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RAPID COMMUNICATIONS PHYSICAL REVIEW B 71, 161304共R兲 共2005兲 Short-range correlations and spin-mode velocities in ultrathin one-dimensional conductors Michael M. Fogler Department of Physics, University of California San Diego, La Jolla, California 92093, USA 共Received 17 December 2004; published 12 April 2005兲 In ultrathin wires positioned on high- dielectric substrates or nearby metallic gates, electrons can form strongly correlated one-dimensional fluids already at rather high electron densities. The density-density correlation function, charge compressibility, spin susceptibility, and electron specific heat of such fluids are calculated analytically. The results are relevant for transport and thermodynamics of carbon nanotube field-effect transistors and semiconductor quantum wires. DOI: 10.1103/PhysRevB.71.161304 PACS number共s兲: 73.21.Hb, 71.10.Pm, 73.22.⫺f There is a long-standing theoretical prediction that onedimensional 共1D兲 electrons do not obey the conventional Fermi-liquid theory but instead form a Luttinger liquid 共LL兲1 whose fundamental degrees of freedom are bosonic modes that separately carry electrons’ charge and spin. Experimental verification of the LL theory has proved to be a challenge. There is, however, evidence for a LL-specific suppression of tunneling into carbon nanotubes 共CNs兲2 and a spin-charge separation in quasi-1D organics.3 Vexing questions remain: 共1兲 Is there a way to demonstrate a truly dramatic departure from the Fermi-liquid in real 1D systems 共aside from the special case of quantum Hall edge states4兲? And if so, 共2兲 What theory should one use to describe such a regime in the presence of long-range Coulomb interactions? Finally, 共3兲 What phenomena can one expect from this deeply nonFermi-liquid state? Below we propose that the desired strong-coupling regime can be realized if an ultrathin wire, e.g., a CN is 共i兲 placed on a high- dielectric substrate or 共ii兲 brought close to a metallic gate. In contrast, the traditional prescription for obtaining a strongly correlated regime is to lower the electron density n thereby increasing the Coulomb coupling constant rs = 1 / 2naB, where aB = ប2 / m*e2 is the effective Bohr radius. Unfortunately, this route quickly runs into the problem of localization by disorder, e.g., random charges on the substrate. The advantage of our proposal is that the Coulomb potential of these charges would be strongly screened, whereas interactions among electrons would be affected much less, as shown below. We begin with the dielectric substrate case. We call a wire of radius R ultrathin if R Ⰶ aB. Experimentally, enormous aB / R ⬃ 104 ratios are achievable in devices that use zigzag CNs 共Ref. 6兲 placed on the SrTiO3 substrate5 共R ⬃ 1 nm, ប2 / m*e2 ⬃ 50 nm, ⬃ 200兲. We show that in such an ultrathin wire Coulomb correlations are enhanced by a large parameter L = ln共aB / R兲 and a correlated regime—Coulomb Tonks gas 共CTG兲—appears in a window L−1 Ⰶ rs Ⰶ 1 of low rs. The CTG can be defined as the state where on all but exponentially large length scales, x ⬍ x* = aB exp共2 / 2rs兲, electrons behave as a gas of impenetrable but otherwise free fermions. At such x the LL theory, being an asymptotic longwavelength theory, has no predictive power, and the alternative method presented below is needed; x* ⬃ 1 m should be achievable in current CN devices.5 The CTG owes its name to a certain similarity it enjoys with the Tonks-Girardeau gas of 1D cold atoms.7 1098-0121/2005/71共16兲/161304共4兲/$23.00 We find that to the leading order in 1 / L Ⰶ 1, the shortrange density