The temperature dependence of conductivity $\sigma (T)$ in the metallic phase
of a two-dimensional electron system in silicon has been studied for different
concentrations of local magnetic moments. The local moments have been induced
by disorder, and their number was varied using substrate bias. The data suggest
that in the limit of $T\to 0$ the metallic behavior, as characterized by
$d\sigma/dT < 0$, is suppressed by an arbitrarily small amount of scattering by
local magnetic moments.Comment: 4 pages, revtex, plus four encapsulated postscript figure

Feng, X. G.

Popovic, Dragana

Washburn, S.

English

arXiv

The temperature dependence of conductivity $\sigma (T)$ in the metallic phase of a two-dimensional electron system in silicon has been studied for different concentrations of local magnetic moments. The local moments have been induced by disorder, and their number was varied using substrate bias. The data suggest that in the limit of $T\to 0$ the metallic behavior, as characterized by $d\sigma/dT < 0$, is suppressed by an arbitrarily small amount of scattering by local magnetic moments. Comment: 4 pages, revtex, plus four encapsulated postscript figure

Popović, Dragana

Carolina Digital Repository

arXiv:cond-mat/9902036v2  [cond-mat.dis-nn]  30 Mar 1999Effect of Local Magnetic Moments on the Metallic Behavior in Two DimensionsX. G. Feng1, Dragana Popović1, and S. Washburn21National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 323102Dept. of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599(August 11, 2018)The temperature dependence of conductivity σ(T ) in the metallic phase of a two-dimensional electronsystem in silicon has been studied for different concentrations of local magnetic moments. The localmoments have been induced by disorder, and their number was varied using substrate bias. Thedata suggest that in the limit of T → 0 the metallic behavior, as characterized by dσ/dT < 0, issuppressed by an arbitrarily small amount of scattering by local magnetic moments.PACS Nos. 71.30.+h, 71.27.+a, 73.40.QvA metal-insulator transition (MIT) has been observedrecently in a variety of two-dimensional (2D) elec-tron [1–4] and hole [5,6] systems but there is still no gen-erally accepted microscopic description of the 2D metallicphase. Some of the relevant properties of the 2D metalinclude: (a) an increase of conductivity σ with decreas-ing temperature T (i. e. dσ/dT < 0) for carrier densitiesns > nc (nc – critical density); and (b) a suppression ofthe dσ/dT < 0 behavior by magnetic field [7]. The lattersuggests the importance of the spin degrees of freedom,which can be probed further by studying the effect oflocal magnetic moments on the transport properties ofthe conduction electrons. Indeed, magnetic impuritieshave been used extensively over the last several decadesto probe the properties of metals, and continue to berelevant today in attempts to understand heavy-fermionmaterials and high-Tc superconductors [8]. In the exper-iment discussed below, the number of local moments isvaried in a controlled way. We show that in the T → 0limit the dσ/dT < 0 behavior is suppressed by an ar-bitrarily small amount of scattering of the conductionelectrons by disorder-induced local moments.We present results obtained on a 2D electron systemin Si metal-oxide-semiconductor field-effect transistors(MOSFETs). In such a device, the disorder is due tothe oxide charge scattering (scattering by ionized impu-rities randomly distributed in the oxide within a few Åof the interface) and to the roughness of the Si-SiO2 in-terface [9]. For a fixed ns, it is possible to change thedisorder by applying the substrate (back gate) bias Vsub.In particular, the reverse (negative) Vsub moves the elec-trons closer to the interface, which increases the disorder.It also increases the splitting between the subbands sincethe width of the triangular potential well at the interfaceis reduced by applying negative Vsub. Usually, only thelowest subband is occupied at low T , giving rise to the2D behavior. In sufficiently disordered samples, however,the band tails associated with the upper subbands [10]can be so long that some of their strongly localized statesmay be populated even at low ns, and act as additionalscattering centers for 2D electrons. Clearly, the