F7360 Characterization of thin films and surfaces Lenka Zajíčková Faculty of Science & CEITEC, Masaryk University, Brno lenkaz@physics.muni.cz 2. Chapter - Introduction to Surface Processes spring semester 2016 Central European Institute of Technology BRNO I CZECH REPUBLIC "2* 3. o 2. Introduction to Surface Processes • 2.1 Thermodynamics of Clean Surfaces • 2.2 Electronic Structure of Clean Surfaces • 2.3 Work Function • 2.4 Thermoemission « 2.5 Gas Adsorption to Surfaces • 2.6 Surface Relaxation & Reconstruction o 2.7 Methods for Preparation of Clean Surfaces F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajickova 3/50 We need to introduce the concept of surface (free) energy, surface tension and surface stress. The essential features of bulk thermodynamics (e.g. Callen, 1985) ► in equilibrium, a one-component system is characterized completely by internal energy, U ► U is a unique function of extensive parameters, entropy S, volume V and particle number of the system N and extensive property can be written as U(XS,XV,XN) = XU(S, V, N) U = U{S, V, N) dil dil dil dU= \ — \ViNdS+ | — |s,A/dV+ | — |s,ydA/, OS ON It defines intensive parameters, the temperature T, pressure p and chemical potential \± dil = TdS- pdV + /id/V. that are function of independent extensive parameters S, V and N. Differentiating the Euler equation U= TS-pV + fiN. we can derive a relation among the intensive variables, the Gibbs-Duhem equation SdT- Vdp+ A/d/i = 0 F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajickovä 4/50 urtace I lension 7 Let's create a surface of area A from the infinite solid by a cleavage process. The total energy of the system must increase by amount proportional to A. The constant of proportionality, 7, is called surface tension ([7] = J/m2) U=TS-pV + fiN + 7 A In equilibrium, at any finite T and p, the semi-infinite solid coexists with its vapor. Gibbs ascribed definite amounts of the extensive variables to a given area of surface. solid CO c CD vapor distance There is nothing unique about the particular choice of the boundary positions V=Vi + V2+Vs, S = Si + S2 + Ss, N = A/i + A/2 + Na. Once the surface volume is chosen, the other surface quantities are defined as excesses. Changes of surface quantities are completely determined by changes in the bulk quantities ASS -ASi - AS2, AVS -AVi -AV2, ANS —AA/-| - AAfe. F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajíčková 5/50 urtace 1 lension 7 aru urtace Energy We can take Gibbs G (used for constant p, T, N processes) or Helmholtz F (used for constant V, T, N processes) free energy dFb = -pdV - SdT + E//z/dA// dGb = - Vdp - SdT + E^/dA/,- No operational need to distinguish between F and G for the purpose of measurements (see R. J. Good, J. Colloid Interface Sci. 59, 1977, 398). At constant V and T According to previous definition the surface tension 7 is the reversible work done in creating unit area of new surface. _ / dFtotal \ 7 V dA )VJ How is it related to the surface free energy fs? dFtotal = fsdA + Ez/i/dA/,- jdA = fsdA + E//z/dA// In one component system, e.g. metal - vapor, we can choose the dividing surface such that dNj = 0 7 and fa are the same F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces Surface Stress a,y and Strain e,y Tensors Lenka Zajíčková 6/50 Consider the effect of small variations in the area of the system, e.g., by stretching. The energy change can be described by linear elasticity theory (Landau & Lifshitz, 1970). Accordingly, J(( dU. JO dU. Jl7 dU. J.l ^.dU. dU = —\ViNiAdS+ — \s,N,AdV+ —\SiViAdN + }^\ — \s,v,Ndejj dS dv I, J de ij where de,y£,y = dA/A is the surface strain tensor. dU= TdS- PdV + vdN + Aj2aijdeij i, j where cr,y is the surface stress in units J/m2 = N/m. Differentiation of Euler eq. U = TS — pV + /j,N + jA and its combination with above equations gives Gibbs-Duhem equation for the whole system including surface A ^2 aij deij = SdT - Vdp + Nd/jb + Ad-y + -ydA ij Using Gibbs-Duhem equations for both the bulk phases separately