Nonlinear Quantum Capacitance

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(7) after eval- ρα(x) = Γα [G G f(E qVα) G G f(E)] , Z 2π − − o o xx uating the Green’s functions, or using the scattering 14 2 √ (5) wavefunctions ψ dσα/dE = ψ /hv, where v E is the velocity and h the Planck| constant.| Hence by∼ solv- r,a ing a quantum scattering problem one obtains σ . Let’s where Go are the equilibrium Green’s functions. Next, α we formally expand Gr and Ga in a power series of the consider a case where ψ is not very sensitive to E, thus dσα(E)/dE bα/(2√E) or σα(E) = bα√E where bα is internal potential U, and expand the Fermi function in ≈ series of the bias voltage Vα. Collecting terms accord- a constant. For this LPDOS, solving Eqs. (8, 9, 10) we ing to the powers of U and Vα, Eq. (5) reduces to the obtain following infinite series which can be exactly summed, 2 2 2 2 2 β β 4(b1 b2)b1b2(V1 V2) 2 C(V1 V2)= − −2 2− − (11) dσα 1 d σα 2 − p 2(b1 b2)(V1 V2) ρα(x)= (Vα U)+ (Vα U) + − − dE − 2 dE2 − ··· xx where σα(E + Vα U) σα(E) (6) ≡ − − 2 2 β =4πab1b2 +2b1b2(b1 + b2)√E. (12) where the quantity dσα/dE is defined as The inset of Fig. (1) shows this electrochemical C-V dσα dE r ∂f a curve as a function of V (V2 V1) for two sets of pa- GoΓα Go . (7) ≡ − dE ≡ − Z 2π ∂E xx rameters bα. The physical reason that C changes with V is because the plates have finite DOS. Indeed, by making The physical significance of the quantity dσα/dE can be DOS very large (b1,b2 ) the voltage dependence of identified as the linear spatial dependent local partial (11) disappears and C becomes→ ∞ purely geometrical. Fur- 2,5 density of states (LPDOS). Expression (7) has been thermore, it can be confirmed that formula (11) recovers 13,5 obtained before . For a conductor which is weakly the linear2 and second order nonlinear5 capacitance co- coupled to external leads, LPDOS gives the local DOS efficients when we take the (V2 V1) 0 limit. of this conductor. Hence the spectral function σα(E) Quantum scanning capacitance− microscopy→ . Our characterizes the local electronic structure of a nano-scale general results suggest a quantum capacitance mi- conductor. croscopy (QSCM). This idea naturally follows from the The nonlinear charge distribution gives the general results presented above: since the electrochemical capac- electrochemical capacitance versus voltage curve, itance varies with bias due to a finite DOS of the conductors involved, we should be able to find the DOS by mea- C = [σ1(E + V1 U1) σ1(E)]/(V1 V2) (8) − − − suring C. Essentially we wish to obtain spectral function σ2(E) or local density of states dσ2/dE as a function of where we have set electron charge q to be unity. To energy for an unknown conductor, from a known σ1(E) determine C we must obtain internal potentials U1 and 15 of our QSCM “tip” which has been calibrated . As the U2 at the two plates. For this purpose we introduce the 6 QSCM tip is scanned along the surface of a nano-scale geometrical capacitance C0 conductor, or along the surface of a planar dielectric layer with nano-conductors buried underneath, experimentally C0 = [σ1(E + V1 U1) σ1(E)]/(U1 U2) (9) − − − one can measure the C(V ) curves at each spatial position. and From Eq.(8), we obtain U1 as a function of potential difference V using the known σ1 and the measured C(V ), C0 = [σ2(E + V2 U2) σ2(E)]/(U1 U2) (10) by solving the equation σ1(1 U1) σ1(1) = V C(V ), − − − − − − where we have set V1 = 0, V2 = V , and E = 1 is set at For a parallel plate capacitor, C0 = A/(4πa) where A is the Fermi energy of the QSCM “tip” (C(V ) is measured 1 the area of the plates and a is their separation. In gen- at Fermi energy of the “tip” EF , e.g. Eq. (11)). Next, eral C0 can be calculated numerically. The two equations From Eq.(9) we obtain U2(V ). With U1 and U2 we finally (9,10) determine the internal potentials U1 and U2 when find σ2(E) from Eq. (10). In particular we can solve the the scattering spectral function σα and C0 are known. spectral function σ2 by representing it into a polynomial: Eqs. (8, 9, 10) gives, for the first time, the general elec- n trochemical C-V curve for quantum capacitors. It is uni- m versal in the sense that system specific parameters only σ2(X)= ymX , (13) =0 appear in the scattering spectral functions of the conduc- mX tors. From these general results, two useful applications m m where the coefficients ym (d σ2/dX )/(m!) are just follow: the linear (m = 1) LPDOS≡ and nonlinear (m > 1) C-V curve. Our general results allow us to predict LPDOS5. They are obtained by solving the following set capacitance-voltage curves. For this application we note of linear algebraic equations which come from Eq. (10), that there are well established methods for calculating

