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Sharply increasing effective mass near the 2D metal–insulator transition
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Physica E 22 (2004) 224 – 227 www.elsevier.com/locate/physe
Sharply increasing e ective mass near the 2D metal–insulator transition A.A. Shashkina;1 , Maryam Rahimia , S. Anissimovaa , S.V. Kravchenkoa;∗ , V.T. Dolgopolovb , T.M. Klapwijkc a Physics
Department, Northeastern University, Boston, MA 02115,USA of Solid State Physics, Chernogolovka, Moscow District 142432,Russia c Department of Applied Physics, Delft University of Technology, Delft 2628 CJ, The Netherlands b Institute
Abstract We report measurements of the e ective mass and Land0e g factor in strongly interacting two-dimensional systems in silicon. A sharp increase of the e ective mass has been observed near the metal–insulator transition, while the g factor remained nearly constant and close to its value in bulk silicon. Furthermore, the enhanced e ective mass has been found to be independent of the degree of spin polarization, which points to a spin-independent origin of the mass enhancement, in contradiction with existing theories. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.30. +h; 73.40. Qv Keywords: Metal–insulator transition; E ective mass; Large g-factor
Recently, there has been a lot of interest in the transport and magnetic properties of dilute two-dimensional (2D) electron systems due to the observation of an unexpected metal–insulator transition (MIT) in zero magnetic ;eld [1]. At su>ciently low electron densities, 2D electron systems become strongly correlated, because the kinetic energy is overpowered by energy of electron–electron interactions; this is an example of the strong-coupling many-body problem for which theoretical methods are still poorly developed. The strength of the interactions is usually characterized by the ratio ∗ Corresponding author. Tel.: +1-617-3732926; fax: +1-6173732943. E-mail address: [email protected] (S.V. Kravchenko). 1 Permanent address: Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia.
between the Coulomb energy and the Fermi energy, rs = Ec =EF , which, assuming that the e ective electron mass is equal to the band mass, in the systems with single-valley spectrum reduces to the Wigner– Seitz radius, 1=(ns )1=2 aB (here ns is the electron density and aB is the Bohr radius in semiconductor). There are several suggested candidates for the ground state of the system, for example, (i)Wigner crystal characterized by spatial and spin ordering [2], (ii)ferromagnetic Fermi liquid with spontaneous spin ordering [3], and (iii)paramagnetic Fermi liquid [4]. In the strongly interacting limit (rs 1), no analytical theory presently exists. According to numeric simulations [5], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. The re;ned numeric simulations [6] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ¡ rs ¡ 35, the ground state
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.254
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
of the system is a strongly correlated ferromagnetic Fermi liquid. A paramagnetic Fermi liquid is realized at yet higher electron densities when the interactions are relatively weak (rs ∼ 1). The e ective mass, m, and Land0e g factor within the Fermi liquid theory are renormalized due to exchange and correlation e ects, with renormalization of the g factor being dominant compared to that of the e ective mass [7]. Alternatively, near the onset of Wigner crystallization, strong increase of the e ective mass is expected [8,9]. In this paper, we report measurements of the e ective mass and g factor in a wide range of electron densities including the immediate vicinity of the MIT. We have investigated “clean regime” which occurs only in very low-disordered samples that remain metallic at electron densities down to ≈ 8 × 1010 cm−2 . It turns out that with decreasing electron density, the value of the e ective mass becomes strongly (by more than a factor of 3) enhanced, while the g factor remains nearly constant and close to its value in bulk silicon. Using tilted magnetic ;elds, we ;nd that the value of the e ective mass does not depend on the degree of spin polarization, which points to a spin-independent origin of the e ective mass enhancement. This is in clear contradiction with existing theories [7–9]. Measurements have been made in a rotator-equipped Oxford dilution refrigerator with a base temperature of ≈ 30 mK on low-disordered (100)-silicon MOSFET samples with peak electron mobilities close to 3 m2 =Vs at 0:1 K; for more on samples and experimental procedure, see Refs. [10,11]. Strong enhancement of the spin susceptibility near the MIT was recently observed in Si MOSFETs [10,12–14]. In principle, either g or m (or both) may be responsible for this e ect. To separate g and m, we have analyzed the data for temperature-dependent conductivity in zero magnetic ;eld using recent theory [15]. According to this theory, deep enough in the metallic state, is a linear function of temperature. (T ) = 1 − A∗ kB T; 0
(1)
where the slope, A∗ , is determined by the Fermi liquid constants, F0a and F1s . This allows us to determine the interaction-enhanced g factor and mass m separately using the data for the slope A∗ and the product gm determined independently from the measurements of
225
Fig. 1. The temperature dependence of the normalized conductivity at di erent electron densities (indicated in units of 1011 cm−2 ) above the critical electron density for the metal–insulator transition. The dashed lines are ;ts of the linear interval of the dependence. The inset shows comparison of 1=A∗ (ns ) (see Eq. (1)) and Bc (ns ) (open and solid circles, correspondingly). The dashed lines are linear ;ts which extrapolate to the critical electron density for the metal–insulator transition.
