Electron tunneling measurements of the density of states (DOS) in ultra-thin
Be films reveal that a correlation gap mediates their insulating behavior. In
films with sheet resistance $R<5000\Omega$ the correlation singularity appears
as the usual perturbative $ln(V)$ zero bias anomaly (ZBA) in the DOS. As R is
increased further, however, the ZBA grows and begins to dominate the DOS
spectrum. This evolution continues until a non-perturbative $|V|$
Efros-Shklovskii Coulomb gap spectrum finally emerges in the highest R films.
Transport measurements of films which display this gap are well described by a
universal variable range hopping law $R(T)=(h/2e^2)exp(T_o/T)^{1/2}$.Comment: 4 figure

Adams, P. W.

Butko, V. Yu.

DiTusa, J. F.

English

arXiv

Electron tunneling measurements of the density of states (DOS) in ultrathin Be films reveal that a correlation gap mediates their insulating behavior. In films with sheet resistance Rln(V) Efros-Shklovskii Coulomb gap spectrum finally emerges in the highest R films. Transport measurements of films which display this gap are well described by a universal variable range hopping law R(T) = (h/2e2)exp(T0/T)1/2. © 2000 The American Physical Society

Butko, V. Y.U.

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 1-1-2000 Coulomb gap: How a metal film becomes an insulator V. Y.U. Butko Louisiana State University J. F. DiTusa Louisiana State University P. W. Adams Louisiana State University Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Butko, V., DiTusa, J., & Adams, P. (2000). Coulomb gap: How a metal film becomes an insulator. Physical Review Letters, 84 (7), 1543-1546. https://doi.org/10.1103/PhysRevLett.84.1543 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:cond-mat/0006025v1  [cond-mat.str-el]  1 Jun 2000Coulomb Gap: How a Metal Film Becomes an InsulatorV.Yu. Butko∗, J.F. DiTusa, and P.W. AdamsDepartment of Physics and AstronomyLouisiana State UniversityBaton Rouge, Louisiana, 70806(October 23, 2018)Electron tunneling measurements of the density of states (DOS) in ultra-thin Be films revealthat a correlation gap mediates their insulating behavior. In films with sheet resistance R < 5000Ωthe correlation singularity appears as the usual perturbative ln(V ) zero bias anomaly (ZBA) in theDOS. As R is increased further, however, the ZBA grows and begins to dominate the DOS spectrum.This evolution continues until a non-perturbative |V | Efros-Shklovskii Coulomb gap spectrum finallyemerges in the highest R films. Transport measurements of films which display this gap are welldescribed by a universal variable range hopping law R(T ) = (h/2e2)exp(To/T )1/2.PACS numbers: 71.30.+h, 72.15.Rn, 73.40.GkIt has been known for some time now that 2D is the lower critical dimension for disordered transport and that evena non-interacting 2D electron gas will be localized in the presence of arbitrarily small disorder in the thermodynamiclimit [1,2]. When these systems are probed at a finite length scale, by a magnetic field for instance, logarithmiccorrections to the Drude conductivity are seen but one does not expect a true metal-insulator transition in 2D oreven a metallic phase in the conventional sense [2]. What is somewhat less well understood are the ramificationsCoulomb interactions and their attendant correlations, particularly in 2D systems with moderate to strong disorder.This issue in particular has now come under considerable scrutiny with the recent discovery of an apparent metal-insulator transition in the dilute 2D electron gas of Si metal-oxide-semiconductor field effect transistors (MOSFET’s)[3,4]. This somewhat surprising discovery runs so counter to conventional wisdom that there has been speculationthat perhaps e− e interaction effects are stabilizing an anomalous 2D metallic phase [5]. Clearly, a systematic studyof the DOS spectrum and corresponding transport characteristics of an increasingly disordered 2D electron system isneeded to help illuminate