Pauli spin susceptibility of a strongly correlated two-dimensional electron liquid A. A. Shashkin∗, S. Anissimova, M. R. Sakr†, and S. V. Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. V. T. Dolgopolov Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia 6 0 T. M. Klapwijk 0 Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2 Thermodynamic measurements reveal that the Pauli spin susceptibility of strongly correlated n two-dimensional electrons in silicon grows critically at low electron densities — behavior that is a J characteristic of the existenceof a phase transition. 3 PACSnumbers: 71.30.+h,73.40.Qv 2 ] l Presently, theoretical description of interacting elec- the critical density nc for the zero-field metal-insulator e tron systems is restricted to two limiting cases: (i) weak transition (MIT) in our samples. The nature of the low- - r electron-electroninteractions(smallratiooftheCoulomb density phase (n < n ) still remains unclear because s χ t s and Fermi energies rs = EC/EF ≪ 1, high electron even in the cleanest of currently available samples, it is . t densities) and (ii) very strong electron-electron inter- masked by the residual disorder in the electron system. a actions (r ≫ 1, very low electron densities). In the m s MeasurementsweremadeinanOxforddilutionrefrig- first case, conventional Fermi-liquid behavior [1] is es- eratoronlow-disordered(100)-siliconsamples with peak d- tablished, while in the second case, formation of the electron mobilities of 3 m2/Vs at 0.1 K and oxide thick- n Wigner crystal is expected [2] (for recent developments, ness 149 nm. These samples are remarkable by the ab- o see Ref. [3]). Numerous experiments performed in both senceofabandtailoflocalizedelectronsdowntoelectron c three- (3D) and two-dimensional (2D) electron systems densities n ≈ 1×1011 cm−2, as inferred from the coin- [ at intermediate interaction strengths (1 . r . 5) have s s cidence of the full spin polarization field obtained from 3 not demonstrated any significant change in properties parallel-field magnetotransport and from the analysis of v compared to the weakly-interacting regime (see, e.g., Shubnikov-de Haas oscillations (the former is influenced 0 Refs. [4, 5]). It was not until recently that qualitative bypossiblebandtailoflocalizedelectrons,while the lat- 0 deviations from the weakly-interacting Fermi liquid be- terisnot;formoredetails,seeRefs.[6,8,9]). Thisallows 1 havior (in particular, the drastic increase of the effec- 9 one to study properties of a clean 2D electron system tiveelectronmasswithdecreasingelectrondensity)have 0 without admixture of local moments [8, 9, 10]. The sec- been found in strongly correlated 2D electron systems 4 ond advantage of these samples is a very low contact re- 0 (rs & 10) [6]. However, these findings have been based sistance (in “conventional” silicon samples, high contact t/ solely on the studies of a kinetic parameter (conductiv- resistancebecomesthemainexperimentalobstacleinthe a ity), which, in general, is not a characteristic of a state lowdensity/lowtemperaturelimit). Tominimizecontact m of matter. resistance, thin gaps in the gate metalization have been - introduced, which allows for maintaining high electron d The2Delectronsysteminsiliconturnsouttobeavery density near the contacts regardless of its value in the n convenient object for studies of the strongly correlated o regime due to the large interaction strengths (r > 10 main part of the sample. s c canbe easilyreached)andhighhomogeneityofthe sam- For measurements of the magnetization, the paral- : v ples estimated (from the width of the magnetocapaci- lel magnetic field B was modulated with a small ac Xi tance minima in perpendicular magnetic fields) atabout field Bmod in the range 0.01 – 0.03 T at a frequency 4 × 109 cm−2 [7]. In this Letter, we report measure- f = 0.45 Hz, and the current between the gate and the r a ments of the thermodynamic magnetization and density two-dimensionalelectronsystemwasmeasuredwithhigh of states in such a low-disordered, strongly correlated precision (∼ 10−16 A) using a