the parallel magnetic ;eld, Bc , corresponding to the onset of full spin polarization [10]. Typical dependences of the normalized conductivity on temperature, (T )= 0 , are displayed in Fig. 1 at di erent electron densities above nc ; the value 0 , which has been used to normalize , was obtained by extrapolating the linear interval of the (T ) dependence to T = 0. As long as the deviation | = 0 − 1| is su>ciently small, the conductivity increases linearly with decreasing T in agreement with Eq. (1), until it saturates at the lowest temperatures. The ns dependences of the inverse slope 1=A∗ , extracted from the (T ) data, are shown in the inset to Fig. 1 together with Bc (ns ) by open and solid circles, correspondingly. Over a wide range of electron densities, the values 1=A∗ and B Bc turn out to be close to each other. The low-density data for 1=A∗ are approximated well by a linear dependence which extrapolates to the critical electron density nc in a way similar to Bc . In Fig. 2, we show the extracted values of g=g0 and m=mb as a function of the electron density (here g0 = 2 and mb are the g factor and the band mass in silicon). In the high ns region (relatively weak interactions), the enhancement of both g and m is relatively small, both values slightly increasing with decreasing electron density in agreement with earlier data [16]. Also,
226
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
Fig. 2. The e ective mass (circles) and g factor (squares), determined from the analysis of the parallel ;eld magnetoresistance and temperature-dependent conductivity, vs. electron density. The dashed lines are guides to the eye. The inset compares the so-obtained e ective mass (dotted line) with the one extracted from the analysis of SdH oscillations (circles).
the renormalization of the g factor is dominant compared to that of the e ective mass, which is consistent with theoretical studies [7]. In contrast, the renormalization at low ns (near the critical region), where rs 1, is much more striking. As the electron density is decreased, the renormalization of the e ective mass overshoots abruptly while that of the g factor remains relatively small, g ≈ g0 , without tending to increase. Hence, the current analysis indicates that it is the e ective mass, rather than the g factor, that is responsible for the drastically enhanced gm value near the metal–insulator transition. Since the procedure for extracting g and m described above relies on theoretically calculated functional form for the slope A∗ [15], we have performed independent measurements of the e ective mass based on the temperature analysis of the amplitude, A, of the weak-;eld (sinusoidal) Shubnikov–de Haas (SdH) oscillations. A typical temperature dependence of A for the normalized resistance, Rxx =R0 (where R0 is the average resistance), is displayed in Fig. 3. To determine the e ective mass, we use the method of Ref. [17] extending it to much lower electron densities and temperatures. We ;t the data for A(T ) using Eq. (2). The fact that the experimental dependence A(T ) follows the theoretical curve justi;es applicability of Eq. (2)
Fig. 3. Amplitude of the weak-;eld SdH oscillations vs. temperature at ns = 1:17 × 1011 cm−2 for oscillation numbers = hcns =eB⊥ = 10 (dots) and = 14 (squares). The value of T for the = 10 data is divided by the factor of 1.4. The solid line is a ;t using Eq. (2).
