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STM Physical Backgrounds
Tunneling Effect
The idea of particles tunneling appeared almost simultaneously with quantum mechanics. In classical mechanics, to describe a system of material points at a certain moment of time, it is enough to set every point coordinates and momentum components. In quantum mechanics it is in principle impossible to determine Fig. 1. Rectangular potential barrier and particle wave simultaneously coordinates and momentum function Ψ components of even single point according to the He-isenberg uncertainty principle. To describe the The stationary Schrodinger equations can be written system completely, an associated complex function as follows is introduced in quantum mechanics (the wavefunction). The wavefunction Ψ , which 2 ⎧ && κ =Ψ+Ψ ,0 при z < ;0 is a function of time and all system particles ⎪ 1 ⎪ 2 position, is a solution of the wave Schrodinger ⎨ && κ 2 =Ψ−Ψ ,0 при ∈ [Lz ];;00 (2) equation. In order to use the system wavefunction, ⎪ && κ 2 =Ψ+Ψ ,0 при > Lz one should determine Ψ 2 rather than Ψ . Then, the ⎩⎪ 1 probability for finding particles in an elementary volume dxdydz is given by Ψ 2 dxdydz . 2mE (2 0 − EUm If particles impinge on a potential barrier of a where κ1 = , κ 2 = – wave h h limited width, the quantum mechanics predicts the −34 effect of particles penetration through the potential vectors, h 1005,1 ⋅⋅= сДж – Planck's constant. barrier even if particle total energy is less than the The solution to the wave equation at z<0 can be barrier height which is unknown in classical expressed as a sum of incident and reflected physics. waves =Ψ 1 −+ 1zikazik )exp()exp( , while solution Lets calculate the transparency of the rectangular at z>L – as a transmitted wave =Ψ zikb )exp( . barrier [1], [2]. Suppose that electrons of potential 1 A general solution inside the potential barrier energy 0
1.1 STM Physical Backgrounds
is just equal to the squared wavefunction module at References z>L because the incident wave amplitude is assumed to be 1 and wave vectors of both incident 1. Sivuhin D.V. A General course of physics. and transmitted waves coincide. Nauka, volume 5, chapter 1, 1988 (In Russian) 2. Goldin L.L., Novikova H.I. The introduction in 2 −1 quantum physics. Nauka, 1988 (In Russian). ⎛ 1 ⎛ k k ⎞ ⎞ * == ⎜ 2 LkchbbD )( ⎜ 2 −+ 1 ⎟ 2 Lksh )( ⎟ (3) ⎜ 2 ⎜ ⎟ 2 ⎟ 4 ⎝ k1 k2 ⎠ ⎝ ⎠ Tunneling Effect in Quasiclassical Approximation If 2 Lk >>1, then both 2 Lkch )( and 2 Lksh )( can be The quasiclassical qualitative condition imply that approximated to 2 Lk 2/)exp( and (3) will be written as de Broglie wavelength of the particle λ is less than characteristic length L determining the conditions of ⎧ 2L ⎫ the problem. This condition means that the particle DED 0 exp)( ⎨−= (2 0 − EUm ⎬ (4) wavelength should not change considerably within ⎩ h ⎭ the length of the wavelength order
−1 dD 2 <<1 (1) ⎡ 1 ⎛ k k ⎞ ⎤ dz ⎜ 2 1 ⎟ where D0 ⎢14 ⎜ −+= ⎟ ⎥ . ⎢ 4 ⎝ k1 k2 ⎠ ⎥ ⎣ ⎦ where = λ 2/ π , λ = π zpz )(/2)( – de Broglie D h wavelength of the particle expressed by way of the Thus, analytical calculation of the rectangular particle classical momentum p(z) [1]. barrier transmission coefficient is rather a simple task. However, in many quantum mechanical Condition (1) can be expressed in another form problems it is necessary to find the transmission taking into account that coefficient of the more complicated shape barrier. In this case, there is no common analytical solution dp d m dU mF for the D coefficient. Nevertheless, if the problem = (2 UWm =−=− (2) parameters satisfy the quasiclassical condition, the dz dz p dz p transmission coefficient can be calculated in a general form. (see chapter Tunneling Effect in where = − / dzdUF means classical force acting Quasiclassical Approximation). upon the particle in the external field.
