Magnetoconductivity of insulating silicon inversion layers
Yeekin Tsui, S. A. Vitkalov, and M. P. Sarachik
Physics Department, City College of the City University of New York, New York, New York 10031, USA T. M. Klapwijk
Department of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands 共Received 23 February 2004; revised manuscript received 8 July 2004; published 24 January 2005兲
The normalized in-plane magnetoconductivity of the dilute strongly interacting system of electrons in silicon metal–oxide–semiconductor field effect transistors scales with B / T for low densities in the insulating phase. Pronounced deviations occur at higher metalliclike densities, where an energy scale kB⌬ emerges which is not
associated with either magnetic field or thermal effects. While the energy scale kB⌬ vanishes at n0approaching from the metallic side, we do not find a finite⌬ which goes to zero approaching n0from the insulating side. DOI: 10.1103/PhysRevB.71.033312 PACS number共s兲: 73.40.Qv, 71.30.⫹h, 73.50.Jt
According to well established theory, no metallic state can exist in two dimensions for noninteracting1 or weakly interacting2electrons共or holes兲 in zero magnetic field in the
limit of zero temperature. In dilute two-dimensional systems where the interactions are known to be strong, however, ex-perimental studies have revealed an unexpected decrease in the resistance as the temperature is lowered, behavior that is generally a characteristic of metals. This metallic behavior has been observed down to the lowest accessible tempera-tures at electron3,4 and hole5–7 densities above some critical
density nc 共or pc兲. For densities down to approximately 1.5nc, the temperature dependence has been attributed to electron-electron scattering in the ballistic limit共kBTⰇប/兲, confirming the importance of strong e-e interactions in de-termining the behavior of these systems.8However, the
be-havior observed at lower densities and the nature of the ap-parent metal–insulator transition are still not understood.9
A very unusual characteristic of dilute, strongly interact-ing electron 共hole兲 systems is their strong response to an in-plane magnetic field: the resistivity increases dramatically with increasing field and saturates to a value above a char-acteristic magnetic field that depends on density and temperature.9 From an analysis of the temperature and
density dependence of the magnetoconductance of silicon metal–oxide–semiconductor field effect transistors
共MOSFETs兲, Vitkalov et al.10have identified an energy scale 共kB⌬兲 which extrapolates to zero at a finite density n0in the
vicinity of nc; this was attributed to an increase in the mag-netic susceptibility⬀共g* m*兲 and the approach to a zero temperature quantum phase transition at n0 共here g* and m*
are the renormalized Lande-g factor and effective mass, re-spectively兲. Shashkin et al.11found similar results; moreover, these authors have claimed that the sharp increase in the susceptibility is associated with an increase in the effective mass while the g value remains essentially constant as the electron density approaches nc.12
These findings suggest critical behavior and the approach to a quantum phase transition where an energy scale kB⌬ vanishes at n0approaching from the metallic side. The
ques-tion thus arises whether there exists an energy scale that goes to zero approaching n0from the insulating side, and whether
this is evidenced as the reemergence of a finite ⌬ as the electron density is reduced in the insulating regime.
In this paper we report the results of an investigation of the magnetoconductivity of the dilute two-dimensional共2D兲 system of electrons in a high-mobility silicon MOSFET in the insulating phase. We find that the in-plane magnetocon-ductivity scales with 共B/T兲 for electron densities near and below n0, and that the parameter⌬ remains zero in the
insu-lator down to the lowest electron densities we were able to measure. Pronounced deviations from simple B / T scaling become evident in the metallic regime at higher densities: in agreement with our earlier findings,10scaling in the metallic
phase requires the inclusion of an additional energy scale
共kB⌬兲. The different behavior of the magnetoconductivity at low and high densities suggests the existence of two distinct phases.
The sample used in these studies is a high-mobility silicon MOSFET共peak⬃20 000/V s at 0.5 K兲. Contact resistances
were minimized by using a split-gate geometry that allows a higher electron density to be established in the vicinity of the contacts than in the 2D system under investigation. Data were taken by standard four-terminal ac techniques for elec-tron densities above 1.2⫻1011cm−2, and by dc techniques
for lower densities. Experiments were performed at tempera-tures between 0.25 and 4 K in magnetic fields up to 10 T; data were taken in the linear regime using small currents to avoid overheating the electrons. The critical density nc is
⬇0.90⫻1011cm−2for this sample.
Figure 1共a兲 shows the resistivity of the silicon MOSFET as a function of electron density at four different tempera-tures: the behavior is insulating for densities below the cross-ing point at 0.9⫻1011cm−2 共resistivity increasing with
de-creasing temperature兲 and metallic above that density. Figure 1共b兲 shows the conductivity as a function of in-plane mag-netic field for different densities at T⬃0.25 K; here the top four curves correspond to densities in the conducting range, the fifth is at or near the critical density, and the remaining are in the insulating regime. Consistent with earlier results in silicon and various materials, the magnetocoductivity satu-rates at a progressively lower applied field as the density decreases. The enormous response to in-plane magnetic field is a typical feature in these 2D systems.13–16
PHYSICAL REVIEW B 71, 033312共2005兲
For a fixed electron density in the metallic regime 共ns
⬎nc兲, the inset to Fig. 2 shows the conductivity as a function of in-plane magnetic field at different temperatures. Follow-ing the procedure used in a previous study,10the normalized magnetoconductivity
norm⬅
共B = 0兲 −共B兲
共B = 0兲 −共B → ⬁兲, 共1兲
is plotted as a function of a scaled magnetic field B / B. Here
Bis a fitting共or scaling兲 parameter chosen to obtain a data collapse. Note that the normalized magnetoconductivity共1兲 is simply the field dependent contribution to the conductivity,
关共B=0兲−共B兲兴, normalized by its full value, 关共B=0兲
−共B→⬁兲兴.