correlations in the CTG are identical to those of a one-component free Fermi gas. The spin correlations are the same as in the 1D antiferromagnet. We show that the CTG possesses a number of properties akin to the rs Ⰷ 1 1D Wigner crystal,8 including a negative compressibility, a high spin susceptibility and electron specific heat, and also anomalous finite-temperature transport9 and tunneling10 properties. Thus, although the 1D Fermi gas, the CTG, and the Wigner crystal are not different thermodynamical phases 共they all are LLs兲, significant quantitative differences in short-range properties of the electron system in these three regimes cause sharp crossover changes in the observables, similar to the boson case.7 Let us proceed to the derivation of these results. The crucial insight comes from the two-body problem. Consider two electrons with a relative momentum q ⬃ 1 / a = 1 / 2rsaB interacting via a model potential U共x兲 = e2 / 共兩x兩 + R兲. We face the following puzzle. If rs Ⰶ 1, we have the inequality e2 / បv Ⰶ 1, so that the kinetic energy greatly exceeds the characteristic Coulomb energy 共v = បq / m* is the relative velocity兲. Naively, one may expect that the Coulomb potential should be a small perturbation. But the reflection coefficient computed in the first Born approximation is equal to i ln共Rq兲 / aBq, which is large in the range of rs that corresponds to the CTG. To resolve this puzzle one has to separate the effects of the tails of the Coulomb potential 共large x兲 and of its sharp increase at the origin 共small x兲. Indeed, let us examine the Schrödinger equation for the wave function of the relative motion − ⬙共 兲 + 1 r − = 0, 4 兩兩 + ␣ 共1兲 where = qx is the dimensionless separation, r = 1 / qaB, and ␣ = Rq. We focus on the case r ⬃ rs Ⰶ 1, ␣ Ⰶ 1. The general solution of Eq. 共1兲 is given by 共兲 = AsWir,1/2共i兩兩 + i␣兲 + BsW−ir,1/2共− i兩兩 − i␣兲, 共2兲 where s = sgn共兲 and W,共z兲 is the Whittaker function.11 The constants As and Bs must be chosen to ensure the continuity of and ⬘ at = 0. The Whittaker function has the asymptotic behavior11 161304-1 ©2005 The American Physical Society RAPID COMMUNICATIONS PHYSICAL REVIEW B 71, 161304共R兲 共2005兲 MICHAEL M. FOGLER Wir,1/2共i兲 ⬃ exp关− i共/2兲 + ir ln共兲 − r兴, Ⰷ r. 共3兲 Due to the logarithm in Eq. 共3兲, which results from the slowly decaying 1 / 兩x兩 tail of the interaction potential, the scattered states are not exactly plane waves. Thus, if t共q兲 denotes the transmission amplitude, its phase depends on the distance x from the origin to the points where the scattered states are measured, t共q兲 ⬀ exp关−ir ln共xq兲兴. However, in a wide range of x, from the classical turning point x ⬃ r / q ⬃ a2 / aB to an exponentially large distance x ⬃ q−1 exp共1 / r兲 ⬃ a exp共1 / rs兲, this dependence of t共q兲 is very slow. The phase shift accumulated over this entire interval of x is small and can be ignored. This is consistent with Coulomb potential being a small perturbation at such x. Note that the point x ⬃ 1 / q, which corresponds to the the characteristic interelectron distance a in the many-body problem, is safely within the indicated range of x. Therefore, for our purposes we can define the transmission coefficient by t共q兲 ⬅ B+ / A− at A+ = 0. With this definition, one can show that t共q兲 is given by 2 共− i␣兲兴−1 , t共q兲 = i exp共− r兲关共d/d␣兲W−ir,1/2 共4兲 ⌿ = exp关W共x1,…,x M 兲兴 ⫻共− 1兲Q⌽共s1,…,s M 兲, iq , iq − c共q兲 c=− h共x兲 = a2具⌿兩共x兲共0兲兩⌿典 − 1. 2 ln共Rq兲 , aB qaB Ⰷ 1. 