Introducing this force, we get Summary Fm h <<1 (3) − In quantum mechanics tunneling effect is p3 particles penetration through the potential barrier even if particle total energy is less than From (3) it is clear that the quasiclassical the barrier height. approximation is not valid at too small momentum − To calculate the transparency of the potential of the particle. In particular, it is deliberately invalid barrier, one should solve Shrodinger equation at near positions in which the particle, according to continuity condition of wavefunction and its classical mechanics, should stop, then start moving first derivative. in the opposite direction. These points are the so − The transparency coefficient of the rectangular called "turning points". Their coordinates z1and z2 barrier decreases exponentially with the barrier are determined from the condition E=U(z). width, when wave vectors of both incident and It should be emphasized that condition (3) itself can transmitted waves coincide. be insufficient for the permissibility of the quasiclassical approach. One more condition should be met: the barrier height should not change much over the length L.
1.1 STM Physical Backgrounds
Let us consider the particles move in the field References shown in Fig. 1 which is characterized by the presence of the potential barrier with potential 1. Landau L.D., Lifshitz E. M. Quantum energy U(z) exceeding particle total energy E and mechanics. Nauka, 1989 (In Russian) meeting all the quasiclassics conditions. In this case 2. Mott N., Sneddon I. Wave mechanics and its points z1and z2 are the turning points. application. Nauka, 1966 (In Russian)
Metal Energy-Band Structure
To derive the formula of tunneling current in the metal-insulator-metal (MIM) system (John G. Simmons formula, see chapter "John G. Simmons Formula"), we must make some assumptions and remind of metal electronic theory fundamentals. Fig. 1. Potential barrier of arbitrary shape − Firstly, let us consider that the solid (metal) is a The approximation technique of the Schrodinger three-dimensional potential well with a plain equation solution when quasiclassical conditions bottom (the so called Sommerfeld model) and are met was first used by Wentzel, Kramers and electrons do not interact with each other. The Brillouin. This technique is known as WKB energy of the electron resting at the bottom of approximation or quasiclassical quantization such potential well is less than vacuum level – method. In this textbook we do not present the the energy of electron resting infinitely far from Schrodinger equations solution for the given case. the solid surface. The well depth U is However, it can be found in [1], [2] and the barrier determined by the averaged field of all positive transparency in this case is ions of the lattice and of all electrons. − Secondly, because all electrons are considered z ⎪⎧ 2 2 ⎪⎫ to be noninteracting, the solution to the ED exp)( ⎨−∞ ∫ − ))((2 dzEzUm ⎬ (4) Schrodinger equation for the system of electrons ⎪ h z ⎪ ⎩ 1 ⎭ is turned to the solution to the Schrodinger equation for single electron moving in the averaged field which allows to use formulas (3) Comparing expressions (3) in chapter in chapter Tunneling Effect or (4) Tunneling Effect for transmission coefficients of in chapter Tunneling Effect in Quasiclassical rectangular barrier (precise solution of Shrodinger Approximation. equation) and (4) for quasiclassical approximation, − Thirdly, in the potential box with the plain we can notice that there is no qualitative difference bottom, the dependence of energy allowed between them. In both cases the transparency values on wave vector allowed values looks like decreases exponentially with the barrier width. points on the parabolic dependence of the energy on the wave vector components Summary 2 2 == h 2/)(2/ mkmpE for the free electron in − If the problem parameters satisfy quasiclassical void. If electrons mass m is isotropic in space, conditions, then transmission coefficient can be then the following expression is valid 2 2 2 2 calculated in a general form using (4). x y z =++= 2/2/2/2/ mpmpmpmpE , − In case of the square barrier there is no where Px, Py, Pz,– axial electron momentum qualitative difference between transmission components. coefficients calculated using quantum mechanics and quasiclassical approximation. In From the statistical physics [1] it is known that in a both cases the transparency decreases system in thermodynamic equilibrium all quantum exponentially with the barrier width. states with the same energy level E are occupied with electrons equally. The mean number of electrons in one quantum state with energy E at
1.1 STM Physical Backgrounds
temperature T is given by the Fermi-Dirac distribution:
1 Ef )( = (1) ⎛ E − μ ⎞ + exp1 ⎜ ⎟ ⎝ BTk ⎠
–1 where kB=B (11600) eV/К – Boltzmann's constant, μ – parameter having the dimension of energy and called chemical potential.