to this strongly interacting electron system. 22 kB T=˝c A(T ) = A0 : (2) sinh(22 kB T=˝c ) The so-determined e ective mass is shown in the inset to Fig. 2. In quantitative agreement with the results obtained by the alternative method described above (the dotted line), the e ective mass sharply increases with decreasing ns . The agreement between the results obtained by two independent methods adds con;dence in our results and conclusions. Our data are also consistent with the data for spin and cyclotron gaps obtained by magnetocapacitance spectroscopy [18]. A strong enhancement of m at low electron densities may originate from spin exchange e ects [7–9]. With the aim of probing a possible contribution from the spin e ects, we have introduced a parallel magnetic ;eld component to align the electrons’ spins. In Fig. 4, we show the behavior of the e ective mass with 2 the degree of spin polarization, p = (B⊥ + B2 )1=2 =Bc . As seen from the ;gure, within our accuracy, the e6ective mass m does not depend on p. Therefore, the m(ns ) dependence is robust, and the origin of the mass enhancement has no relation to the electrons’ spins and exchange e ects. Under the conditions of our experiments, the interaction parameter, rs , is larger by a factor of 2m=mb than the Wigner–Seitz radius and reaches approximately
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
227
polarization and, therefore, its increase is not related to spin exchange e ects, in contradiction with existing theories. The corresponding strong rise of the spin susceptibility may be a precursor of a spontaneous spin polarization; unlike in the Stoner scenario, the latter originates from the enhancement of the e ective mass rather than the increase of the g factor. Our results show that the dilute 2D electron system in silicon behaves well beyond the weakly interacting Fermi liquid. This work was supported by the National Science Foundation grants DMR-9988283 and DMR-0129652, the Sloan Foundation, the Russian Foundation for Basic Research, the Russian Ministry of Sciences, and the Programme “The State Support of Leading Scienti;c Schools”. Fig. 4. The e ective mass vs. the degree of spin polarization for the following electron densities in units of 1011 cm−2 : 1.32 (dots), 1.47 (squares), 2.07 (diamonds), and 2.67 (triangles). The dashed lines are guides to the eye.
50, which is above the theoretical estimate for the onset of Wigner crystallization. As has already been mentioned, two approaches to calculate the renormalization of m and g have been formulated. The ;rst one exploits the Fermi liquid model extending it to large rs . Its main outcome is that the renormalization of g is large compared to that of m [7], which is in obvious contradiction to our data. The other theoretical approach either employs analogy between a strongly interacting 2D electron system and He3 [8] or applies Gutzwiller’s variational method [19] to Si MOSFETs [9]. It predicts that near the crystallization point, the renormalization of m is dominant compared to that of g and that the e ective mass tends to diverge at the transition. Although the sharp increase of the mass is in agreement with our ;ndings, the expected dependence of m on the degree of spin polarization is not con;rmed by our data: the model of Ref. [8] predicts the increase of the e ective mass with increasing spin polarization, whereas the prediction of the other model [9] is the opposite. In summary, we have found that in very dilute twodimensional electron systems in silicon, the e ective mass sharply increases with decreasing electron density, while the g factor remains nearly constant and close to its value in bulk silicon. The enhanced e ective mass does not depend on the degree of the spin