共5兲 Thus, there exists a window of momenta, 1 / aB Ⰶ q Ⰶ L / aB, where the Coulomb barrier is effectively impenetrable 共opaque兲,12 兩t共q兲兩 Ⰶ 1, due to the strong backscattering at an exponentially short approach distance, x ⬃ q−1 exp共−qaB兲. We conclude that while the tails of the Coulomb potential act as a small perturbation at momenta q ⬃ kF ⬅ n and distances x ⬃ a, which are the most relevant for the many-body problem, the strong short-range repulsion yields the effective hard-core constraint for the charge dynamics. Therefore, in the first approximation the Coulomb potential is equivalent to a very thin and high barrier, i.e., to a ␦ function of a large strength. Equation 共5兲 supports this identification because up to O共1 / L兲 terms, t共q兲 coincides with the transmission amplitude for the potential U共x兲 = 共ប2/m*兲c共kF兲␦共x兲. 共8兲 共9兲 as in the spin-polarized In the GIF 共W = 0兲 h共x兲 is the Fermi gas with the Fermi momentum kF = n, same18 h共x兲 = − sin2共kFx兲/共kFx兲2 , 共10兲 h̃共q兲 = − a + 共2kF − q兲q/共2kFn兲, 共11兲 where x, q−1 Ⰶ L, 共z兲 is the step function, and henceforth the tilde denotes the Fourier transform. The tails of the Coulomb potential cause a correction ␦h共x兲 to Eq. 共10兲. From the analysis of the two-body problem, we expect that at not too small x, ␦h共x兲 admits a diagrammatic expansion in rs. Since the term 共−1兲Q⌽ does not affect the dynamics, this expansion has identically the same form as for one-component fermions, so that the standard calculation yields, in the leading order ␦h̃ ⯝ − 共6兲 Note that in the opposite limit r Ⰶ 1 of a low-energy scattering the tails of the Coulomb potential cannot be ignored. The point of interest x ⬃ q−1 resides deeply inside a classically forbidden region of the Coulomb barrier where the wave function depends exponentially on x. This regime will not be important in what follows, but for future reference we quote the counterpart of Eq. 共5兲, 兩t共q兲兩 ⯝ 共 / L兲exp共− / qaB兲. Let us now apply the above ideas to the analysis of the many-body problem. Consider the limiting case R = + 0 first. We have c = ⬁ and t共q兲 = 0 for all q; hence, the ground-state wave function ⌿ has a node whenever coordinates x j of any particles coincide, 1 艋 j 艋 M. Using an argument similar to Lieb-Mattis theorem,13 one can show that ⌿ has no other nodes. Thus, it must have the form 共7兲 where Q1 through QM are the indices in the ordered list of the electron coordinates 0 ⬍ xQ1 ⬍ ¯ ⬍ xQM ⬍ L 共periodic boundary conditions are assumed兲, 共−1兲Q is the parity of the corresponding permutation, s j is the spin of Qjth electron, and ⌽ is the spin part of the wave function. The factor exp共W兲 incorporates the effect of weak 1 / x tails of the interaction. Apart from this, ⌿ coincides with the ground state of electrons with infinitely strong ␦-function repulsion,18 i.e., the gas of impenetrable but otherwise free fermions 共GIF兲. Below we focus on correlation functions for which W is not needed directly. Due to the strict impenetrability built into ⌿, particle exchanges are forbidden, and so neither the parity factor 共−1兲Q nor spin are dynamical degrees of freedom. 共So, at R = + 0 extra assumptions are needed to fix ⌽兲. All correlation functions are slaved to those of the density operator 共x兲, e.g., the two-point cluster function, which entails 共cf. Ref. 11兲 t共q兲 = 兿 sin L 共xQi − xQj兲 Qi⬎Qj ⬃ ␦h共2kF兲 + ⯝ rs q 2kF ln , 2 k Fn q q Ⰶ 2kF , 共12兲 rs兩z兩 2 ln 兩z兩, 2n 2kF − q → 0, 2kF 共13兲 8 rs kF4 , 2 n q4 z= q Ⰷ 2kF . 共14兲 As one can see, ␦h共x兲 is a small correction to h共x兲 关Eq. 共10兲兴 up to an exponentially large distance x* ⬃ a exp共2 / 2rs兲. We show below that these first-order results for ␦h共x兲 smoothly match at x ⬃ x* with the asymptotic long-distance behavior of h共x兲 computed from the LL theory, which is supposed to resum the perturbative series to all orders. We conclude that at rsa Ⰶ x Ⰶ x* the first-order perturbation theory applies and that the charge correlations in the CTG are indeed no different from those of free spinless fermions. 