the crucial interplay between disorder and correlations as the system is brought from theweakly to the strongly localized regime. In the present Letter we present such a study using ultra-thin Be films withlow temperature sheet resistances ranging from R = 500Ω to 2.6MΩ.The theory of interaction effects in disordered electronic systems has for the most part been developed in twoextreme limits. In the weak disorder/interaction 2D limit it is known that the primary effect of e − e interactions isto produce a logarithmic suppression of the density of states (DOS) at the Fermi energy [6], δN ∼ −ln(V ). This iscommonly known as the zero bias anomaly (ZBA) and has been well established in a number of different systems viatunneling measurements of the DOS [7–9]. This depletion of the DOS is perturbative and results in a weakly metallicln(T ) transport conductivity [2]. In the opposite limit, i.e. strongly insulating regime, Efros and Shklovskii [10,11]have shown that the Coulombic interactions can produce a non-perturbative gap in the DOS, which is commonlyknown as the Coulomb gap. The 3D Efros-Shklovskii Coulomb gap, which has a quadratic energy dependence, hasonly recently been observed [12,13]. Interestingly, the 2D Coulomb gap is expected to be linear in energy [11],N(eV ) =α(4πǫoκ)2|eV |e4, (1)where κ is the relative dielectric constant, ǫo is the permittivity of free space, and α is a constant of order unity. DespiteEq.(1) having been in the literature for more than 20 years now, there has been no direct spectroscopic verificationof its peculiar linear energy dependence due to the intrinsic technical difficulties associated with measuring the DOSin thin insulating films [14]. Nevertheless a direct measurement of this gap is important in that it is believed to playa crucial role in the electron transport [15] and thermoelectric [16] properties of highly disordered 2D metals andsemiconductors. For instance, it is well known that insulating films typically obey a modified variable range hoppinglaw of the formR(T ) = Ro exp (To/T )ν, (2)where R is the film sheet resistance, and Ro is a constant. In the case of a flat DOS near the Fermi energy, the filmsimply obeys Mott’s variable range hopping law with ν = 1/3 [17]. If there is a simple gap in the DOS then To isthe gap energy for fixed range hopping and ν = 1. Finally, if the DOS spectrum is given by Eq.(1) then one expectsν = 1/2, Ro to be of the order of the quantum resistance RQ = h/e2 [18] and1To = 2.8e2/(kB4πǫoκξ), (3)where kB is the Boltzman constant and ξ is the localization length [11]. Thus when the ν = 1/2 hopping form isobserved, the DOS spectrum is simply assumed to be that of Eq.(1). In this Letter we examine this assumptionby presenting the first systematic electron tunneling study of the DOS in uniformly disordered metal films whosetransport properties range from that of weakly metallic to strongly insulating. We show that with increasing filmresistance the DOS evolves from an essentially flat spectrum with a perturbative ln(V ) ZBA to that of a Coulomb gapgiven by Eq.(1). We also show that the emergence of the Coulomb gap coincides with the emergence of the ν = 1/2hopping behavior of Eqs.(2) and (3).The Be films used in the present study ranged in thickness from 1.5 - 2.0nm with corresponding sheet resistancesR = 500Ω−3MΩ at T = 50mK. They were deposited by thermally evaporating 99.5% pure beryllium powder onto firepolished glass substrates held at 84K. The evaporations were made in a 4x10−7 Torr vacuum at a rate ∼ 0.30nm/s.The film area was 1.5mm x 4.5mm. Scanning force micrographs of the films’ exposed oxide surface did not revealany salient morphological features down to the 0.7nm resolution of the instrument. In fact, the films seemed to be as”smooth” as the fire-polished glass on which they were deposited. In addition, a transmission electron microstructuralanalysis of 15nm thick Be films deposited on cleaved NaCl crystals at 84K revealed that