7] V. M. Pudalov et al., Phys. Rev. Lett. 70, 1866 (1993). [8] A. A. Shashkin et al., Phys. Rev. Lett. 87, 266402 (2001). [9] R. Heemskerk, Ph.D. thesis, University of Groningen, The Netherlands, 1998 (unpublished); R. Heemskerk and T. M. Klapwijk, Phys. Rev. B 58, R1754 (1998).

[10] The normalized noise power SRf  SV; f=V2did not

depend on the excitation current, indicating that the measured voltage fluctuations result from resistance (ex-cess) noise.

[11] M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988); 65, 829 (1993); M. B. Weissman et al., J. Magn. Magn. Mater. 114, 87 (1992), and references therein.

[12] G. T. Seidler and S. A. Solin, Phys. Rev. B 53, 9753 (1996); K. M. Abkemeier, Phys. Rev. B 55, 7005 (1997). [13] R. Leturcq et al., Phys. Rev. Lett. 90, 076402 (2003).

[14] ncB  0 are known [1] to be sample dependent because

of the differences in the amount of disorder.

[15] A. L. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975). [16] D. Belitz et al., Rev. Mod. Phys. 66, 261 (1994). [17] R. Fletcher et al., Semicond. Sci. Technol. 16, 386 (2001). [18] V. T. Dolgopolov and A. Gold, JETP Lett. 71, 27 (2000);

I. F. Herbut, Phys. Rev. B 63, 113102 (2001).

[19] V. Dobrosavljevic´, D. Tanaskovic´, and A. A. Pastor, Phys. Rev. Lett. 90, 016402 (2003).

[20] J. S. Thakur and D. Neilson, Phys. Rev. B 54, 7674 (1996); 59, R5280 (1999).

[21] A. A. Pastor and V. Dobrosavljevic´, Phys. Rev. Lett. 83, 4642 (1999).

[22] S. Chakravarty et al., Philos. Mag. B 79, 859 (1999). [23] D. Dalidovich et al., Phys. Rev. B 66, 081107 (2002).

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 T1.5 (K1.5) σ (e 2/h) B= 2 T (a) 10–2 10–1 10–3 10–2 10–1 δn=ns/nc(B)–1 σ(n s ,B,T=0) (e 2/h) B=1 T, µ=1.4±0.2 B=2 T, µ=1.4±0.1 B=3 T, µ=1.6±0.1 (b)

FIG. 5. (a) h Ti in the MSED for ns1010cm2 

11:9; 11:6; 11:3; 11:2; 11:0; 10:9; 10:7 from top; B  2 T; ncB  2T  10:67  1010cm2. (b) h T  0i /  n. P H Y S I C A L R E V I E W L E T T E R S week ending 4 JUNE 2004 VOLUME92, NUMBER 22 226403-4 226403-4

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