A study of the scaled normalized magnetoconductivity at various temperatures and densities allows a determination of the parameter B as a function of temperature and density.
Consistent with our earlier analysis,10B
satisfies the empiri-cal formula
B= An
冑
⌬2+ T2, 共2兲where the fitting parameter Anvaries by less than 15% in the range of densities of our experiments. The result of this analysis is shown in Fig. 3, where⌬ is plotted as a function of electron density ns.
The quantity⌬ enters on an equal footing with the tem-perature T and represents an energy scale 共kB⌬兲 associated with B. In agreement with published results, the plot shown FIG. 1. 共a兲 Resistivity of a silicon MOSFET at four
tempera-tures, as labeled, as a function of electron density. The curves cross at the critical density nc= 0.9⫻1011cm−2. 共b兲 Conductivity as a
function of applied in-plane magnetic field at different densities, as labeled; T =⬃0.27 K. The top four magnetoconductance curves are at metallic densities, and the bottom three are insulating.
FIG. 2. Normalized magnetoconductivity as a function of the scaled field at different temperatures at electron density ns= 1.2 ⫻1011cm−2. The inset shows the共unnormalized兲 conductivity vs in-plane magnetic field at various temperatures.
FIG. 3. The parameter⌬ determined from measured data using Eq. 共2兲 vs electron density ns. The open circles represent data of Vitkalov et al.共see Ref. 10兲 and closed symbols denote data re-ported in this paper; the solid line is a guide to the eye. Note that each point at a given electron density is obtained by scaling many magnetoconductivity curves at different temperatures. The dashed line represents an estimate of the uncertainty associated with pos-sible heating of the electron system. The inset shows B vs tem-perature T for density ns= 0.82⫻1011cm−2, illustrating that B
= AnT and⌬=0 for this density.
BRIEF REPORTS PHYSICAL REVIEW B 71, 033312共2005兲
in Fig. 3 indicates that the energy共kB⌬兲 extrapolates to zero at a finite electron density n0 close or equal to the critical density ncthat signals the change in temperature dependence of the conductivity from metallic to insulating.10,17 That ⌬
= 0 at or below n0, and hence, B= AnT, is illustrated in the inset to Fig. 3, where we have used B as a free fitting parameter to collapse the curves; the parameter⌬ equals zero with an estimated accuracy of 5 – 10 mK.
The vanishing energy 共kB⌬兲 signals a diverging correla-tion time scale ⬃ប/共kB⌬兲, suggesting that the system is approaching a quantum phase transition in the limit T→0.
The results of our current measurements are denoted by black squares in Fig. 3. For the metallic densities above nc, the measurements provide additional and more precise data that support our earlier conclusions. At low densities, our measurements provide important new information: the en-ergy scale 共kB⌬兲 remains zero in the insulator down to the lowest density measured, n = 0.76⫻1011cm−2. This implies
that the normalized magnetoconductivity scales strictly with
B / T in the insulating phase关see Eq. 共2兲兴.
This is shown explicitly in Figs. 4 and 5. Figure 4 shows the normalized magnetoconductivity as a function of B / T at various different temperatures for two electron densities, one below and one above nc: the magnetoconductivity curves collapse onto one curve in the insulator shown in frame共a兲 while this is clearly not true for the metallic density shown in frame 共b兲. Figure 5 shows the normalized
magnetoconduc-tivity as a function of B / T at temperature 0.5 K for various densities across the transition. The data collapse onto a single curve for densities up to about 0.9⫻1011cm−2, and devia-tions become progressively more pronounced as the electron density is increased into the metallic phase.
Experiments on MOSFETs18,19and other two-dimensional
materials indicate that the magnetoconductivity is directly related to the degree of polarization of the electrons,共holes兲, and therefore, to the magnetization. This one-to-one corre-spondence suggests that if the conductivity scales with B / T, then so does the magnetization. We note that the magnetiza-tion of a set of localized, noninteracting magnetic moments is a function of B / T. However, this is quite unlikely to pro-vide an adequate description for dilute 2D silicon MOSFETs, where electron interactions are an order of magnitude larger than the Fermi energy and provide the dominant energy in the problem. B / T behavior is predicted in the insulating phase by Agrinskaya and Kozub, who calculated the effect of on-site spin correlations on the hopping conductivity.20
To summarize, the normalized in-plane magnetoconduc-tivity of the dilute strongly interacting system of electrons in silicon MOSFETs scales with B / T in the insulating regime for densities below n0. Deviations occur at higher densities
which we attribute to an energy scale kB⌬ that is not associ-ated with either magnetic field or thermal effects. While the energy scale kB⌬ vanishes at n0approaching from the
metal-lic side, we do not find a finite ⌬ which goes to zero ap-proaching n0from the insulating side. The breakdown of B / T
scaling above n0suggests that distinct phases exist above and
below n0 at or near the electron density nc that signals the change from insulating to metallic temperature dependence of the conductivity.
We thank B. Spivak and V. T. Dolgopolov for useful dis-cussions. This work was supported by DOE-FG02-84-ER45153 and NSF Grant No. DMR-0129581.
FIG. 4. The normalized magnetoconductivity关see Eq. 共1兲兴 vs B / T for共a兲 insulating density ns= 0.82⫻1011cm−2and共b兲 metallic density n = 1.0⫻1011cm−2.
FIG. 5. The normalized magnetoconductivity关see Eq. 共1兲兴 vs B / T for several electron densities at T = 0.5 K. Note that the mag-netoconductivity scales with B / T for insulating densities ns⬍ncand
does not scale for metallic densities.
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