References [1] E. Abrahams, S.V. Kravchenko, M.P. Sarachik, Rev. Mod. Phys. 73 (2001) 251; S.V. Kravchenko, M.P. Sarachik, Rep. Prog. Phys. 67 (2004) 1. [2] E. Wigner, Phys. Rev. 46 (1934) 1002. [3] E.C. Stoner, Rep. Prog. Phys. 11 (1947) 43. [4] L.D. Landau, Sov. Phys. JETP 3 (1957) 920. [5] B. Tanatar, D.M. Ceperley, Phys. Rev. B 39 (1989) 5005. [6] C. Attaccalite, S. Moroni, P. Gori-Giorgi, G.B. Bachelet, Phys. Rev. Lett. 88 (2002) 256601. [7] N. Iwamoto, Phys. Rev. B 43 (1991) 2174; Y. Kwon, D.M. Ceperley, R.M. Martin, Phys. Rev. B 50 (1994) 1684; G.-H. Chen, M.E. Raikh, Phys. Rev. B 60 (1999) 4826. [8] B. Spivak, Phys. Rev. B 64 (2001) 085317. [9] V.T. Dolgopolov, JETP Lett. 76 (2002) 377. [10] A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. Lett. 87 (2001) 086801; A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. B 66 (2002) 073303. [11] A.A. Shashkin, M. Rahimi, S. Anissimova, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. Lett. 91 (2003) 046403. [12] S.V. Kravchenko, A.A. Shashkin, D.A. Bloore, T.M. Klapwijk, Solid State Commun. 116 (2000) 495. [13] S.A. Vitkalov, M.P. Sarachik, T.M. Klapwijk, Phys. Rev. B 65 (2002) 201106(R). [14] V.M. Pudalov, M.E. Gershenson, H. Kojima, N. Butch, E.M. Dizhur, G. Brunthaler, A. Prinz, G. Bauer, Phys. Rev. Lett. 88 (2002) 196404. [15] G. Zala, B.N. Narozhny, I.L. Aleiner, Phys. Rev. B 64 (2001) 214204. [16] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. [17] J.L. Smith, P.J. Stiles, Phys. Rev. Lett. 29 (1972) 102. [18] V.S. Khrapai, A.A. Shashkin, V.T. Dolgopolov, Phys. Rev. Lett. 91 (2003) 126404. [19] W.F. Brinkman, T.M. Rice, Phys. Rev. B 2 (1970) 4302.
Physica E 22 (2004) 224 – 227 www.elsevier.com/locate/physe
Sharply increasing e ective mass near the 2D metal–insulator transition A.A. Shashkina;1 , Maryam Rahimia , S. Anissimovaa , S.V. Kravchenkoa;∗ , V.T. Dolgopolovb , T.M. Klapwijkc a Physics
Department, Northeastern University, Boston, MA 02115,USA of Solid State Physics, Chernogolovka, Moscow District 142432,Russia c Department of Applied Physics, Delft University of Technology, Delft 2628 CJ, The Netherlands b Institute
Abstract We report measurements of the e ective mass and Land0e g factor in strongly interacting two-dimensional systems in silicon. A sharp increase of the e ective mass has been observed near the metal–insulator transition, while the g factor remained nearly constant and close to its value in bulk silicon. Furthermore, the enhanced e ective mass has been found to be independent of the degree of spin polarization, which points to a spin-independent origin of the mass enhancement, in contradiction with existing theories. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.30. +h; 73.40. Qv Keywords: Metal–insulator transition; E ective mass; Large g-factor
Recently, there has been a lot of interest in the transport and magnetic properties of dilute two-dimensional (2D) electron systems due to the observation of an unexpected metal–insulator transition (MIT) in zero magnetic ;eld [1]. At su>ciently low electron densities, 2D electron systems become strongly correlated, because the kinetic energy is overpowered by energy of electron–electron interactions; this is an example of the strong-coupling many-body problem for which theoretical methods are still poorly developed. The strength of the interactions is usually characterized by the ratio ∗ Corresponding author. Tel.: +1-617-3732926; fax: +1-6173732943. E-mail address: [email protected] (S.V. Kravchenko). 1 Permanent address: Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia.