161304-2 RAPID COMMUNICATIONS PHYSICAL REVIEW B 71, 161304共R兲 共2005兲 SHORT-RANGE CORRELATIONS AND SPIN-MODE… This result suffices to demonstrate that the compressibility of the CTG is negative. We define the inverse compressibility by ⑂−1 = d2 / dn2, where 共n兲 is the energy density of the system. To make it finite the Hartree term 共interaction with a neutralizing uniform background兲 must be subtracted away. This is an important difference from the case of short-range interactions. To the leading order in rs, is equal to the kinetic energy of the spin-polarized Fermi gas plus the potential energy evaluated using Eq. 共11兲 for h̃共q兲. This simple calculation gives ⑂−1 = − 冋 冉 冊 册 2e2 2rs 2 L− + ln −␥ , 4rs 共15兲 which is indeed negative at 1 / L Ⰶ rs Ⰶ 1. Here ␥ is the Euler constant.11 With our definition of ⑂, its negative sign does not imply any instability of the CTG towards, e.g., phase separation. The phase separation would cost a large Hartree charging energy that would outweigh any gain due to negative ⑂−1 term.14 Let us briefly make a connection with the LL theory. The above result for ⑂ enters the LL machinery through the charge stiffness parameter K共q兲 = 冑 共/4兲បv Ũ共q兲 + ⑂ −1 , v= បn . m* 共16兲 Unlike the “classical” definition,1 in a Coulomb LL 共Refs. 8, 16, and 17兲 K depends not only on ⑂ but also on the interaction potential Ũ共q兲 ⬃ −2共e2 / 兲ln Rq. This is again because the total-energy cost of the charge buildup is the sum of the negative ⑂ term 共correlation energy15兲 and the large postitive Hartree term 共electrostatic energy兲. K shows up 关through vc共q兲 = vF / K共q兲兴 in the dispersion = vc共q兲q of charge mode. It also determines the low-q behavior of the density correlation function h̃共q兲 + a ⬃ − K共q兲q/共kFn兲, q → 0. 共17兲 We notice that Eqs. 共12兲 and 共17兲 match at q ⬃ 1 / x* and take it as evidence that Eq. 共17兲 applies at q ⬍ 1 / x*, while Eq. 共12兲 is valid at q ⬎ 1 / x*. Thus, the first-order perturbation theory is sufficient for computing h共x兲 at x ⬍ x*. The LL theory can also be used to calculate the crossover from Eq. 共13兲 to a different type of singularity in the immediate vicinity of q = 2kF. If desired, one can use this to study in detail how the asymptotic exp关−冑rs ln共x / x*兲兴 decay8,16,17 of the “2kF” 共2a-periodic兲 oscillations of h共x兲 is recovered at large x. So far, we have discussed the case of an infinitely thin wire. If R is finite, ⌿ has nodes only at the coincident positions of the same-spin particles. However, due to the opacity of the Coulomb barrier in the CTG, h共x兲 is perturbed very slightly. A much more important difference is that particle exchanges become allowed and the spin acquires some dynamics. Thus, it becomes meaningful and interesting to determine ⌽. Since particle exchanges are still highly suppressed, only those between nearest neighbors are relevant. In the CTG they are determined by the two-body transmission amplitude t共q兲. Recall that the orbital part of the scat- tered wave depends on the total spin S = S1 + S2 of the two colliding particles, = ei/2 ± e−i/2 + 关t共q兲 − 1兴共ei兩兩/2 ± ei兩兩/2兲, 共18兲 where the upper 共lower兲 sign is for the singlet 共triplet兲. To the order O共t兲, exactly the same asymptotic scattered wave would result from the exchange coupling Ueff = 共S1S2 − 1 / 4兲U共x兲 if at all k such that 兩t共k兲兩 Ⰶ 1 we have Ũ共k兲 = 2共ប2 / im*兲t共k兲k + const, as can be readily verified via Born approximation. Although the form of the short-range potential U共x兲 in the real space is not unique, this has no significance to the first order in the small parameter t. Therefore, the spin-spin interaction is captured by the Hamiltonian H = ប2 m* 冕 冉 冊 1 dk sks−k − k−k Im t共k兲k, 4 2 共19兲 where the integration is to be done up to the ultraviolet cutoff kmax ⱗ c共kF兲 and sk, k are the harmonics of the spin and charge densities at wave vector k. The physical idea expressed by Eq. 