negativeVsub reduces this type of scattering by depopulating theupper subband. The effect of scattering by electrons lo-calized deep in the tails of the upper subband was firstobserved as an enhancement of the mobility µ at low nswith negative Vsub [11], and was subsequently studied inmore detail by other groups using different measurementsand techniques [12]. More recently, we have used one neg-ative value of Vsub to enhance µ (reduce the disorder) atintermediate values of ns, and observed the change fromdσ/dT > 0 to dσ/dT < 0 for ns > nc [2]. Here, how-ever, we present a systematic study of this process as thedisorder is varied using Vsub. We show clearly that thebare value of high T (Drude) mobility is not sufficientto predict the sign and the magnitude of dσ/dT at lowT but rather that it is the type of the disorder that isrelevant [13]. In particular, we show that scattering byelectrons localized in the tail of the upper subband hasa much more profound effect on dσ/dT than potentialscattering due to oxide charges and surface roughness.This is attributed to spin flip scattering by electrons inlocalized states that are singly populated due to a strongon-site Coulomb repulsion, and act as local magnetic mo-ments. Large on-site Coulomb interaction has been welldocumented in systems similar to ours, such as electronsin quantum dots [16], and other materials with stronglylocalized states [17]. For typical localization lengths of∼100 Å in Si MOSFETs [9,18], the on-site Coulomb re-pulsion is ∼10 meV. Therefore, such states will be singlyoccupied at low ns [9].Our measurements were carried out on n-channel SiMOSFETs with the oxide charge density of 3×1010cm−2,determined using standard techniques [9,19]. Other de-tails of the sample structure are given in Ref. [2]. For afixed Vsub, ns was controlled by the gate voltage Vg anddetermined in a standard fashion [9,19]. σ(Vg) was mea-sured at temperatures 0.3 < T < 4.5 K for ns of up to3× 1012cm−2 and for –50 V≤ Vsub ≤+1 V. The effect ofVsub on µ at 4.2 K was found [14] to be consistent withearlier work and our interpretation. In particular, forns < nmax (nmax ∼ 5× 1011cm−2 is the density where µreaches its maximum), an increase of µ is observed [14]10 1 2 3 4 5051015+1V–1V–4V–8V–16V–32V–50V(a)T (K)σ (e2 /h)0 2 4 6 8 10051015(b)T2 (K2)σ (e2 /h)FIG. 1. Temperature dependence of the conductivity σ forns = 3.0 × 1011cm−2. (a) The data are shown for differentvalues of Vsub as given on the plot. (b) The data are plottedvs. T 2, and shown for Vsub = +1,−0.5,−1,−2,−4,−6,−8 Vgoing from bottom to top.with the negative Vsub as a result of the decreased scatter-ing by local moments from the upper subband, and con-sistent with early work [11]. For ns > nmax, µ decreaseswith (negative) Vsub, consistent with the fact that surfaceroughness scattering is the dominant source of disorderin this range of ns [9]. This is a result of an increasedproximity of the 2D electrons to the interface and, pos-sibly, a reduction in scattering by local moments fromthe upper subband. The latter could be due to a smallernumber of local moments being present in a sample athigh ns (for a given Vsub) because of an improved screen-ing by 2D electrons. For sufficiently high negative Vsub(−Vsub > 35 V), the 4.2 K mobility decreases with Vsubfor all ns, suggesting that the upper subband has beencompletely depopulated and that the further increase inVsub leads only to increasing disorder due to potentialscattering from roughness at the Si-SiO2 interface.Fig. 1(a) shows some typical results for σ(T ) in thens < nmax range as a function of Vsub. The metallic be-havior, such that dσ/dT < 0, spreads out towards lowerT with the increasing negative Vsub, i. e. as the scatter-ing by local moments is reduced, and it also spreads outtowards higher values of ns (not shown). In other words,σ(T ) displays a maximum at T = Tm, such that Tm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70123456(1/2 Ns + Nd)–2/3 (1011/cm2)–2/3Tm (K)ns = 2x1011/cm2ns = 3x1011/cm2ns = 5x1011/cm2dσ/dT < 0 dσ/dT > 0FIG. 2. Position of the maximum Tm in σ(T ) for differentvalues of ns as a function of the inverse subband splitting.