The chemical potential μ is defined by the normalization condition
∞ Fig. 1. Diagram of MIM system in equilibrium. )()( = ndEEfEn (2) j1 and j2 – work function of the left and right metals, respectively ∫ e 0 where ne – number of conduction band electrons per volume unit (concentration), 3 1 ⎛ 2m ⎞ dEEn = π )2(2)( 3 3kd = ⎜ ⎟ dEE – ∫∫∫ 2 ⎜ ⎟ [], +dEEE 2π ⎝ h ⎠ number of elecron states per volume unit in the energy range from E to E + dE. n(E) function is called the energetic density of states.
In a metal at the temperature close to absolute zero, electrons occupy all quantum states with energies up to the level μ0 = μ )0( called the Fermi level. All quantum states above the Fermi level are not occupied with electrons. In a metal, the chemical Fig. 2. Model of MIM system then positive potential is applied to potential slightly depends on temperature, therefore the right metal it can be approximated by μ0 at T = 0: Summary 2 h 322 T )( 0 =≈ πμμ ne )3( (3) To derive the formula of tunneling current in the 2m MIM system (chapter "John G. Simmons Formula") we assume that: where m – electron mass. If ne is expressed in − 3214 − Sommerfeld model is correct for all solid C.G.S. units, then μ ⋅= 1036,0 eVn . 0 e bodies. − Each electron from the solid move in averaged If a system is in thermal equilibrium and consists of field of all positive ions of the lattice and of all some sub-systems, the Fermi levels of subsystems electrons. should be the same (Fig. 1). If voltage V is applied − Isotropic square-law of dispersion as for free between two subsystems (solids), the Fermi level of electron in void is correct (1). the solid connected to the positive pole decreases, − In a metal at → 0KT , electrons occupy all while for the other solid it increases. The difference quantum states with energies up to the Fermi in Fermi levels between solids is eV (Fig. 2). level μ (3). All quantum states above the 0 Fermi level are not occupied with electrons.
1.1 STM Physical Backgrounds
References
1. Landau L.D., Lifshitz E. M. Statistical physics. Nauka, 1976 (in Russian) 2. Kittel Ch. Introduction in solid-state physics. Nauka, 1978 (in Russian)
1.1 STM Physical Backgrounds
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Tunnel Current in MIM System
Consider two metal electrodes with an insulator of John G. Simmons Formula thickness L between them. If electrodes are under the same potential, the system is in thermodynamic Now, using the Sommerfeld model equilibrium (see chapter "Metal Energy-Band (see chapter "Metal Energy-Band Structure") and WKB Structure") and Fermi levels of electrodes coincide approximation (see chapter "Tunneling Effect in (Fig. 1). However, if electrodes are under different Quasiclassical Approximation") and assuming that T = potentials, current flow between them is available. 0, potential barrier is of arbitrary shape and the Fig. 2 shows the energy diagram of electrodes with mass of electrons is isotropic in space, we can applied bias energy eV. Potential barrier width for derive an expression for the tunneling current electrons occupying the Fermi level is denoted as δz flowing in a metal-insulator-metal (MIM) system. = z2 – z1. Consider that all the current flowing in the system is due to the tunneling effect.
Probability D(Ez) of the electron transmission through the po-tential barrier of height U(z) is given by expression (4) in chapter "Tunneling Effect in Quasiclassical Approximation". For the number of electrons N1 tunneling through the barrier from electrode 1 into electrode 2, we can write [1], [2]
∞ ∞ ∞ p N = z +− )())(1)(( dpdpdpEDeVEfEf = 1 ∫∫∫ 33 1 2 zyxz ∞− ∞− ∞− 4π h m
E∞ = )()( dEpnED Fig. 1. Diagram of MIM system in equilibrium. ∫ zzz j1 and j2 – work function of the left and right metals, respectively 0 (1)
where
1 ∞ ∞ pn )( = +− ))(1)(( dpdpeVEfEf (2) z 33 ∫∫ 1 2 yx 4π h ∞− ∞−
and Em – maximum energy of tunneling electrons.
