Measurements of conductance $G$ on short, wide, high-mobility Si-MOSFETs
reveal both a two-dimensional metal-insulator transition (MIT) at moderate
temperatures (1 $<~ T <$ 4~K) and mesoscopic fluctuations of the conductance at
low temperatures ($T~ <$ 1~K). Both were studied as a function of chemical
potential (carrier concentration $n_s$) controlled by gate voltage ($V_g$) and
magnetic field $B$ near the MIT. Fourier analysis of the low temperature
fluctuations reveals several fluctuation scales in $V_g$ that vary
non-monotonically near the MIT. At higher temperatures, $G(V_g,B)$ is similar
to large FETs and exhibits a MIT. All of the observations support the
suggestion that the MIT is driven by Coulomb interactions among the carriers.Comment: 4 pages, LaTeX, physica.sty (slightly modified prabib.sty), Submitted
  to the 1997 Conference on Electronic Properties of Two-Dimensional System

Abrahams

Ando

Dobrosavljević

Dragana Popović

K.P. Li

Lifshitz

Popović

Pudalov

S. Washburn

English

arXiv

Crossref

Conductance fluctuations near the two-dimensional metal–insulator transition

Measurements of conductance G on short, wide, high-mobility Si-MOSFETs reveal both a two-dimensional metal-insulator transition (MIT) at moderate temperatures (1 &lt; T &lt; 4 K) and mesoscopic fluctuations of the conductance at low temperatures (T &lt; 1 K). Both were studied as a function of chemical potential (carrier concentration ns) controlled by gate voltage (Vg) and magnetic field B near the MIT. Fourier analysis of the low temperature fluctuations reveals several fluctuation scales in Vg that vary non-monotonically near the MIT. At higher temperatures, G(Vg,B) is similar to large FETs and exhibits a MIT. All of the observations support the suggestion that the MIT is driven by Coulomb interactions among the carriers

Li, K.P.

Popović, Dragana

Washburn, S.