(The left hand side on the x-axis corresponds to large valuesof the negative Vsub, and the right hand side corresponds tolow values of Vsub.) The metallic behavior with dσ/dT < 0 isobservable at T > Tm. Tm extrapolates to zero (dashed lines)for Vsub ∼ −40 V, corresponding to the subband splitting ofthe order of 30 meV.shifts to lower T with the (negative) Vsub. As (−Vsub) isincreased beyond 35 V, the form of σ(T ) is no longer verysensitive to changes in Vsub even though the disorder dueto potential scattering increases. In addition, by compar-ing the data for Vsub = −50 V and –1 V, for example, itis obvious that dσ/dT < 0 behavior is more pronounced(Tm is lower) when scattering by local moments is re-duced even though the 4.2 K mobility is lower [13]. Thisdemonstrates clearly the need to distinguish between dif-ferent types of disorder, a fact that has been overlookedin some theoretically proposed phase diagrams [20,21].For T < Tm, σ decreases with decreasing T and, in fact,follows a T 2 form at the lowest T [Fig. 1(b)]. Such σ(T )is often considered to be a signature of local magneticmoments, and results from the Kondo effect [8]. Here itrepresents a direct evidence for the existence of local mo-ments in our samples. A detailed study of this regime hasbeen presented elsewhere [22]. Fig. 1(b) also shows thatthe range of T (T < Tm) where local moments dominatetransport becomes smaller as their number is reduced byincreasing negative Vsub.The position of the maximum Tm in σ(T ) is shownin Fig. 2 for different values of ns as a function ofthe inverse subband splitting (Nd is the depletion layercharge density, which increases with the reverse Vsub) [9].The metallic behavior with dσ/dT < 0 is observable atT > Tm [see Fig. 1(a)]. For a given ns, Tm extrapolatesto zero for a finite value of the subband splitting. Thisvalue is slightly higher for higher ns, consistent with thefact that the number of local moments in the upper sub-band is also slightly higher (Fermi energy EF is higher)and, therefore, one needs to apply more Vsub in order todepopulate the upper subband. From the data shown in2–55 –50 –45 –40 –35 –30 –25 –20 –15 –10 –5 01.01.11.21.31.41.5Vsub (V)nc (1011 /cm2 )0.600.700.800.901.00σ c (e2/h)ncσc<––––>–50 –40 –30 –20 –10 01.61.71.81.92.02.1Vsub (V)zνFIG. 3. Critical density nc and critical conductivity σc de-termined from the data at T > Tm as a function of Vsub. Thedisorder due to potential scattering increases with the nega-tive Vsub. Inset: critical exponents zν vs. Vsub obtained fromthe same data. Dashed lines guide the eye.Fig. 2, it follows that Tm goes to zero for Vsub ∼ −40 V inagreement with the measurement of the 4.2 K mobility asdiscussed above. Vsub ∼ −40 V corresponds to the sub-band splitting of the order of 30 meV [9]. The extent ofthe tail of the upper subband derived in this way is con-sistent with earlier work [12]. Our data, therefore, showthat the 2D metal with dσ/dT < 0 can exist at T = 0only in the absence of scattering by disorder-induced lo-cal moments. This is similar to the behavior observed ina magnetic field [7], and consistent with some theoreticalmodels [23–26].In the presence of scattering by local moments,dσ/dT < 0 is observable at T > Tm, where Tm, of course,can be arbitrarily small. For a fixed Vsub, σ(ns, T ) forT > Tm exhibits [2] all of the properties of a 2D MIT.Fig. 3 shows the values of nc and the (apparent) [27]critical conductivity σc determined only from the dataat T > Tm, where scattering by local moments is not sig-nificant, as a function of Vsub. Both nc and σc increasemonotonically with the reverse Vsub, i. e. with an increasein disorder due to potential scattering. The increase ofnc and σc with disorder observed here on a single sam-ple is in agreement with the same conclusion reached bycomparing MOSFETs with different peak mobilities [28].Fig. 3 inset shows the dependence of the critical expo-nents zν (z– dynamic exponent, ν– correlation lengthexponent) on Vsub. zν was determined by scaling thedata in the vicinity of nc, where σ(ns, T ) = σcf(T/δzνn )[δn = (ns − nc/nc)] [2,27].Fig. 4(a) shows σ(T ) for Vsub = −40 V for a smallrange of ns close to nc. Based on the analysis discussedabove, we are confident that for this value of Vsub thereare no local moments associated with the upper subbandin the sample. Indeed, the σ(T ) curves do not exhibita maximum in the measured range of T , except