the films were composed of anultra-fine base structure that was interspersed with 5− 15nm Be nanocrystallites. Electron diffraction measurementsshowed no diffraction from the metallic base structure suggesting that it was amorphous. Similarly the oxide (BeO)produced a broad, continuous diffraction ring indicating its grain size was < 1nm [19].The tunnel junctions were formed by exposing the films to atmosphere for 0.1 - 3 hours in order to form a nativeoxide, then a 20nm thick Ag counter-electrode was deposited directly on top of the film with the oxide serving asthe tunnel barrier. The junction area was 0.7mm x 0.7mm. This technique produced tunnel junction resistances oforder RTJ ∼ 1kΩ to 1000kΩ depending upon the exposure time and other factors. We were always careful to ensurethat RTJ >> Rfilm at low temperatures. Our lowest resistance Be films superconducted [19] which allowed us to testthe integrity of the junctions by measuring the dc I-V characteristics at temperatures well below the superconductingtransition temperature Tc. The sub-gap impedance of a ”good” junction was always greater than 108Ω [20].Shown in the inset of Fig.1 is the normal state conductance of the 2600Ω film as a function of ln(T ). This samplehad a superconducting transition temperature Tc = 0.33K which was suppressed by the application of a magneticfield oriented along the film surface. (Tc was a monotonically decreasing function of R.) This sample had the expectedln(T ) weakly insulating behavior with some rounding below 100mK. In stark contrast, the main body of Fig. 1 showsthe activated-like behavior of the 2.6MΩ film. The solid line is a least squares fit to the data. The linearity of thedata in Fig.1 shows unequivocally that the hopping exponent ν = 1/2. Furthermore the slope and intercept of the fitdetermine the parameters To ≃ 1.6K and Ro ≃ RQ/2 in Eq.(2). It appears that Ro is universal in that it is of orderthe quantum resistance.In principle we can also use Eq.(3) along with the measured value of To to calculate a localization length ξ. Howeverthe relative dielectric constant κ in Eq.(3) is unknown for a highly disordered metal film. Alternatively, if one insteadassumes a reasonable value of ξ ∼ 1nm and takes To ∼ 1K then Eq.(3) predicts κ ∼ 104. This value seems reasonablein that it lies between the metallic and insulating limits of κ ∼ ∞ and κ ∼ 10 respectively. This is not an issue in 3Dsemiconducting systems where the dielectric constant is well defined in the insulating phase. In any case, it is evidentthat Fig.1 is consistent with the existence of the Coulomb gap described by Eq.(1). Accordingly, if the Coulomb gapis indeed measurable in 2D, then it will be seen in a sample such as that of Fig.1.Electron tunneling provides one of the most direct experimental measures of the DOS spectrum of an electronicsystem. The usual tunnel junction geometry consists an electronically ”inert” metallic counter-electrode which isseparated from the film by a highly insulating barrier, usually a few angstroms of oxide. When a potential is placedacross the barrier electrons tunnel from the counter-electrode and into the film. At low temperatures the tunnelcurrent is proportional to both the counter-electrode’s and the film’s DOS [21]. For the technique to be useful inhighly resistive films, however, care must be taken to insure that the impedance of the tunnel junction is much higherthan that of the film. Otherwise, some of the measured voltage drop will actually occur in the film itself, since thetunnel current must be drained off through the film. Additionally, there is a legitimate concern that screening effectsassociated with the close proximity of the highly conducting counter-electrode might ”wash out” any Coulombicfeatures in the DOS. Fortunately, this has been shown not to be the case in recent measurements of the Coulomb gapin 3D Si:B [13].In order to demonstrate the resolution of our tunnel junctions and the 2D nature of