Integration of expression (2) can be performed in polar coordinates. Because in the model under 222 2 consideration += ppp yxr , = rr 2/ mpE and total energy is = + EEE , changing variables Fig. 2. Model of MIM system with an arbitrary shape potential rz barrier. Positive potential is applied to the right metal = pp rx cosθ , = pp ry sinθ , we get
1.2 Tunnel Current in MIM System
2π ∞ 1 ED z exp)( { A z ϕμδ )( −+−∞ Ez z } (8) pn )( = +− ))(1)(( ddppeVEfEf θ = z 33 ∫∫ 1 2 rr 4π h 00 where ϕ – average barrier height relative to Fermi m ∞ = +− ))(1)(( dEeVEfEf z2 32 ∫ 1 2 r 1 2π h 0 level of the negative electrode; φ = φ )( dzz ; δ ∫ z z1 (3) 2m A = 2β 2 , β – dimensionless factor defined in Substituting (3) in (1), we obtain h the Appendix (A6). m E∞ ∞ N = (1)(()( ++−+ ))dEeVEEfEEfdEED 1 32 zz ∫∫ 1 rz 2 rz r At T = 0 K 2π h 0 0 (4) ⎧eV, при E []μ −∈ eV ;;0 ⎪ z The number of electrons N2 tunneling back from me ⎪ ξ Ez )( = 32 ⎨μ − Ez , при z []−∈ eVE ;μμ (9) electrode 2 into electrode 1 is calculated in the same 2π h ⎪ way. According to (4) from chapter "Tunneling ⎩⎪ ,0 при Ez > μ Effect in Quasiclassical Approximation", the potential barrier transparency in the given case will Introducing (8) and (9) into (7), we obtain be such as if positive voltage V is applied to μ −eV electrode 1 relative to electrode 2. In this case me ⎪⎧ J = eV exp A φμδ dEE +−+− 32 ⎨ ∫ []z zz E 2π ⎪ m ∞ ∞ h ⎩ 0 N = ()( +−++ ))(1)( dEEEfeVEEfdEED 2 32 zz ∫∫ 2 rz 1 rrz μ 2π h 0 0 ⎪⎫ μ exp)( AE φμδ −+−−+ dEE (10) (5) ∫ z [ z ] zz ⎬ μ −eV ⎭⎪ Net electrons flow N through the barrier is obviously N = N1 – N2. Let us denote Integrating (10), we get me ∞ ξ E )( = +− ))(1)(( dEeVEfEf α 1 z 32 1 2 r J = exp A φφδφ exp φδ +−+−− eVAeV 2π ∫ 2 { ( z ) ()[ z ]} h 0 δ z (11) me ∞ ξ E )( = −+ ))(1)(( dEEfeVEf 2 z 32 ∫ 2 1 r where = e 4/ βπα 22 . 2π h 0 h
me ∞ Thus, expression (11) approximates the tunneling ξ z eVE ),( ξξ 21 =−= [1 2 +− )()( ]dEeVEfEf r current in the MIM system for arbitrary barrier 2π 32 ∫ h 0 shape.
(6) Summary Then, the tunneling current density J is − The general expression (7) to calculate the E∞ tunneling current in the MIM system was = ξ ),()( dEeVEEDJ (7) ∫ zz z derived in this chapter. 0 − The analytic approximate solution (11) of According Fig. 2, U(z) can be written in the tunneling current in the MIM system was form μ += ϕ zzU )()( . Then, integrating (4) calculated. from chapter "Tunneling Effect in Quasiclassical Approximation" and using expression (A5) from Appendix, we get 1.2 Tunnel Current in MIM System
References where the correction factor is 1. Burshtein E., Lundquist S. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian) z2 1 2 β 1−= )( − dzfzf (A6) 2. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 2 ∫[] 8 f δ z 1793. z1 3. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 238.