current-voltage converter 2D electron system in silicon. We concentrate on the and a lock-in amplifier. The imaginary (out-of-phase) metallic regime where conductivity σ ≫ e2/h. We have currentcomponentisequaltoi=(2πfCBmod/e)dµ/dB, foundthatinthis system, the spinsusceptibility ofband where C is the capacitance of the sample and µ is the electrons (Pauli spin susceptibility) becomes enhanced chemical potential. By applying the Maxwell relation by almost an order of magnitude at low electron den- dM/dn = −dµ/dB, one can obtain the magnetization s sities, growing critically near a certain critical density M from the measured i. A similar technique has been n ≈ 8×1010 cm−2: behavior that is characteristic in appliedbyPruset al.[11]toa2Delectronsysteminsili- χ the close vicinity of a phase transition. The density n conwith highlevelofdisorder,inwhich casethe physics χ is coincident within the experimental uncertainty with of local moments has been mainly studied. As discussed 2 3 0.4 -15i (10A) 012 insulator metal 112mM (10/cm)B 01..01550insulatornmetals2 (10114 cm-26) 0001...482mmd/dB ()B md/dB1 m B (a) B654 =TTT 7 Tm)/dB (/teslaB0.2 (b) cc/0012340ns (n12c011 c4136m. 5TT -T2)6 3 T md 0 1 -1 the onset of complete -0.4 2 T -B 7 T 4 T spin polarization 1.5 T 6 T 3 T 5 T 2 T 0 1 2 3 4 5 6 7 n (1011 cm-2) -0.2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 s n (1011 cm-2) n (1011 cm-2) s s FIG. 1: Imaginary current component in the magnetization experimentasafunctionoftheelectrondensityinamagnetic FIG. 2: (a) The experimental dµ/dB as a function of elec- field of 5 T and T = 0.4 K. Grey area depicts the insulating tron density in different magnetic fields and T =0.4 K. The phase. Magnetizationversusnsisdisplayedintheinset. Note curvesareverticallyshifted forclarity. Grey areadepictsthe that the maximum M is coincident within the experimental insulatingphase. Notethattheonsetoffullspinpolarization uncertainty with µBns. in our experiment always takes place in the metallic regime. (b)Scalingofthedµ/dBcurves,normalizedbymagneticfield magnitude, at high electron densities. The dashed line rep- below, the data analysis and interpretation is not quite resents the “master curve”. Spin susceptibility obtained by correct in Ref. [11]; in particular, Prus et al. do not dis- integrating the master curve (dashed line) and raw data at tinguish between the Pauli spin susceptibility of band B= 1.5, 3, and 6 T is displayed in the inset. electrons and the Curie spin susceptibility of local mo- ments. For measurements of the thermodynamic density of at which the electrons start to fill the upper spin sub- states, a similar circuit was used with a distinction that band — is given by the condition dµ/dB = 0 (M(ns) the gate voltage was modulated and thus the imaginary reaches a maximum), as indicated by the black arrow in current component was proportional to the capacitance. thefigure. Itisimportantthatovertherangeofmagnetic Thermodynamic density of states dn /dµ is related to fieldsusedintheexperiment(1.5–7tesla),themaximum s magnetocapacitance via 1/C = 1/C0 +1/Ae2(dns/dµ), M coincides within the experimental uncertainty with where C0 is the geometric capacitanceandAis the sam- µBns thus confirming that all the electrons are indeed ple area. spin-polarizedbelownp. Note howeverthat the absolute Atypicalexperimentaltraceofi(ns)inaparallelmag- value of dµ/dB at ns .nc is reduced in the experiment. netic field of 5 T is displayed in Fig. 1. The inset We attribute this to smearing of the minimum in i(ns) shows magnetization M(n ) in the metallic phase ob- caused by possible influence of the residual disorder in s tained by integrating dM/dn = −dµ/dB with the in- the electron system, which is crucial in and just above s tegration constant M(∞) = Bχ0, where χ0 is the Pauli the insulating phase, in contrast to the clean metallic spin susceptibility of non-interacting electrons. A nearly regime we focus on here. Another reason for the reduc- anti-symmetric jump of i(n ) about zero on the y-axis tion in dµ/dB is the electron-electron interactions (due s (markedbytheblackarrow)separatesthehigh-andlow- to, e.g., the enhanced effective mass). density regions in which the signal is positive and neg- InFig.2(a),weshowasetofcurvesfortheexperimen- ative (M(n ) is decreasing and increasing), respectively. tal dµ/dB versus electron density in different magnetic s Such a behavior is expected based on simple considera- fields. Experimental results in the range of magnetic tions. Atlowdensities,allelectronsarespin-polarizedin fields studied do not depend, within the experimental amagneticfield,soforthesimplecaseofnon-interacting noise,ontemperaturebelow0.6K(downto0.15Kwhich 2D electrons one expects dµ/dB = −µ (at n → 0, wasthelowesttemperatureachievedinthisexperiment). B s deepintheinsulatingregime,the capacitanceofthesys- Theonsetoffullspinpolarizationshiftstohigherelectron tem vanishes and, therefore, the measured current ap- densities with increasing magnetic field. Grey area de- proaches zero). At higher densities, when the electrons pictstheinsulatingphase,whichexpandssomewhatwith start to fill the upper spin subband, M(n ) starts to de- B (formoreonthis,seeRef.[12]). Notethattherangeof s crease, and dµ/dB is determined by the renormalized magnetic fields used in our experimentis restricted from Pauli spin susceptibility χ and is expected to decrease below by the condition that dµ/dB crosses zero in the with n due to reduction in the strength of electron- metallic regime. In Fig. 2(b), we show how these curves, s electron interactions. Finally, in the high-density limit, normalized by magnetic field, collapse in the partially- the spin susceptibility approaches its “non-interacting” polarizedregime onto a single “master curve”. The exis- value χ0, and dµ/dB should approach zero. The onset tence of such scaling verifies proportionality of the mag- of complete spin polarization — the electron density n netization to B, confirming that we deal with Pauli spin p 3 624 7 (a) )0.008 (b) 0.6 10 C (pF)623 B7 T= 9.9 T (0) - C(B)] / C(0000...000000246 99876. 9TTTT T cc/03456 mB (meV)Bc00..2400 0.5nnsc n(c110111 .c5m-22) 2.502468B (tesla)c 4 T C 5 T 2 0 T [ 0 4 T n 622 1 c 1 2 3 4 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 n (1011 cm-2) n (1011 cm-2) n (1011 cm-2) s s s FIG. 3: (a) Magnetocapacitance versus electron density for FIG. 4: Dependenceof the Pauli spin susceptibility on elec- different magnetic fields. (b) Deviation of the C(ns) depen- trondensityobtained byall threemethodsdescribed intext: dences for different magnetic fields from the B =0 reference integral of the master curve (dashed line), dµ/dB = 0 (cir- curve. The traces are vertically shifted for clarity. The onset cles), and magnetocapacitance (squares). The dotted line is of full spin polarization is indicated by arrows. aguidetotheeye. Alsoshownbyasolidlineisthetransport data of Ref. [7]. Inset: polarization field as a function of the electron density determined from the magnetization (circles) and magnetocapacitance (squares) data. Thesymbolsize for susceptibility of band electrons, and establishes a com- themagnetization datareflectstheexperimentaluncertainty, mon zero level for the experimental traces. Integration andtheerrorbarsforthemagnetocapacitancedataextendto of the master curve over n yields the spin susceptibil- s the middle of the jump in C. The data for Bc are described ity χ = M/B, as shown in the inset to Fig. 2(b). Also byalinearfitwhich extrapolatestoadensitynχ closetothe shown is the spin susceptibility obtained by integration critical density nc for the B=0 MIT. of raw curves at B = 1.5, 3, and 6 tesla, which, within the experimental error, yield the same dependence. Thismethodofmeasuringthespinsusceptibility,being larization at the beginning of the interaction-broadened the most direct, suffers, however,from possible influence jump, as indicated by arrows in the figure. In case the of the unknown diamagnetic contribution to the mea- residualdisorderdoescontributetothejumpbroadening, sureddµ/dB,whicharisesfromthefinitewidthofthe2D we extend error bars to the middle of the jump, which