the Be films, we have plottedin Fig. 2 the tunneling conductance of a R = 530Ω superconducting film, Tc = 0.55K, in a parallel magnetic fieldjust below its parallel critical field Hc|| = 1.2T . Note that we can resolve two peaks on either side of V = 0. Thesepeaks result from the Zeeman splitting of the usual BCS DOS peak and indicate that the superconductivity was2spin-paramagnetically limited [22]. The inset of Fig. 2 shows the normal state tunneling conductance of the samefilm in a supercritical parallel field. The normal state displays the expected ln(V ) DOS anomaly which is consistentwith the the ln(T ) transport behavior shown in the inset of Fig.1.In Fig.3 we show the evolution of the DOS with increasing disorder by plotting tunneling spectra of Be films withT = 50mK sheet resistances of R = 530Ω, 2600Ω, 16000Ω, and 2.6MΩ. The 530Ω and 2600Ω samples superconductedwith Tc = 0.55K and Tc = 0.33K respectively. For those samples a parallel magnetic field was applied in order tosuppress the superconducting state and a standard ac lock-in technique was used to measure tunneling conductanceG directly. In the higher resistance films, the conductance was obtained by numerically differentiating dc I-V curves.Note that in each data set there is a significant ZBA. In the 530Ω and 2600Ω films the ZBA was of the weak disorderform δN/N ∼ −ln(V ). However, as is clearly evident in Fig. 3, the ZBA was no longer perturbative in films withR > 104Ω. An exponential growth in the ZBA is shown in Fig. 4 where we have plotted G(0)/G(15mV ) as a functionof R. By making a linear fit to the data we find that G(0) ∼ exp(−R/Rc) where Rc = 6000Ω ≈ RQ/4.The most interesting attribute of the data in Fig.3 is that the conductance traces become more linear at the highestresistances studied. The energy dependence of the 16000Ω curve represents the DOS spectrum of a film somewherebetween the weak and strong disorder limits and is correspondingly neither ln(V ) nor |V | in form. In fact, thisspectrum is best described by a V 1/2 dependence. In contrast, the 2.6MΩ spectrum is quite linear and symmetric.For a consistency check, we also measured the dc I-V characteristic of the films over the same current and temperatureranges that were used in the tunneling measurements. The nonlinearity in these transport I-V’s were negligible incomparison to the tunneling I-V’s indicating that the traces in Fig.3 are representative of the tunneling behavior. Thesolid line through the 2.6MΩ data is a least squares fit to the form G(V ) = β|V | where β is an adjustable parameter.We believe that the linear behavior of this data represents the Efros-Shklovskii gap as given by Eq.(1). Unfortunately,we were unable to determine an absolute normalization of the tunneling conductance. The spectrum remained linearup to our maximum biases of ∼ 100mV . Consequently we cannot make a quantitative comparison between Eq.(1)and the tunneling data. Nevertheless, the observation of a linear spectrum in films that show the expected hoppingtransport behavior is compelling.In conclusion, we have made the first direct spectroscopic measurements of the Efros-Shklovskii Coulomb gap ina 2D system. We find the the gap emerges from an exponential growth in the ZBA as the film sheet resistance isincreased. The details of the activated transport behavior of films showing the gap are in excellent agreement withtheory and have a universal prefactor of the order of the quantum resistance. In principle, spectroscopic studies of theCoulomb gap in films deposited in situ would allow for an absolute calibration of the tunneling conductance since avariety of film thickness could be investigated using a single tunnel junction. Such measurements could then be usedin conjunction with transport measurements to extract the two primary microscopic parameters of the theory, κ andξ, via Eqs.(1) and (3).We gratefully acknowledge discussions with Boris Shklovskii, Vladimir Dobrosavljevic, Boris Altshuler. This workwas supported by the NSF under Grant No.’s DMR 9972151 and DMR 97-02690.