between the Coulomb energy and the Fermi energy, rs = Ec =EF , which, assuming that the e ective electron mass is equal to the band mass, in the systems with single-valley spectrum reduces to the Wigner– Seitz radius, 1=(ns )1=2 aB (here ns is the electron density and aB is the Bohr radius in semiconductor). There are several suggested candidates for the ground state of the system, for example, (i)Wigner crystal characterized by spatial and spin ordering [2], (ii)ferromagnetic Fermi liquid with spontaneous spin ordering [3], and (iii)paramagnetic Fermi liquid [4]. In the strongly interacting limit (rs 1), no analytical theory presently exists. According to numeric simulations [5], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. The re;ned numeric simulations [6] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ¡ rs ¡ 35, the ground state
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.254
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
of the system is a strongly correlated ferromagnetic Fermi liquid. A paramagnetic Fermi liquid is realized at yet higher electron densities when the interactions are relatively weak (rs ∼ 1). The e ective mass, m, and Land0e g factor within the Fermi liquid theory are renormalized due to exchange and correlation e ects, with renormalization of the g factor being dominant compared to that of the e ective mass [7]. Alternatively, near the onset of Wigner crystallization, strong increase of the e ective mass is expected [8,9]. In this paper, we report measurements of the e ective mass and g factor in a wide range of electron densities including the immediate vicinity of the MIT. We have investigated “clean regime” which occurs only in very low-disordered samples that remain metallic at electron densities down to ≈ 8 × 1010 cm−2 . It turns out that with decreasing electron density, the value of the e ective mass becomes strongly (by more than a factor of 3) enhanced, while the g factor remains nearly constant and close to its value in bulk silicon. Using tilted magnetic ;elds, we ;nd that the value of the e ective mass does not depend on the degree of spin polarization, which points to a spin-independent origin of the e ective mass enhancement. This is in clear contradiction with existing theories [7–9]. Measurements have been made in a rotator-equipped Oxford dilution refrigerator with a base temperature of ≈ 30 mK on low-disordered (100)-silicon MOSFET samples with peak electron mobilities close to 3 m2 =Vs at 0:1 K; for more on samples and experimental procedure, see Refs. [10,11]. Strong enhancement of the spin susceptibility near the MIT was recently observed in Si MOSFETs [10,12–14]. In principle, either g or m (or both) may be responsible for this e ect. To separate g and m, we have analyzed the data for temperature-dependent conductivity in zero magnetic ;eld using recent theory [15]. According to this theory, deep enough in the metallic state, is a linear function of temperature. (T ) = 1 − A∗ kB T; 0
(1)
where the slope, A∗ , is determined by the Fermi liquid constants, F0a and F1s . This allows us to determine the interaction-enhanced g factor and mass m separately using the data for the slope A∗ and the product gm determined independently from the measurements of
225
Fig. 1. The temperature dependence of the normalized conductivity at di erent electron densities (indicated in units of 1011 cm−2 ) above the critical electron density for the metal–insulator transition. The dashed lines are ;ts of the linear interval of the dependence. The inset shows comparison of 1=A∗ (ns ) (see Eq. (1)) and Bc (ns ) (open and solid circles, correspondingly). The dashed lines are linear ;ts which extrapolate to the critical electron density for the metal–insulator transition.
the parallel magnetic ;eld, Bc , corresponding to the onset of full spin polarization [10]. Typical dependences of the normalized conductivity on temperature, (T )= 0 , are displayed in Fig. 1 at di erent electron densities above nc ; the value 0 , which has been used to normalize , was obtained by extrapolating the linear interval of the (T ) dependence to T = 0. As long as the deviation | = 0 − 1| is su>ciently small, the conductivity increases linearly with decreasing T in agreement with Eq. (1), until it saturates at the lowest temperatures. The ns dependences of the inverse slope 1=A∗ , extracted from the (T ) data, are shown in the inset to Fig. 1 together with Bc (ns ) by open and solid circles, correspondingly. Over a wide range of electron densities, the values 1=A∗ and B Bc turn out to be close to each other. The low-density data for 1=A∗ are approximated well by a linear dependence which extrapolates to the critical electron density nc in a way similar to Bc . In Fig. 2, we show the extracted values of g=g0 and m=mb as a function of the electron density (here g0 = 2 and mb are the g factor and the band mass in silicon). In the high ns region (relatively weak interactions), the enhancement of both g and m is relatively small, both values slightly increasing with decreasing electron density in agreement with earlier data [16]. Also,
226
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
Fig. 2. The e ective mass (circles) and g factor (squares), determined from the analysis of the parallel ;eld magnetoresistance and temperature-dependent conductivity, vs. electron density. The dashed lines are guides to the eye. The inset compares the so-obtained e ective mass (dotted line) with the one extracted from the analysis of SdH oscillations (circles).