John G. Simmons Formula in a Case of Small, • Appendix Intermediate and High Voltage (Field Emission Mode)
Let us integrate an arbitrary function zf )( from According chapter John G. Simmons Formula, the z1 to z2. approximate expression for the tunneling current in the MIM system can be written as [1]: z2 )( dzzf (A1) ∫ α z J = { exp( A z φφδφ exp)() []z φδ +−+−− eVAeV } 1 2 δ z
Defining f as (1)
z 1 2 2m = )( dzzff (A2) where = e 4/ βπα 22 , A = 2β , φ – average δ ∫ h 2 z z1 h
barrier height, δ z – barrier width, V– voltage where f – average value of a function f on the between electrodes. interval from z1 to z2, δ z = − zz 21 . Then equation (A1) can be rewritten as Small voltage
z2 z2 [])( − fzf At low voltages φ >> eV , expression (1) can be )( fdzzf 1+= dz (A3) ∫∫ f simplified [1] z1 z1 Considering a Taylor series expansion of the 3 φγ V integrand (A3) in and neglecting []− /))(( ffzf J = exp(−A φδ ) (2) δ z and higher order members, we get z z z 2 2 2 ⎪⎧ )( − fzf )( − fzf ⎪⎫ 2me [][] where γ = 22 . Since φ >> eV , we can )( fdzzf ⎨1+= − 2 ⎬dz 4βπ ∫∫ ⎪ 2 f 8 f ⎪ h z1 z1 ⎩ ⎭ consider that φ doesn't depend on V. Thus, in the (A4) case of small applied voltage, the tunneling current proportionate to V. Energy diagram of the MIM The second term in (A4) vanishes upon integration, system then φ >> eV is shown on Fig. 1. therefore (A4) can be expressed as
z2 )( = fdzzf δβ (A5) ∫ z z1
1.2 Tunnel Current in MIM System
φγ 3 J = exp(− z +σφδ VVA ))( (3) δ z
Ae)( 2 Ae2 where σ 2 −= 23 z 3296 zφδδφ
High voltage – Field emission mode
The case when eV > φ2 corresponds to energy diagram shown in Fig. 3 and to the following
δ z = Lφ /(φ −φ211 + eV) , =φφ 1 2/ .
Fig. 1. Potential barrier in the MIM system then V ~ 0.
φ1 and φ2 – work function of the left and right metals, respectively
In this case, as shown on Fig. 1, δ z = L and
+= φφφ 21 2/)(
Intermediate voltage
If eV < φ2 , then δ z = L and ( φφφ 21 −+= eV 2/) (Fig. 2).
Fig. 3. Potential barrier in the MIM system then eV > φ2 .
φ1 and φ2 – work function of the left and right metals, respectively.
Substituting δ z and φ into equation (1), we obtain
23 ⎧ Fe ⎪ ⎡ β 23 22 m ⎤ J = 2 2 ⎨exp⎢− φ1 ⎥ − 8 h 1βφπ ⎩⎪ ⎣ eF h ⎦ (4) Fig. 2. Potential barrier in the MIM system then eV < φ2 . ⎛ 2eV ⎞ ⎡ β 22 2eVm ⎤⎪⎫ ⎜1+ ⎟exp − φ 23 1+ φ and φ – work function of the left and right metals, ⎜ ⎟ ⎢ 1 ⎥⎬ 1 2 ⎝ φ1 ⎠ ⎣ eF h φ1 ⎦⎭⎪ respectively
In [2] it is shown, that for this case the tunneling where = / LVF – electric field strength. current-voltage relation is given by
1.2 Tunnel Current in MIM System
Summary At high applied voltage (eV φ1 +> μ) the Fermi level of electrode 2 is lower than the conduction − Depend upon magnitude of applied voltage, band bottom of electrode 1. Under such conditions, formula (1) can be simplified (2)–(5). electrons can not tunnel from electrode 2 into − It is possible to describe the experimental electrode 1 because of lack of empty states. An tunneling current dependences by approximated inverse situation is for electrons tunneling from expressions (2)–(5) in accordance with electrode 1 into empty states of electrode 2. This magnitude of applied voltage. process is similar to autoelectronic emission from a metal into vacuum. Thus, since eV φ1 +> μ , the second summand in (4) can be neglected and for the References current we get 1. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 3 Fe 2 ⎡ β 22 m φ 23 ⎤ 1793. J = exp⎢− 1 ⎥ (5) 2. John G. Simmons. J. Appl. Phys. - 1963. - V. 34 8 βπ 22 φ e F h 1 ⎣ h ⎦ 238. 3. Dobretzov L.N., Gomounova M.V. Emission where coefficient β = 23/24. This result agrees electronics. Nauka, 1966 (in Russian) qualitatively with an analytical expression for the field emission current density [3].