Carolina Digital Repository

arXiv:cond-mat/9708168v1  [cond-mat.str-el]  21 Aug 1997March 22, 2018Conductance Fluctuations Near the Two-Dimensional Metal-InsulatorTransitionK.P. Li1, Dragana Popović2, and S. Washburn11Dept. of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC27599-3255, USA2National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USAAbstractMeasurements of conductance G on short, wide, high-mobility Si-MOSFETs reveal both a two-dimensionalmetal-insulator transition (MIT) at moderate temperatures (1 < T < 4 K) and mesoscopic fluctuations of theconductance at low temperatures (T < 1 K). Both were studied as a function of chemical potential (carrierconcentration ns) controlled by gate voltage (Vg) and magnetic field B near the MIT. Fourier analysis of thelow temperature fluctuations reveals several fluctuation scales in Vg that vary non-monotonically near theMIT. At higher temperatures, G(Vg, B) is similar to large FETs and exhibits a MIT. All of the observationssupport the suggestion that the MIT is driven by Coulomb interactions among the carriers.Keywords: mesoscopic transport, metal-insulator transition, two-dimensional electron system1. Introduction and methodIn spite of wide-spread acceptance of the view thattwo-dimensional electron systems cannot undergometal-insulator transitions (MIT) [1,2], there arenow solid experimental and theoretical reasons to ac-cept the opposite viewpoint [3–5]. In high-mobility(µ > 1m2/V·sec) silicon metal-oxide-semiconductorfield-effect transistors (MOSFETs) [6], where thedisorder (fluctuations in the impurity potential en-ergy W ) is small enough, a MIT is observed. Atzero magnetic field B, the conductivity fits the scal-ing equations appropriate for quantum phase tran-sitions [8] as a function of carrier concentration andas a function of voltage bias [3,4]. For a system inwhich the Coulomb energy U = e2/r dominates (weestimate here that U ≫ EF >∼ W ), we expect thescreening length to be particularly important and toexhibit anomalies near the MIT. U should increaseas the MIT is approached from the insulating sideas separation of the electrons decreases, and fromthe metallic side as the screening becomes less effi-cient [6]. Therefore, U should be maximum near theMIT.In very short FETs, where L is comparable to rel-evant length scales in the quantum transport prob-lem, we expect to see anomalies associated with thecompeting length scales [7,8]. In addition, thereare complications associated with the reduced avail-able phase space in the small samples. As T → 0,we expect G to be dominated by resonance tunnel-ing through a few localized states [9] or by hop-ping along chains of such sites [10]. Studies ofthe typical conductance 〈lnG〉 at low temperatureshave already provided counter-intuitive results [11].At higher temperatures, we expect thermal averag-ing to remove (average away) many of the fluctua-tions and leave mainly the effects of bulk conductiv-ity. Here the generic signatures (scaling of σ(T, Vg),(σ(Vg , B)−σ(Vg , 0) > 0) of the MIT appear in largesamples and will appear in ours.Our samples were short (L < 4 µm), wide (w >1101 102 1030.20.30.40.50.6T zν / [(ns–nc)/nc]σ ( e2/h )1 2 3 40.000.050.10ns (1015/m2)q (1/Ω/T2 )FIG. 1. Scaled conductance data for a 1.25 µm long11.5 µm wide MOSFET for 1.9 K< T < 4.0 K. Eachsymbol represents conductance data from a particulartemperature. Inset: coefficient q for the parabolic G(B)for the short MOSFET (▽) at T = 0.7 K and represen-tative data from a large sample with L = 400 µm (+) atT = 1.4 K.11 µm) n-channel MOSFETs on the (100) surfaceof Si (≈ 3 × 1020 acceptors/m3, 50 nm gate ox-ide, oxide charge < 1014 m−2). High peak mobil-ities (≃ 1 m2/V·sec) indicate that there is muchless disorder than in samples used to “confirm” [2]the non-interacting scaling picture. The conduc-tance G(Vg, B, T ) was measured by standard lock-in techniques in a dilution refrigerator in a shieldedenclosure at low source-drain bias (VSD < 2µV ).For each measurement we infer a conductivity σ =(L/w) exp < lnG >, and we find several anomaliesin the region of conductivities where the MIT occurs.2. Metal-insulator transitionThe MIT appears in our samples as a crossing ofthe G(Vg) curves at nc ≃ 1.7 × 1015/m2 for a fam-ily of curves at different temperatures. We forgodisplaying the bare G(T ) which are less informativeand leap immediately to the scaling plot of σ vs.