electronlayer[13]. Toverifythatthisinfluenceisnegligi- yieldsanupper boundaryforthe onsetoffullspinpolar- ble in our samples, we employ another two independent ization. methodstodetermineχ. Thesecondmethodisbasedon markingtheelectrondensitynp atwhichdµ/dB =0and In Fig. 4, we show the summary of the results for the which corresponds to the onset of complete spin polar- Pauli spin susceptibility as a function of n , obtained s ization, as mentioned above. The so-determined polar- using all three methods described above. The excel- ization density np(B) can be easily converted into χ(ns) lentagreementbetweenthe resultsobtainedbyallofthe via χ = µBnp/B. Note that in contrast to the value methods establishes that a possible influence of the dia- of dµ/dB, the polarization density np is practically not magnetic shift is negligible [14] and, therefore, the valid- affected by possible influence of the diamagnetic shift. ityofthedataincludingthoseatthelowestelectronden- The third method for measuring n and χ, insensitive sities is justified. There is also good agreement between p to the diamagnetic shift, relies on analyzing the magne- these results and the data obtained by the transport ex- tocapacitance, C. Experimental traces C(n ) are shown periments of Ref. [7]. This adds credibility to the trans- s in Fig. 3(a) for different magnetic fields. As the mag- port data and confirms that full spin polarizationoccurs netic field is increased, a step-like feature emerges on at the field B ; however,we note againthat evidence for c the C(n ) curves and shifts to higher electron densities. the phase transition can only be obtained from thermo- s This feature corresponds to the thermodynamic density dynamic measurements. The magnetization data extend ofstatesabruptlychangingwhenthe electrons’spinsbe- to lower densities than the transport data, and larger come completely polarized. To see the step-like feature values of χ are reached, exceeding the “non-interacting” moreclearly,inFig.3(b)wesubtracttheC(ns)curvesfor value χ0 by almost an order of magnitude. The Pauli different magnetic fields from the reference B =0 curve. spin susceptibility behaves critically close to the critical The fact that the jumps in C (as well as in dµ/dB) are density n for the B = 0 metal-insulator transition [15]: c washed out much stronger than it can be expected from χ∝n /(n −n ). This is in favorof the occurrence of a s s χ possibleinhomogeneitiesintheelectrondensitydistribu- spontaneousspinpolarization(eitherWignercrystal[16] tion(about4×109cm−2 [7])pointstotheimportanceof or ferromagnetic liquid) at low n , although in currently s electron-electron interactions. Since the effects of inter- available samples, the formation of the band tail of lo- actions are different in the fully- and partially-polarized calized electrons at n . n conceals the origin of the s c regimes, it is natural to mark the onset of full spin po- low-density phase. In other words, so far, one can only 4 reach an incipient transition to a new phase. dence characteristicof local moments, and the extracted The dependence B (n ), determined from the magne- spin susceptibility in their sample has a Curie tempera- c s tization and magnetocapacitance data, is represented in turedependence[9]. Thisisthecaseevenathighelectron the inset to Fig. 4. The two data sets coincide and are densities, where metallic behavior might be expected in- describedwellbyacommonlinearfitwhichextrapolates stead. Such effects are absent in our samples: the spin to a density n close to n . We emphasize that in the susceptibility (in the partially-polarized system) is inde- χ c low-field limit (B <1.5 tesla), the jump in dµ/dB shifts pendent of the magnetic field and temperature, confirm- to the insulating regime, which does not allow us to ap- ing that we deal with Pauli spin susceptibility of band proach closer vicinity of n : based on the data obtained electrons. χ in the regime of strong localization, one would not be In summary,the Paulispin susceptibility has been de- abletomakeconclusionsconcerningpropertiesofaclean termined by measurements of the