Thus, using formulas (2)–(5), we can compute the tunnel current at given system parameters and plot current-voltage characteristics. Fig. 4 shows theoretical tunneling current-applied voltage plot in case of carbon electrode 1 (φ1 = 4,7 эВ) and platinum electrode 2 (φ2 = 5,3 эВ) at δ z = 5 Å and contact area S = 10–17 m2.
Fig. 4. Current-voltage characteristic for carbon electrode 1 and platinum electrode 2 at δ z = 5 Å and contact area 10–17 m2. Parts of J(V) curve correspond to the following expressions: AB – (22), BC – (23), CD – (24), DE – (25).
1.2 Tunnel Current in MIM System
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"Observed" Physical Quantities in STM
Using expression (1) and VJ )( curve at constant Current-Voltage Characteristic tip-sample separation δ z , it is possible to compute Measurement of relation between tunneling current the density of electronic states: and probe-sample voltage is carried out in VJ )( dJ spectroscopy mode. The VJ )( spectroscopy is μξ −∞ eV )( (2) eVd )( based on the dependence of tunneling current on number of electron states N, forming a tunneling Thus, inspection of VJ )( and its derivative contact of conductors, in the energy range from the Fermi level μ to μ − eV (Fig. 1), which at T = 0 eVddJ )(/ curves allows to investigate energy gives (see (7) in chapter "John G. Simmons Formula") levels distribution with atomic resolution. It is possible to determine a conductivity type, in μ particular for semiconductors – to detect the valence =∞ ξ )( dEENJ (1) band, conductivity band and impurity band. [1]–[3]. ∫ zz μ−eV According to (2) and (3) from chapter "John G. Thus the tunneling current dependence J(V) at Simmons Formula in a Case of Small, Intermediate and High Voltage (Field Emission Mode)" tunneling constant tip-sample separation δ z represents an allocation of torn bonds as well as other electron conductivity = / dVdJG does not depend on states corresponding different energies, i.e. energy applied voltage V in case eV <<φ . band structure of either tip or surface. Function ξ E )( , which was introduced in (6) of chapter "John φγ z G = exp − A φδ (3) G. Simmons Formula", depends on electron state ( z ) δ z density of phase space plane which is normal to tunneling direction at given Ez . at eV < φ2 relation between G and V is parabolic
G ≈ φγ exp( A )(+− 31 σφ V 2 ) (4)
On Fig. 2, 3 experimental dependences J(V), G(V), which were measured for Pt and HOPG samples and Pt-Ro probe using STM Solver P47, is shown. Experimental data are in good agreement with the theoretical predictions (1)–(4).
Fig. 1. Model of MIM system with an arbitrary shape potential barrier. Positive potential is applied to the right metal
1.3 "Observed" Physical Quantities in STM
Summary
− Tunneling current-voltage characteristic represents number of electron states and their distribution in energy spectrum of electrodes which creates tunneling contact. − Differential conduction G is proportional an electron state density. For metals at low voltages G does not depend on applied voltage (3). At intermediate voltages the relation between G and applied voltage is parabolic (4). − Experimantal current-voltage and differential Fig. 1a. Experimental (points) and theoretical (solid line) characteristic are in good agreement with dependences J(V) for Pt theory.
References
1. G. Binnig., H. Rohrer. Scanning tunneling microscopy. Helv. Phys. Acta. - 1982, - V. 55 726. 2. A. Burshtein, S. Lundquist. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian). 3. E. Wolf. Electron tunneling spectroscopy principles. Kiev: "Naukova Dumka", 1990, 454 p. (in Russian).
Fig. 1b. Experimental dependence G(V) for Pt Current-Distance Characteristic
Measurement of relation between tunneling current and tip-sample distance is carried out in
I δ z )( spectroscopy mode. According to (11) from chapter "John G. Simmons Formula", in absence of a condensate, a typical current-height relation is an exponential current decay (Fig. 1) with a characteristic length of a few angstrom [1], [2].