possiblywhere shown in Fig. 4(a). Such a structure might, in fact,0 1 2 30.00.51.01.52.0(b)T (K)σ (e2 /h)0123(a)σ (e2 /h)FIG. 4. Temperature dependence of conductivity for nsfrom 1.02 × 1011cm−2 (bottom curve) up, in steps of0.1×1011cm−2 and (a) Vsub = −40 V, (b) Vsub = −50 V. Thearrows indicate the peaks in σ(T ) for ns(1011cm−2) = 1.42and 1.52 for Vsub = −50 V, and possible peaks forns = 1.32× 1011cm−2 at both Vsub.be due to local moments associated with the lowest sub-band. From the theoretical point of view, the possibilityof disorder-induced local moment formation in stronglyinteracting electronic systems has been suggested sincethe early developments of the theory of interacting disor-dered systems [29,24] and has been studied, subsequently,using different models [30]. In order to test this idea, wehave increased Vsub to −50 V, thus increasing the dis-order, which is now due only to scattering by the oxidecharges and the surface roughness. Fig. 4(b) shows that,for the same ns, the values of σ are reduced as expectedfor higher disorder. More importantly, we observe theappearance of the peak in σ(T ) for those ns for which itwas clearly absent in Fig. 4(a). Also, a weak structure(peak) in σ(T ) for ns = 1.32 × 1011cm−2 has shifted tohigher T . This increase of Tm, and the dependence of Tmon ns as shown in Fig. 4(b), are qualitatively the sameas what was observed by increasing the number of lo-cal moments associated with the upper subband. Theseresults strongly suggest that an increase in the poten-tial (non-magnetic) scattering has led to the formation ofthe (additional) local moments in the system. We havealso observed similar peaks in σ(T ) in different Si MOS-3FETs at T < 0.4 K but, without a systematic study suchas this one, it would have been impossible to determinetheir origin. We speculate that local moments might ex-ist in other materials as well but that the correspondingTm might be experimentally inaccessible in high-mobilitydevices.Back gate bias was used recently in a 2D hole sys-tem [15] to study the effect of the spin-splitting due tothe spin-orbit interaction and the inversion asymmetryof the confining potential [31]. It was found that themagnitude of the dσ/dT < 0 behavior was reduced asthe spin-splitting decreased, i. e. as the confining poten-tial became more symmetric. In our samples, we observethe opposite: the triangular confining potential becomesmore symmetric with the application of the reverse Vsub,and that is exactly when the dσ/dT < 0 behavior ap-pears. Therefore, even if the effect of the spin-orbit in-teraction exists in our samples, it does not drive the MIT.The reverse Vsub also reduces the average spatial ex-tent ∆z of the inversion layer charge density in the di-rection perpendicular to the interface (typically, ∆z ≈20 − 30 Å) [9], leading to an increase of the effectiveCoulomb interaction [32]. Since rs (rs – the average inter-carrier separation in units of the effective Bohr radius)is already fairly large (rs ∼ 15) for ns close to nc (givenin Fig. 3), we expect that the further increase in theCoulomb interaction for a fixed ns would only lead to aninsulating behavior (see, e. g. Refs. [20,21]) and not tothe metallic behavior with dσ/dT < 0, as observed. Weconclude that the effect of Vsub on the effective Coulombinteraction is not the dominant effect in our samples.Our study shows that the 2D metal with dσ/dT < 0can exist in the T → 0 limit only in the absence of scat-tering by local magnetic moments. Our results emphasizethe key role of the spin degrees of freedom in the physicsof the low density 2D electron system.The authors are grateful to V. Dobrosavljević and A.B. Fowler for helpful discussions. This work was sup-ported by NSF Grant No. DMR-9796339.[1] 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).[2] D. Popović, A. B. Fowler, and S. Washburn, Phys. Rev.Lett. 79, 1543 (1997).[3] S. J. Papadakis, et al., Phys. Rev. B 57, R15068 (1998).[4] E. Ribeiro, et al., Phys. Rev. Lett. 82, 996 (1999); Y.Hanein, et al., preprint, cond-mat/9808251.[5] J. Lam, et al., Phys. Rev. B 56, R12741 (1997); P. 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Effect of Local Magnetic Moments on the Metallic Behavior in Two Dimensions

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