the renormalization of the g factor is dominant compared to that of the e ective mass, which is consistent with theoretical studies [7]. In contrast, the renormalization at low ns (near the critical region), where rs 1, is much more striking. As the electron density is decreased, the renormalization of the e ective mass overshoots abruptly while that of the g factor remains relatively small, g ≈ g0 , without tending to increase. Hence, the current analysis indicates that it is the e ective mass, rather than the g factor, that is responsible for the drastically enhanced gm value near the metal–insulator transition. Since the procedure for extracting g and m described above relies on theoretically calculated functional form for the slope A∗ [15], we have performed independent measurements of the e ective mass based on the temperature analysis of the amplitude, A, of the weak-;eld (sinusoidal) Shubnikov–de Haas (SdH) oscillations. A typical temperature dependence of A for the normalized resistance, Rxx =R0 (where R0 is the average resistance), is displayed in Fig. 3. To determine the e ective mass, we use the method of Ref. [17] extending it to much lower electron densities and temperatures. We ;t the data for A(T ) using Eq. (2). The fact that the experimental dependence A(T ) follows the theoretical curve justi;es applicability of Eq. (2)
Fig. 3. Amplitude of the weak-;eld SdH oscillations vs. temperature at ns = 1:17 × 1011 cm−2 for oscillation numbers = hcns =eB⊥ = 10 (dots) and = 14 (squares). The value of T for the = 10 data is divided by the factor of 1.4. The solid line is a ;t using Eq. (2).
to this strongly interacting electron system. 22 kB T=˝c A(T ) = A0 : (2) sinh(22 kB T=˝c ) The so-determined e ective mass is shown in the inset to Fig. 2. In quantitative agreement with the results obtained by the alternative method described above (the dotted line), the e ective mass sharply increases with decreasing ns . The agreement between the results obtained by two independent methods adds con;dence in our results and conclusions. Our data are also consistent with the data for spin and cyclotron gaps obtained by magnetocapacitance spectroscopy [18]. A strong enhancement of m at low electron densities may originate from spin exchange e ects [7–9]. With the aim of probing a possible contribution from the spin e ects, we have introduced a parallel magnetic ;eld component to align the electrons’ spins. In Fig. 4, we show the behavior of the e ective mass with 2 the degree of spin polarization, p = (B⊥ + B2 )1=2 =Bc . As seen from the ;gure, within our accuracy, the e6ective mass m does not depend on p. Therefore, the m(ns ) dependence is robust, and the origin of the mass enhancement has no relation to the electrons’ spins and exchange e ects. Under the conditions of our experiments, the interaction parameter, rs , is larger by a factor of 2m=mb than the Wigner–Seitz radius and reaches approximately
A.A. Shashkin et al. / Physica E 22 (2004) 224 – 227
227
polarization and, therefore, its increase is not related to spin exchange e ects, in contradiction with existing theories. The corresponding strong rise of the spin susceptibility may be a precursor of a spontaneous spin polarization; unlike in the Stoner scenario, the latter originates from the enhancement of the e ective mass rather than the increase of the g factor. Our results show that the dilute 2D electron system in silicon behaves well beyond the weakly interacting Fermi liquid. This work was supported by the National Science Foundation grants DMR-9988283 and DMR-0129652, the Sloan Foundation, the Russian Foundation for Basic Research, the Russian Ministry of Sciences, and the Programme “The State Support of Leading Scienti;c Schools”. Fig. 4. The e ective mass vs. the degree of spin polarization for the following electron densities in units of 1011 cm−2 : 1.32 (dots), 1.47 (squares), 2.07 (diamonds), and 2.67 (triangles). The dashed lines are guides to the eye.