Fig. 2a. Experimental (points) and theoretical (solid line) dependences J(V) for HOPG
Fig. 1. Theretical I δ z )( curve for Pt sample and Pt-Ro probe
Inspecting an experimental I δ z )( curve, it is possible to estimate the potential barrier height φ . Fig. 2b. Experimental dependence G(V) for HOPG If tip-sample bias V is small enough, then
1.3 "Observed" Physical Quantities in STM
according to chapter "John G. Simmons Formula in a Case of Small, Intermediate and High Voltage (Field Emission Mode)", tunneling current can be written as
φγ VS I = exp(− A z φδ ) (1) δ z
2me 2m where γ = 22 , A = 2β 2 , S – contact 4βπ h h area, m – free electron mass, e – elementary charge, h – Planck's constant.
Fig. 2. Experimental curve (semilogarithmic scale). I δ z )( Solid line – approximation I = (− δ 11/10exp47,0 ). From (1) φ can be expressed through some z analytical function of /ln dId δ . Finding the () z Substituting value of approximated straight line natural logarithm of (1) and differentiate the result −1 slope dId δ −= 09,0/)ln( nm in (3), we obtain, by δ one can obtain z z that φ ≈ 8,0 eV . Theoretical value of φ , in case, Id 1)ln( then both electrodes are produced from Pt, equals (2) −−= A φ φ = 3,5 eV . Thus, experimental value of φ is lower d z δδ z by a factor about 6 than theoretical one. Most probably, the main reason of this difference is If δ >> /1 A φ , then φ can be expressed from (2) z condensate presence on electrodes surface. Even for by following way fresh surfaces of pyrolitic graphite at maximum
2 2 /)ln( dId δ values, the corresponding values of φ ⎛ Id )ln(1 ⎞ ⎛ Id )ln( ⎞ z φ = ⎜ ⎟ ≈ 10−20 ⎜ ⎟ eV (3) are less than several tenths of eV. These values are 2 ⎜ ⎟ ⎜ ⎟ A ⎝ dδ z ⎠ ⎝ dδ z ⎠ deliberately less than those known from high vacuum and low temperature STM experiments for –1 where ()/ln dId δ z is experessed in m . the same samples and tips [3]. Values of φ derived from experiments in air are close to those obtained Emphasize, that in most cases the condition using STM configured for electrochemical measurements in situ when liquid polar medium δ z >> 1 A φ realizes practically always. For exists between sample and tip [3]. A condensate in instance, if φ = 4eV , then expression (3) will be the STM operating in air is evidently the analogue correct at δ z >> 5,0 Å . of such a medium. Thus, the condensate occurence on the sample surface results in the STM image
Experimental I δ z )( curve for Pt-flim which was quality deterioration and values of φ measured using Pt-Ro probe in STM (Solver P47) understatement. on air, is shown on Fig. 2.
Frequently I δ z )( spectroscopy is used for a determination of tip quality (sharpness).
Experimental I δ z )( curves, which are measured at investigation of HOPG surface using "good" and "poor" sharp tip of Pt-Ro probe, is shown on Fig. 3, 4.
1.3 "Observed" Physical Quantities in STM
Reference
1. John G. Simmons. J. Appl. Phys. – 1963. – V. 34 1793. 2. G. Binnig., H. Rohrer. Helv. Phys. Acta. – 1982, – V. 55 726. 3. S. Yu. Vasilev, A. V. Denisov. Journal of technical physics. – 2000, – vol. 70, num. 1 (in Russian). 4. NT-MDT. Solver P47 users guide.
Measurements of the Electronic States Density
In chapter Current-Voltage Characteristic we Fig. 3. I δ z )( spectroscopy for a "good-shape" STM tip considered the spectroscopy of electronic states. Meanwhile, at given eV it is possible to measure the electronic states distribution across the sample surface.