(T/|δn|)zν , where δ = (ns −nc)/nc is the scaled dis-tance from the critical point and ν and z are expo-nents describing the scaling of the spatial and tem-poral correlations, respectively, in the 2DES. Fig-ure 1 contains conductance against a scaled tem-perature for one of our samples (L = 1.25 µmand w = 11.5 µm). It is clear that the scalingof the data provides two branch curves that con-tain all of the data as expected from the theoreti-cal arguments [5,8] and in accord with previous ex-periments [3,4,12] in many systems. We point outthat nc = 1.7× 1015/m2 agrees well with all resultson large samples [3,4,12,13], but that the exponentzν = 16 ± 6 is much larger than has been observedbefore.In addition to G(Vg , 0, T ), we have studiedG(Vg, B, T ) for 0 < B < 2 T. It was learned re-cently that the application of B perpendicular tothe sample plane allows the differentiation betweenterms associated with generic (non-interacting) weaklocalization and with (Hartree) electron-electron col-lisions. G(B) can be decomposed as pf(B) − qB2,with f(B), p, q > 0, since the weak localizationterm is expected to be positive and Hartree term isexpected be negative and quadratic up to substan-tial fields. The coefficient q has a distinct minimumnear the MIT [4]. (A parallel B has a much moredramatic effect in the metallic region [3,13].) The co-efficient of the Hartree terms for one of our samplesare plotted in the inset of Fig. 1 along with datafrom a much larger MOSFET [4]. Another MOS-FET with L = 1.5 µm yielded very similar resultsand q for both samples were not temperature de-pendent down to T = 0.04 K. There is a clear (butsofter) minimum near the MIT in the short MOS-FETs.For the short MOSFET, the curve for q is muchflatter – at least in the metallic region, which leadsus to speculate that the correlation range from theCoulomb interactions that are generating the metal-lic behavior are being cut off by the sample lengthand therefore “quenching” the more dramatic metal-lic behavior seen in the large samples. This sugges-tion of a cut-off is supported by our observations thatthe low temperature (T < 1.5 K) curves of G(Vg) arelargely temperature independent and do not obeythe scaling law above and in Fig. 1. From analogywith non-interacting weak localization ideas, we ex-pect the coherence lengths Lϕ for the quantum cor-relations to grow as the temperature decreases be-cause thermal and inelastic processes are weaker [8],so that eventually L < Lϕ and the MIT is “short-circuited”.3. Conductance fluctuationsAs illustrated in Fig. 2, G can fluctuate on upto three different scales in Vg [11,14]. The dif-ferent scales are apparent in the power spectrumS(1/∆Vg) [related by the Wiener-Khinchin equa-tions to the autocorrelation function C(∆Vg)] of thefluctuations δ lnG = lnG(Vg) − 〈lnG(Vg)〉 as threeseparate slopes in the low, intermediate and high“frequency” regions. More than one thousand suchtraces have been recorded and analyzed for this ex-20.66 0.67 0.68-0.5-0.4FIG. 2. Representative power spectrum S(f) of fluc-tuations δ lnG for L = 1.5 µm; W = 11.5 µm;T = 0.045 K. Dashed lines illustrate the three differ-ent decay rates of the spectrum (from the left) vl, viand vh. The raw data for G(Vg) appear in the inset.σ = 5.40 × 10−1e2/h.periment. As discussed below, certain features of theconductance fluctuations behave regularly near theMIT and some exhibit anomalies that might presagecritical behavior in some of the transport variables,as energy and length scales fluctuate in ways thatare completely unexpected from the point of viewassociated with non-interacting electrons.4. Fluctuation correlation scalesThe different regions of exponential decay, S(f) =S(0) exp(−2πvf), correspond to contributions fromdistinct Lorentzians C(∆Vg) of widths ∼ vx (withx=i, h or l). The voltage scales vx are related to thetypical spacing of the fluctuations for the particu-lar transport process that leads to that Lorentzian,and so to typical energies or lengths in the transportproblem. The 1/vx are separate fits to the differentparts of S(f). The different regions of the spectrumare separated by other characteristic voltage scales.The voltage scale for cross-over between vl and therest of the spectrum, does not depend on Vg or Gv0 = 0.45± 0.15 mV. Similarly vh ≃ 50 µV indepen-dent of measurement parameters. The remainingcorrelation scales, however, vary non-monotonicallyby substantial amounts near the MIT. For example,vi emerges only in short samples at low temperaturesnear the MIT. Else it vanishes.Figure 3(b) contains vh (lower curves) and10–4 10–3 10–2 10–1 100 101050100150200250σ (e2/h)vi, vh  ( µV )(b)L=1.0 µm W=11.5 µmT=0.010 K T=0.045 K T=0.336 K T=0.555 K024681012vl (mV)(a)L=1.0 µm, T=0.045 K L=1.5 µm, T=0.045 K L=2.0 µm, T=0.023 K L=2.0 µm, T=0.023 KFIG. 3. (a) Correlation voltage vl for several samplesat low temperatures. The various lines guide the eye.One sample (▽) was 162 µm wide, and the rest were11.5 µm wide. (b) Correlation voltages vh (open sym-bols) and vi (solid symbols) vs. σ for several MOSFETs.Dashed lines guide the eye.vi(upper curves) for several MOSFETs. The highestfrequency scale vh is essentially constant throughoutthe transition region. This is typical of the behaviorexpected for all energy and length scales in the non-interacting pictures proposed in the original