共19兲 is very similar to that behind the familiar antiferromagnetic coupling 共4t2 / U兲共SiSi+1 − nini+1 / 4兲 in a large-U 1D Hubbard model.19 Since the spins are slow degrees of freedom, we can average H over the fast orbital motion to obtain the usual 1D S = 1 / 2 Heisenberg model H = J 兺j S jS j+1, J= បv 22 冕 h̃共k兲Im t共k兲kdk. 共20兲 Therefore, the spin wave function ⌽ for the ground and excited states are given by the appropriate Bethe ansatze.20 Substituting Eq. 共5兲 into Eq. 共20兲 and keeping only terms O共1 / L兲, we obtain J= 2 ប 2 n 3a B , 3 m* L + ln rs 1 Ⰶ rs Ⰶ 1. L 共21兲 In the CTG, J Ⰶ EF ⬅ ប2kF2 / 8m*, as expected. The velocity of spin excitations in the 1D Heisenberg model is 共Ref. 20兲 v = 共 / 2兲共Ja / ប兲 and the spin susceptibility per unit length is = 共gB / 2兲2共2 / បv兲. This implies that of the CTG exceeds that of the Fermi gas 0 共where v = v / 2兲 by the large factor / 0 = v / 2v ⬇ rsL Ⰷ 1. The low-temperature electron specific heat of a CTG is determined by the velocities of the charge and the spin modes18 and is dominated by the latter, Ce = 共 / 3兲共kBT / បv兲 ⬃ 共2 / 2兲共 / e2兲共rs2 / L兲kBT. This is large compared to Ce in the Fermi gas because of the smallness of v. Obviously, v is relevant for many observables. To verify that our strategy for finding v is correct, we made sure that it reproduces the known exact results18,19 for the 1D Hubbard model and for electrons with the contact interaction 共6兲, v ⯝ 共3/3兲共បn2/m*c兲, n Ⰶ c. 共22兲 A gated wire. Properties similar to those of the CTG may also be exhibited by a modestly thin 1D wire, R ⱗ aB, if, instead of a high- dielectric, it is positioned a small distance D away from a metallic gate. In that case one can model the interaction potential by U共x兲 = e2 / 兩x兩 − e2 / 冑x2 + 4D2 at 兩x兩 Ⰷ R. 161304-3 RAPID COMMUNICATIONS PHYSICAL REVIEW B 71, 161304共R兲 共2005兲 MICHAEL M. FOGLER This model was studied recently by Häusler et al.21 who surmised that K → 1 / 2 and v ⬀ n2 / ln共D / R兲 at low n. In the regime a Ⰷ D2 / aB Ⰷ aB the validity of these statements can be examined in a controlled fashion. The interaction potential is short range, U共x兲 ⬀ x−3 at x ⬃ a, and opaque, so the system is in the GIF limit. After a straightforward calculation of t共q兲 one finds that c in Eq. 共5兲 has to be replaced by22 c = A1 再冑 冋 aB exp D2 D ⌫共1/8兲 ⌫共5/8兲 + 2aB ⌫共5/8兲 ⌫共9/8兲 册冎 , 共23兲 where coefficient A1 ⬃ 1 depends on the behavior of U共x兲 at x ⬃ R, and ⌫共z兲 is Euler’s gamma function.11 Since the interaction potential is now short range, subtraction of the Hartree term is no longer necessary. We redefine ⑂−1 + Ũ共q = 0兲 → ⑂−1 and obtain ⑂−1 = 2e2aBn. Equation 共16兲 then implies that K ⯝ 1 / 2; however, the spin velocity, which can be found by substituting Eq. 共23兲 into Eq. 共22兲, differs from the surmise of Ref. 21. Experimental manifestations. The predicted large difference of spin and charge mode velocities 共vc and v兲 can be verified by momentum-resolved photoemission3 or tunneling.23 Since CN is currently the best candidate for realizing the Coulomb Tonks gas regime, we have to mention here that the bands of a pristine CN have a twofold valley degeneracy. This can be accommodated into the model via effective spin-1 / 2 operators Ti. Each two-body term in Eq. D. M. Haldane, J. Phys. C 14, 2585 共1981兲. Bockrath et al., Nature 共London兲 397, 598 共1999兲; Z. Yao et al., ibid. 402, 273 共1999兲. 3 R. Claessen et al., Phys. Rev. Lett. 88, 096402 共2002兲. 4 A. M. Chang, Rev. Mod. Phys. 75, 1449 共2003兲. 5 A. Javey et al., Nat. Mater. 