The electronic states density distribution measurement is performed in parallel with surface topography imaging in the = constJ mode. Instead
of constant tip bias V0 , the alternating voltage
= 0 + sinωtbVV is applied between sample and tip, where sinωtb – the alternating signal having
amplitude < ∞ 0 + NNJ ≈ (1) Fig. 4. I δ z )( spectroscopy for a "poor-shape" STM tip μ Tip quality criterion is following: 1) if tunneling where = ξ()dEEN current drops twice at tip-sample distance less 0 ∫ zz μ−eV0 than 3 Å, then tip quality is very good; 2) if this distance is about 10 Å, then atomic resolution can and N = ξ μ − sin)( ωtebeV . be still obtained on HOPG using such STM tip; 3) ~ 0 if the current drops at distance equals or more than 20 Å, then this STM tip should be replaced or sharpened [4]. Summary − It is possible to estimate electron work-function of investigated material using I δ z )( spectroscopy (3). − The difference between experimental and table values of work-function is connected with the condensate presence on electrodes surface. Fig. 1. Diagram of MIM system, when applied voltage is − In practice I δ z )( curve is used for a modulated as += sinωtbVV determination of STM tip quality (sharpness). 0 1.3 "Observed" Physical Quantities in STM Thus, the total current flowing through the The work function distribution measurement tunneling gap is equal to += JJJ ~0 , where J ~ – is performed in parallel with surface topography imaging in the = constJ mode. In this case, alternating component. Because J is held constant 0 the Z-axis piezo tube motion is determined not only during the scan and = consteb , the alternating by the feedback signal but also by application of an tunneling current amplitude is proportional to the alternating signal producing motion law electronic states density ξ(μ − eV0 ). Hence, Δ = ωtaZ )cos( . Accordingly, the tip-sample measuring the alternating current amplitude during separation is δ = δ + ωta )cos( , where parameter scan allows for the mapping of the electronic states zz 0 being a << δ z0 , δ z0 – tip-sample separation held density in standard units. Since < The frequency ω , as mentioned above, should be much more than the reciprocal feedback integrator time constant and be limited by maximum permissible scan frequency. Summary − Modulation of applied voltage V results in oscillations of tunneling current J. Amplitude of such oscillations depends on electron properties of electrodes, which create a tunneling contact. − Using this method, it is possible to measure distribution of electron state density on investigated sample surface. Fig. 1. Diagram of MIM system when tip-sample distance is modulated as δ δ zz 0 += ωta )cos( References If voltage applied between tip and sample is small V ≈ 0, then according to designations introduced, 1. G. Binnig, H. Rohrer. Helv. Phys. Acta. – 1982, expression (2) from chapter 1.2.2 can be – V. 55 726. transformed to the following 2. A. Burshtein, S. Lundquist. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian). φγ V 3. E. Wolf. Electron tunneling spectroscopy J = exp()− ()δφ z0 + ωtaA )cos( ≈ principles. Kiev: "Naukova Dumka", 1990, 454 p. δ z0 + ωta )cos( (in Russian). 0 []1−≈ ωφ taAJ )cos( (1) Work-Function Distribution Study where 0 = JJ δ z0 )( . In chapter Current-Distance Characteristic we considered that it is possible to estimate average Thus, total current flowing through the tunneling electron work-function of electrodes using current- gap in this case is equal to += JJJ ~0 , where J ~ – distance experimental curves. However, these alternating tunneling current. Because J is held measurements give work-function values only in 0 constant during the scan, the alternating tunneling small region where one electrode is located over current amplitude is proportional to the square root another. In STM there is another method which is of the tip and the sample work function half-sum. able to measure distribution of work-function along Assuming that tip work function is constant during all investigated sample surface. 1.3 "Observed" Physical Quantities in STM scanning, the J ~ amplitude will depend only on the studied surface work function. The frequency ω , as mentioned above, should be much more than the reciprocal feedback integrator time constant and be limited by maximum permissible scan frequency. Summary − Modulation of tip-sample distance results in oscillations of tunneling current J. − Using this method it is possible to measure the distribution of work-function along all investigated sample surface. References 1. G. Binnig., H. Rohrer. Scanning tunneling microscopy. Helv. Phys. Acta. - 1982, - V. 55 726. 2. A. Burshtein, S. Lundquist. Tunneling phenomena in solid bodies. Mir, 1973 (in Russian). 3. E. Wolf. Electron tunneling spectroscopy principles. Kiev: "Naukova Dumka", 1990, 454 p. (in Russian). 1.3 "Observed" Physical Quantities in STM