scalingmodel [1], where all quantities should behave regu-larly: ”smooth” and monotonic in energy. Insteadof remaining “flat” or regular near the MIT, vi risesby 400% and then falls back to the original value. Asimilar anomaly on a completely different absoluteenergy (or length) scale (and presumably from a dif-ferent transport process) appears in vl [Fig. 3(a)].To relate the measured fluctuation scales to en-ergy or length scales in the sample physics is sim-ple in the naive sense. A non-interacting model ofthe device electrostatics provides that [6] dµ/dVg =(dµ/dns)(dns/dVg) = (1/D(E))(C/e) with C =6.92 × 10−4 F/m2 as the gate capacitance per areaand D(E) as the density of states. It is mainly ow-ing to lack of a reliable model for D(E) that seriouscorrespondence between the electrostatics (which weexpect to be largely reliable even near the MIT)and the electron energies in the MOSFETs cannot be provided. Non-interacting pictures lead toD(E) ≃ 1.6×1018 m−2eV−1 for high Vg and consid-erably reduced as σ falls below the MIT. This places30 20 402 1 0G  (        S  )x10 -6B  ( T )0 100 200 300C o u n t sVg0.62 <       < 0.64VFIG. 4. Distribution function of G(Vg, B) for a samplewith L = 3.5 µm and w = 11.5 µm (so σ(0) = 0.17e2/h)at T = 0.04 K. Conductance fluctuations appear “ontop of” the positive and negative components of G(B)discussed above in the context of MIT at higher temper-atures.vh in the 1 µeV range, which is about the same asthe level spacing for the electrons, if the effectivesample area is a few µm2, which is quite reasonable.The lack of critical behavior is also in line with thisassociation.The other energy scales are correspondinglylarger. Without substantial theoretical insight forthis problem, we are not able to assign them reliablyto particular energy scales or to any specific physics.One plausible sugestion is that vi, which grows fromand retreats into vh near the MIT, is a disturbancein the energy level spacing for some of the states,perhaps as separate chains of localized states beginto exchange carriers as the screening length passesthrough a maximum.5. Magnetoconductance fluctuationsThe fluctuations in G(B) have been studied indepth. Generally speaking we find that the fluc-tuations are superimposed on the high-temperatureG(B) discussed above. An example of the fluctua-tions appears in a topographical display of G(Vg , B)in Fig. 4. Signatures of both positive and negative(p and q) components of G(B) are apparent in spiteof the presence of sizable fluctuations.6. ConclusionWe have studied the 2d MIT in short Si-MOSFETs, where the sample length L competeswith the length scales controlling the MIT. We findseveral anomalies in fluctuation correlation scales atlow temperatures T < 1 K and a “quenched” MITat higher temperatures in which development of themetallic phase has been short-circuited as L limitsthe range of the important quantum correlations.AcknowledgementsThe authors are grateful to B.L. Altshuler, V. Do-brosavljević, A.B. Fowler, and B.B. Mandelbrot foruseful discussions. This work was supported by NSFGrant DMR-9510355.References[1] E. Abrahams et al., Phys. Rev. Lett. 42, 673 (1979).[2] D.J. Bishop et al., Phys. Rev. Lett. 44, 1153 (1980);M.J. Uren, et al., J. Phys. C: Solid St. Phys. 13,L985 (1980).[3] S.V. Kravchenko et al., Phys. Rev. B 50 8039(1994); S.V. Kravchenko et al., Phys. Rev. B 51,7038 (1995); S.V. Kravchenko et al., Phys. Rev.Lett. 77, 4938 (1996); D. Simonian et al., preprintcond-mat/9704071.[4] D. Popović, A.B. Fowler, and S. Washburn, Phys.Rev. Lett. 79, 1543 (1997).[5] V. Dobrosavljević, E. Abrahams, E. Miranda, andSudip Chakravarty, Phys. Rev. Lett. 79, 455 (1997).[6] T. Ando, A.B. Fowler, and F. Stern, Rev. Mod.Phys. 54, 437 (1982).[7] P. A. Lee and T. V. Ramakrishnan, Rev. Mod.Phys. 57, 287 (1985).[8] S.L. Sondhi, S.M. Girvin, J.P. Carini, and D. Sha-har, Rev. Mod. Phys. 69, 315 (1997); T.R. Kirk-patrick and D. Belitz, in Electron Correlations inthe Solid State, ed. N.H. March (Imperial CollegePress, London, 1998), preprint cond-mat/9707001.[9] I.M. Lifshitz and V.Ya. Kirpichenkov, JETP 67,1276 (1979).[10] L.I. Glazman and K.A. Matveev, Zh. Eksp. Teor.Fiz. 94, 332 (1988) [Sov. Phys. JETP 67, 1276(1988)].[11] D. Popović, A.B. Fowler, and S. Washburn, Phys.Rev. Lett. 67, 2870 (1991).[12] K. Ismail, J.O. Chu, D. Popović, A.B. Fowler, and S.Washburn, submitted to Phys. Rev. Lett., preprintcond-mat/9707061.[13] V.M. Pudalov, G. Brunthaler, A. Prinz, and G.Bauer, JETP Lett. 65, 932 (1997).[14] D. Popović and S. Washburn, Phys. Rev. B, RapidComm. (Oct. 15, 1997).4

https://cdr.lib.unc.edu/downloads/nc580x093

Conductance Fluctuations Near the Two-Dimensional Metal-Insulator
  Transition

Conductance Fluctuations Near the Two-Dimensional Metal-Insulator Transition

Abstract

Similar works

Full text

Available Versions

Crossref

Carolina Digital Repository