thermodynamic mag- metallic electronsystem which we are interested in here. netization and density of states in a low-disordered, Clearly, the fact that the linear B (n ) dependence per- c s strongly correlated 2D electron system in silicon. It is sistsdowntothelowestelectrondensitiesachievedinthe foundto behavecriticallynearthe zero-fieldMIT,which experiment confirms that we always deal with the clean is characteristic of the existence of a phase transition. metallic regime. Finally, we would like to clarify the principal differ- We gratefully acknowledge discussions with S. ence between our results and those of Ref. [11]. In the Chakravarty, D. Heiman, N. E. Israeloff, R. S. sample usedby Pruset al., the criticaldensity n forthe Markiewicz,andM.P.Sarachik. Oneofus(SVK)would c B = 0 MIT was considerably higher than in our sam- liketothankB.I.Halperinforsuggestingthismethodto ples caused by high level of disorder, and the band tail measure spin susceptibility. We would also like to thank of localized electrons was present at all electron densi- A. Gaidarzhy and J. B. Miller for technical assistance ties [11]. As a result, the crucial region of low electron andC.M.MarcusandP.Mohantyforanopportunityto densities, in which the critical behavior of the Paulispin use their microfabrication facilities. This work was sup- susceptibility occurs, falls within the insulating regime ported by NSF grant DMR-0403026, PRF grant 41867- where the physics of local moments dominates [8, 9, 10]. AC10,the RFBR, RAS, and the Programme“The State Indeed, Prus et al. have found sub-linear M(B) depen- Support of Leading Scientific Schools”. [*] Permanent address: Institute of Solid State Physics, dalov, Phys.Rev.B 67, 205407 (2003). Chernogolovka, Moscow District 142432, Russia. [12] A. A. Shashkin, S. V. Kravchenko, and T. M. Klapwijk, [†] Presentaddress: DepartmentofPhysicsandAstronomy, Phys. Rev.Lett. 87, 266402 (2001). UCLA,Los Angeles, CA 90095, U.S.A. [13] To deal with this problem, the “subtraction of the [1] L. D.Landau, Sov.Phys. JETP 3, 920 (1957). diamagnetic contribution” was suggested in Ref. [11]. [2] E. Wigner, Phys. Rev. 46, 1002 (1934). The diamagnetic contribution ∆ was determined in the [3] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 partially-polarized regime as the difference between the (1989);G.Benenti,X.Waintal,andJ.-L.Pichard,Phys. direct magnetization data (“Mag”) and the data ob- Rev. Lett. 83, 1826 (1999); C. Attaccalite, S. Moroni, tained from Shubnikov-de Haas oscillations (“SdH”): P.Gori-Giorgi, andG.B.Bachelet,Phys.Rev.Lett. 88, ∆ =Mag−SdH. The experimental data were then cor- 256601 (2002). rected by ∆. We find this procedure meaningless as it [4] H. von L¨ohneysen, Advances in Solid State Physics 30, essentiallyresultsinreplacingthemagnetizationdataby 95 (1990). Shubnikov-de Haas data: Mag− ∆ = Mag −(Mag− [5] T. Okamoto, K. Hosoya, S. Kawaji, and A. Yagi, Phys. SdH)=SdH. In fact, the difference between magnetiza- Rev.Lett. 82, 3875 (1999); J. Zhu,H.L.Stormer, L.N. tion and Shubnikov-deHaas data in their experiment is Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. likelytobeduetothepresenceofabandtailoflocalized Lett.90, 056805 (2003). electrons at all electron densities in their sample. [6] S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. [14] Since the diamagnetic shift decreases with increasing ns 67, 1 (2004); A. A. Shashkin, Physics-Uspekhi 48, 129 and/or decreasing B, it may in principle be noticeable (2005). at the lowest ns and highest B used in the experiment. [7] A.A.Shashkin,S.V.Kravchenko,V.T.Dolgopolov,and Comparinglow-andhigh-fieldcurvesshowninFig.2(a), T. M. Klapwijk, Phys. Rev.Lett. 87, 086801 (2001). weestimatethatatns ∼1×1011 cm−2 andB=7T,the [8] V. T. Dolgopolov and A. Gold, Phys. Rev. Lett. 89, contribution of the diamagnetic shift is less than 0.2µB. 129701 (2002). [15] The critical density for the MIT was determined from [9] A.GoldandV.T.Dolgopolov,J.Phys.:Condens.Matter transport measurements (see Refs. [6, 12]). 14, 7091 (2002). 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