50, which is above the theoretical estimate for the onset of Wigner crystallization. As has already been mentioned, two approaches to calculate the renormalization of m and g have been formulated. The ;rst one exploits the Fermi liquid model extending it to large rs . Its main outcome is that the renormalization of g is large compared to that of m [7], which is in obvious contradiction to our data. The other theoretical approach either employs analogy between a strongly interacting 2D electron system and He3 [8] or applies Gutzwiller’s variational method [19] to Si MOSFETs [9]. It predicts that near the crystallization point, the renormalization of m is dominant compared to that of g and that the e ective mass tends to diverge at the transition. Although the sharp increase of the mass is in agreement with our ;ndings, the expected dependence of m on the degree of spin polarization is not con;rmed by our data: the model of Ref. [8] predicts the increase of the e ective mass with increasing spin polarization, whereas the prediction of the other model [9] is the opposite. In summary, we have found that in very dilute twodimensional electron systems in silicon, the e ective mass sharply increases with decreasing electron density, while the g factor remains nearly constant and close to its value in bulk silicon. The enhanced e ective mass does not depend on the degree of the spin
References [1] E. Abrahams, S.V. Kravchenko, M.P. Sarachik, Rev. Mod. Phys. 73 (2001) 251; S.V. Kravchenko, M.P. Sarachik, Rep. Prog. Phys. 67 (2004) 1. [2] E. Wigner, Phys. Rev. 46 (1934) 1002. [3] E.C. Stoner, Rep. Prog. Phys. 11 (1947) 43. [4] L.D. Landau, Sov. Phys. JETP 3 (1957) 920. [5] B. Tanatar, D.M. Ceperley, Phys. Rev. B 39 (1989) 5005. [6] C. Attaccalite, S. Moroni, P. Gori-Giorgi, G.B. Bachelet, Phys. Rev. Lett. 88 (2002) 256601. [7] N. Iwamoto, Phys. Rev. B 43 (1991) 2174; Y. Kwon, D.M. Ceperley, R.M. Martin, Phys. Rev. B 50 (1994) 1684; G.-H. Chen, M.E. Raikh, Phys. Rev. B 60 (1999) 4826. [8] B. Spivak, Phys. Rev. B 64 (2001) 085317. [9] V.T. Dolgopolov, JETP Lett. 76 (2002) 377. [10] A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. Lett. 87 (2001) 086801; A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. B 66 (2002) 073303. [11] A.A. Shashkin, M. Rahimi, S. Anissimova, S.V. Kravchenko, V.T. Dolgopolov, T.M. Klapwijk, Phys. Rev. Lett. 91 (2003) 046403. [12] S.V. Kravchenko, A.A. Shashkin, D.A. Bloore, T.M. Klapwijk, Solid State Commun. 116 (2000) 495. [13] S.A. Vitkalov, M.P. Sarachik, T.M. Klapwijk, Phys. Rev. B 65 (2002) 201106(R). [14] V.M. Pudalov, M.E. Gershenson, H. Kojima, N. Butch, E.M. Dizhur, G. Brunthaler, A. Prinz, G. Bauer, Phys. Rev. Lett. 88 (2002) 196404. [15] G. Zala, B.N. Narozhny, I.L. Aleiner, Phys. Rev. B 64 (2001) 214204. [16] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. [17] J.L. Smith, P.J. Stiles, Phys. Rev. Lett. 29 (1972) 102. [18] V.S. Khrapai, A.A. Shashkin, V.T. Dolgopolov, Phys. Rev. Lett. 91 (2003) 126404. [19] W.F. Brinkman, T.M. Rice, Phys. Rev. B 2 (1970) 4302.