2 n 7 m S. Datta, Electronic Transport in Mesoscopic Systems, U2(Vj ) U1(Vj )=4πa ym [(1 + Vj U2(Vj )) 1] (Cambridge University Press, New York, 1995). − =0 − − 8 mX Baigeng Wang, Jian Wang and Hong Guo, Phys. Rev. Lett. (14) 82, 398 (1999). 9 A. P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B where j = 1, 2, ,n. Hence by making experimental ··· 50, 5528 (1994). measurements at n different voltages Vj , we obtain the 10 M. P. Anantram and S. Datta, Phys. Rev. B 51, 7632 functional form of σ2(E) from (13). Fig. (1) demon- (1995). strates the principle of QSCM. We use Eq. (11) as the 11 Recently Stafford investigated the I-V curves of a tunneling experimentally measured C-V curve (the inset) to simu- system within NEGF formalism, by including the internal late a measurement. Then using the QSCM “measured” potential using a linear capacitive charging model. C. A. σ2(E) from Eq. (13), we plot the local density of states Stafford, Phys. Rev. Lett. 77, 2770 (1996). 12 dσ2/dE versus energy E. The solid line is the exact Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. dσ2/dE = b2/(2√E) and the dots are the QSCM re- 70, 2601 (1993); T. K. Ng, Phys. Rev. Lett. 70, 3635 sult. We used 10 voltages in solving Eq. (14) and the (1993); C. Bruder and H. Schoeller, Phys. Rev. Lett. 72, outcome is quite good, while using 3 voltages it already 1076 (1994); S. König, H. Schoeller, and G. Schön, Phys. Rev. Lett. 76, 1715 (1996). represents a rough trend. 13 In summary, we have developed a general nonlinear T. Gramespacher and M. Büttiker, Phys. Rev. B 56, 13026 (1997). DC theory which is applied to investigate, for the first 14 time, the full nonlinear charge distribution in nano-scale M. Büttiker and T. Christen, in Quantum Transport in Semiconductor Submicron Structures conductors. We have derived an exact expression of the , edited by B. Kramer, (Kluwer Academic Publishers, Dordrecht, 1996, pp263). electrochemical capacitance versus external bias voltage 15 The calibration can be done in many ways. For example one curve for quantum capacitors. This result is generic in can prepare two identical conductors thus σ1(E)= σ2(E), the sense that all system specific information are included and solve for σ1. in the scattering local density of states. Hence the C-V 16 In the convenstional SCM a classical geometrical capaci- formula has a wide range of applicability. By invert- tance is assumed. See, for example, A.N. Erickson, D.M. ing this formula, we propose a novel QSCM. The QSCM Adderton, Y.E. Strausser and R.J. Tench, in Solid State extends the ability of the usual scanning capacitance 16 Technology, pp. 125, (June, 1997); and J. Electronic Mate- microscopy : QSCM includes the quantum corrections rials, 25, 301 (1996). to capacitance in mapping out the the spatial charge distribution; and it gives the local density of states as a function of electron energy. We believe such an idea can FIGURE CAPTION be readily implemented in a scanning apparatus using tiny tips as the calibrated conductor, thus allowing mea- Fig.(1) Operation of QSCM: comparison of “measured” surements of electronic properties for other conductors LPDOS to the exact one. Solid lines: exact; solid at the nano-scale. circles: fitted LPDOS using 10 voltages; solid tri- Acknowledgments. We gratefully acknowledge sup- angles: fitted LPDOS using 3 voltages. Upper set port by a RGC grant from the SAR Government of Hong of curves are for b2 = 0.002, lower set b2 = 0.001. Kong under grant number HKU 7112/97P, and a CRCG The QSCM tip has been fixed with b1 = 0.003. grant from the University of Hong Kong. H. G is sup- 1 The energy unit is the Fermi energy EF of the tip. ported by NSERC of Canada and FCAR of Québec. Inset: the electrochemical capacitance versus voltage curve for the two sets of b2: upper curve is for 1 b2 =0.002. The unit of V is EF /e.

1 T.P. Smith, B.B. Goldberg, P.J. Stiles, and M. Heiblum, Phys. Rev. B 32, 2696 (1985). 2 M. Büttiker, J. Phys. Condens. Matter 5, 9361 (1993). 3 Recently Blanter and Büttiker discussed the nonlinear charge effects and nonlinear capacitance as related to shot noise in resonant quantum wells. Ya. M. Blanter and M. Büttiker, cond-mat/9807254. 4 J. Wang, et.al., Phys. Rev. Lett. 80, 4277 (1998). 5 Z.S. Ma, J. Wang and H. Guo, to appear in Phys. Rev. B. (1999). 6 T. Christen and M. Büttiker, Phys. Rev. Lett. 77 143 (1996).