1, 241 共2002兲; B. M. Kim et al., Appl. Phys. Lett. 84, 1946 共2004兲. 6 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes 共Imperial College Press, London, 1998兲. 7 B. Paredes et al., Nature 共London兲 429, 277 共2004兲. 8 H. J. Schulz, Phys. Rev. Lett. 71, 1864 共1993兲. 9 K. A. Matveev, Phys. Rev. Lett. 92, 106801 共2004兲; Phys. Rev. B 70, 245319 共2004兲. 10 K. Penc, F. Mila, and H. Shiba, Phys. Rev. Lett. 75, 894 共1995兲; V. V. Cheianov and M. B. Zvonarev, ibid. 92, 176401 共2004兲; G. A. Fiete and L. Balents, ibid. 93, 226401 共2004兲. 11 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., edited by A. Jeffrey and D. Zwillinger 共Academic, San Diego, 2000兲. 12 For an early general discussion, see M. Andrews, Am. J. Phys. 1 F. 2 M. 共20兲 should then be replaced by J共SiSi+1 + 1 / 4兲共2TiTi+1 + 1 / 2兲. The resultant H remains integrable24 and one finds that now v = 共3 / 6兲共បn2 / m*c兲. Transport is another powerful probe of the CTG and Wigner crystal regimes. In the temperature window J Ⰶ kBT Ⰶ EF, with J ⬃ 1 K 共a crude estimate兲, ballistic conductance G through a CN should be unusual. Adopting the theory of Ref. 9 to the two-valley case, we find G = e2 / h, which is 1 / 4 of the noninteracting result, i.e., the “0.25 anomaly.” On the other hand, G measured through a CN with resistive 共tunneling兲 contacts may display an unusual power-law decrease10 with T. The negative compressibility would reduce the charging energy of finite wires, and so Eq. 共15兲 for ⑂ can be tested by careful Coulomb blockade experiments on the CN quantum dots. Finally, the strong electron correlations can also be observed25 in imaging microscopy: instead of the 4a-periodic Friedel oscillations of a Fermi gas, charge density near boundaries of defects would oscillate with period a, i.e., four times smaller, in the CTG. With further descrease of n, these oscillations would become strongly anharmonic as appropriate for a pinned Wigner crystal. Detailed quantitative predictions for all such measurements will be discussed elsewhere. This work is supported by the C. & W. Hellman Fund and the A. P. Sloan Foundation. I am indebted to L. S. Levitov for many valuable insights into the problem and to D. Arovas, G. Fiete, W. Häusler, and E. Pivovarov for discussions and comments. 44, 1064 共1976兲, and references therein. Lieb and D. Mattis, Phys. Rev. 125, 164 共1962兲. 14 For a review, see O. V. Dolgov, D. A. Kirznits, and E. G. Maksimov, Rev. Mod. Phys. 53, 81 共1981兲. 15 At the CTG-Fermi gas boundary, r ⬃ L−1, ⑂−1 matches the famils iar Fock correction to K. 16 R. Egger and A. O. Gogolin, Eur. Phys. J. B 3, 281 共1998兲. 17 D. W. Wang, A. J. Millis, and S. Das Sarma, Phys. Rev. B 64, 193307 共2001兲. 18 P. Schlottmann, Int. J. Mod. Phys. B 11, 355 共1997兲. 19 M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 共1990兲. 20 D. C. Mattis, The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension 共World Scientific, Singapore, 1993兲. 21 W. Häusler, L. Kecke, and A. H. MacDonald, Phys. Rev. B 65, 085104 共2002兲, and references therein. 22 Equation 共23兲 was independently derived in Ref. 9共b兲. 23 O. M. Auslaender et al., Nature 共London兲 295, 825 共2002兲. 24 Y.-Q. Li, M. Ma, D.-N. Shi, and F.-C. Zhang, Phys. Rev. B 60, 12 781 共1999兲. 25 A. Bachtold et al., Phys. Rev. Lett. 84, 6082 共2000兲. 13 E. 161304-4