∗ Permanent address: Ioffe Physical Technical Institute (PTI), Russian Academy of Sciences, Polytekhnicheskaya St., 26,194021, St. Petersburg, Russia.[1] E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).[2] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).[3] S.V. Kravchenko, W.E. Mason, G.E. Bowker, J.E. Furneaux, V.M. Pudalov, and M. D’Iorio, Phys. Rev. B 51, 7038 (1995).[4] D. Simonian, S.V. Kravachenko, M.P. Sarachik, and V.M. Pudalov, Phys. Rev. Lett. 79, 2304 (1997).[5] V. Dobrosavljevic, E. Abrahams, E. Miranda, and S. Chakravarty, Phys. Rev. Lett. 79, 455 (1997).[6] B.L. Altshuler, A.G. Aronov, M.E. Gershenson, and Yu.V. Sharvin, Sov. Sci. Rev. A. Phys. Vol. 9, 223 (1987).[7] Yoseph Imry and Zvi Ovadyahu, Phys. Rev. Lett. 49, 841 (1982).[8] M.E. Gershenson, V.N. Gubankov, M.I. Falei, Pis’ma Zh. Eksp. Teor. Fiz. 41, 435 (1985) [JETP Lett. 41, 535 (1985)].[9] A.W. White, R.C. Dynes, and J.P. Garno, Phys. Rev. B 31, 1174 (1985).[10] A.L. Efros and B.I. Shklovskii, J. Phys. C 8, L49 (1975).[11] B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors, (Springer, New York, 1984).[12] J.G. Massey and Mark Lee, Phys. Rev. Lett. 75, 4266 (1995).[13] J.G. Massey and Mark Lee, Phys. Rev. Lett. 77, 3399 (1996).[14] H.B. Chan, P.I. Glicofridis, R.C. Ashoori, and M.R. Melloch, Phys. Rev. Lett. 79, 2867 (1997).3[15] Shih-Ying Hsu and J.M. Valles, Jr., Phys. Rev. Lett. 74, 2331 (1995); Y. Shapir and Z. Ovadyahu, Phys. Rev. B 40, 12441(1989); Youzhu Zhang and M.P Sarachik, Phys. Rev. B 43, 7212 (1991).[16] J.J. Lin, J.L. Cohn, and C. Uher in Anderson Localization, ed. by T. Ando and H. Fukuyama, p. 186 (Springer-Verlag,New York, 1988).[17] N.F. Mott and G.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed. (Clarendon, Oxford, 1979).[18] R.Berkovits and B.I.Shklovskii, J. Phys. Condens. Mat. 11, 779 (1998).[19] P.W. Adams, P. Herron, and E.I. Meletis, Phys. Rev. B 58, 2952 (1998).[20] M. Tinkam, Introduction to Superconductivity (McGraw-Hill, New York, 1996).[21] E.L. Wolf, Principles of Electron Tunneling Spectroscopy (Oxford Univ. Press, Oxford, 1985).[22] P. Fulde, Adv. Phys. 22, 667 (1973).FIG. 1. Semi-log plot of the resistance of the 2.6MΩ Be film as a function of T−1/2. The solid line is a linear fit to the datafor which we get To = 1.6K and Ro ≈ h/(2e2), see Eq.(2). Inset: Conductance of the 2600Ω film as a function of ln(T ). Thesolid line is a guide to the eye.FIG. 2. Tunnel conductance in the superconducting state of the 530Ω film in a parallel magnetic field H|| = 1.1T . Thecritical parallel field was Hc|| = 1.2T . The tunneling spectrum demonstrates that we can resolve the Zeeman splitting of theusual BCS DOS. Inset: Normal state tunneling conductance of the same film as a function of ln(V ) at H|| = 2.5T .FIG. 3. Tunnel conductances normalized to G(15mV) for Be films with T = 50mK resistances of R = 530Ω, 2600Ω, 16000Ωand 2.6MΩ (top to bottom). The solid lines are a best fit to the form G(V ) = β|V |, where β is an adjustable parameter. The2.6MΩ film had a RTJ = 0.6MΩ junction. To insure that Rfilm << RTJ we took the 2.6MΩ data at 700mK, see Fig. 1. Theother curves were measured at 50mK.FIG. 4. Semi-log plot of the normalized zero bias tunneling conductance as a function of R. The solid line is linear fit to thedata.4This figure "CGaF1.png" is available in "png" format from:http://arxiv.org/ps/cond-mat/0006025v1This figure "CGaF2.png" is available in "png" format from:http://arxiv.org/ps/cond-mat/0006025v1This figure "CGaF3.png" is available in "png" format from:http://arxiv.org/ps/cond-mat/0006025v1This figure "CGaF4.png" is available in "png" format from:http://arxiv.org/ps/cond-mat/0006025v1

Coulomb gap: How a metal film becomes an insulator

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Coulomb Gap: How a Metal Film Becomes an Insulator

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