VOLUME83, NUMBER 1 PHYSICAL REVIEW LETTERS 5JULY1999Charged Impurity-Scattering-Limited Low-Temperature Resistivityof Low-Density Silicon Inversion LayersS. Das Sarma and E. H. HwangDepartment of Physics, University of Maryland, College Park, Maryland 20742-4111(Received 14 December 1998)We calculate within the Boltzmann equation approach the charged impurity-scattering-limited low-temperature electronic resistivity of low-density n-type inversion layers in Si MOSFET structures. Wefind a rather sharp quantum to classical crossover in the transport behavior in the 05Ktemperaturerange, with the low-density, low-temperature mobility showing a strikingly strong nonmonotonictemperature dependence, which may qualitatively explain the recently observed anomalously strongtemperature dependent resistivity in low-density, high-mobility MOSFETs.PACS numbers: 73.40.Qv, 71.30.+h, 73.50.BkSeveral recent publications on low-temperature resis-tivity measurements [1–4] in various low-density two-dimensional (2D) systems report the observation of ananomalously strong temperature dependence as a functionof carrier density, which has been interpreted as evidencefor a zero-temperature two-dimensional metal-insulatortransition (2D MIT), which is considered to be forbiddenin two dimensions (at least for a noninteracting 2D sys-tem) by the one parameter scaling theory of localization[5]. A number of theoretical papers [6–8] have appearedin the literature providing many possible resolutions of thisseemingly unanticipated (but apparently ubiquitous) phe-nomenon. In this Letter, we propose a possible theoreticalexplanation for (at least a part of ) the observed phenom-ena. Our explanation is quantitative, microscopic, andphysically motivated. Although our theory is quite generaland generic (and thus applicable to all the systems [1–4]exhibiting the so-called 2D MIT), we specifically con-sider here the electron inversion layer in Si metal-oxide-semiconductor field-effect transistor (MOSFETs), which isboth the original system in which the 2D MIT was firstreported [1] and is also the most exhaustively experimen-tally studied [1–3] system in this context. It is importantto emphasize that, in contrast to much [6] of the existingtheoretical work on the subject, our theory does not ad-dress the existence (or not) of a zero temperature 2D MIT,but addresses the issue of quantitatively understanding thestrikingly unusual finite temperature experimental resultson the effective “metallic” side of the transition.We first summarize the key experimental features of the2D MIT phenomenon (focusing on Si MOSFETs), empha-sizing the specific aspects addressed in our theory. Ex-perimentally one finds a “critical density” (nc) separatingan effective metallic behavior (for density ns. nc) froman effective “insulating” behavior (ns, nc). We con-centrate entirely on the effective metallic behavior in thisLetter since a 2D metal is “unusual” according to theconventional theory [5] and a 2D insulator is not. Theexperimental insulating behavior (for ns, nc) is quiteconventional for a strongly localized semiconductor, andcan be understood using standard transport models [7,8].The effective metallic behavior is characterized by a strongdrop in the temperature dependent resistivity, r共T 兲, at lowtemperatures (0.1 K # T # 13K) and at low densities(ns$ nc). This novel and dramatically strong tempera-ture dependence of r共T 兲, where r共T兲 may drop by a factorof 210 at low electron densities as temperature decreasesfrom 2 K to 100 mK, is one of the most significant experi-mental observations we qualitatively explain in this Let-ter. In addition the experimental resistivity, r共T , ns兲,asa function of temperature and electron density, shows anapproximate “scaling” behavior r共T , ns兲⯝r共T兾T0兲 withT0⬅ T0共ns兲 indicating consistency with quantum critical-ity. Our theoretical results show the same scaling behav-ior with our calculated T0共ns兲 having very similar densitydependence as the experimental observation. There areinteresting aspects of the magnetic field and the electricfield dependence of the observed resistivity, which we donot address here, concentrating entirely on the behavior ofr共T , ns兲 in the ns$ ncmetallic regime. It is this “anoma-lous metallic” behavior (in the sense of a very strong metal-lic temperature dependence of the resistivity in a narrowdensity range above nc) which has created the recent in-terest in the 2D MIT phenomena since in general, the tem-perature dependent resistivity of a metal should saturateas it enters the low-temperature Bloch-Grüneisen regimewithout manifesting any strong temperature dependence.Our theory, which provides good qualitative agreementwith the existing experimental data on the metallic (ns.nc) side of the transition, is based on two essential assump-tions: (1) transport is dominated by charged impurity scat-tering centers (with a density of Niper unit area) which arerandomly distributed at the interface; (2) the MIT at ns苷ncis characterized by a “freeze-out” of free carriers dueto impurity binding—the free carrier density responsiblefor metallic transport is thus (ns2 nc) for ns. nc, andon the insulating side, ns, nc, the free carrier density (atT 苷 0) is by definition zero. Some justifications for theseassumptions have been provided in Ref. [7], although ourcurrent model transcends the specific scenario envisioned164 0031-9007兾 99兾83(1)兾164(4)$15.00 © 1999 The American Physical SocietyVOLUME83, NUMBER 1 PHYSICAL REVIEW LETTERS 5JULY1999in Ref. [7] and is more general. In contrast to Ref. [7],we do not specify any particular mechanism for the carrierfreeze-out and accept it as an experimental fact. We notethat we could extend our model and go beyond the abovetwo assumptions, for example, by making the effective freecarrier density n 苷 共ns2 nc兲u共ns2 nc兲 1 na共T兲, wherena共T兲 is a thermally activated contribution to the carrierdensity (this relaxes the second assumption), and/or byintroducing additional scattering mechanisms such as theshort-range surface roughness scattering (this relaxes thefirst assumption). These extensions beyond our two essen-tial approximations will undoubtedly produce better quan-titative agreement between our theory and experiment (atthe price of having unknown adjustable parameters). We,however, refrain from such a generalized theory, becausewe believe that the