it can be reduced to quantities describing only the surface =^ Gibbs adsorption equation Ad-y + SsdT - Vsdp + Afed/z + Aj2hsij ~ °ijdeij) = 0 ij F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces Lenka Zajíčková 7/50 Gibbs adsorption equation At first glance it seems that there 5 independent variables, 7, p, p, T and e in Adj + Ss T - Vsdp + Nsdp + A ^(jSy ~ a—' w c /4d7 + SsdT + A^(j6jj - *ij)deij = 0 ' /'J p "vapor distance F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajíčková 8/50 It follows from Gibbs adsorption equation that e surface tension and stress are identical only if d^/de = 0 only if the system is free to rearrange in response to a perturbation, i. e. in a liquid. If d^f/de < 0 atomic dislocations and elastic buckling of the surface can be expected. Estimation of 7 - energy cost/unit area to cleave a crystal, i. e. break bonds ... from bulk cohesive energy Ecoh « 3eV, fractional number of bonds broken Zs/Z « 0.25, areal density of surface atoms A/s « 1015 cm-2 CleaveX^/ Energy 2yA Area A 7 = Ecoh Ns 1.2 J/nť material 7 [J/m2] mica gold PTFE 4.5 « 1 0.019 The surface tension can be regarded as an excess free energy/unit area. High energy surfaces tend to reduce energy by adsorption of contaminants from environment. F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajíčková 9/50 Annisotropy of 7 an The tendency to minimize surface energy is a defining factor also in the morphology of surfaces and interfaces: ► ► rum 1.060 1-050 ■ 1.040-1.030 1.020 1.010 1.000 Spherical equilibrium shape in an isotropic liquid (in the absence of gravity), In crystalline solids, the surface tension depends on the crystallographic orientation minimization of / 7(/?/c/)d/A(/?/c/) even if it implies a larger surface area. [01] [ln] . Iff \ \ 200°C 250 °C 275 °C 300 °C V f 0 10 1 {100} 20 30 t 40 50 60 I {111} 1 1 70 80 90 f t {110} anisotropy of 7 relative to {111} for lead Heyraud, Metois, Surf. Sci. 128, 334 (1983) I step width ... n.a, step free energy ... f3 tension in direction [01].. .7(0) for large n... 6 ~ tan 6 = a/{n.a) = 1 /n tension in direction >y(0) = 7(0) + ^=7<0) + £|*| na a F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajíčková 10/50 Equilibrium shapes can be calculated but it is easier to use a graphics method, the Wulff construction. The surface tension is plotted in polar coordinates vs. the angle. The minimization: construction the surface from the inner envelope of planes perpendicular to the radius vector. Faceting is energetically favored for crystals, shown example is lead at 473 K Heyraud, Metois, Surf. Sci. 128, 334 (1983). \ It is important to note that the formation of the equilibrium shape requires sufficient mobility (or fast kinetics), not just thermodynamics. F7360 Characterization of thin films and surfaces: 2.1 Thermodynamics of Clean Surfaces ka Zajíčková 11/50 I ougnenmg irans At finite T, the discussion must be supplemented to include entropy effects. a) c) ■'Y//YYY V / / fig. a At very low T any given facet is microscopically flat with only a few thermally excited surface vacancies. fig. b At higher T more and more energetic fluctuations in the local height of the surface can occur leading to delocalized interface with long wavelength variations in height (step free energy (3 i). At certain roughening temperature, Tr, (3 < 0, i. e. the facets disappear and only a smoothly rounded macroscopic morphology remains (phase transition for T = Tr). F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces ka Zajíčková 12/50 acnes Hi 71* Knowing the positions of all the atoms in the semi-infinite crystal (described by the set of R) and ignoring ion motion (Born-Oppenheimer approx.), the Hamiltonian for N electrons is N fr2X72 /=1 2/7? N 7p2 1 N P2 N A N EE^- + iE" = E"/ + iE v,. The wave function describing the set of N electrons has the general form where Sz/ is electron spin in a chosen z-direction, with measurable values \h and — \h. Approximate methods for systems with many electrons: ► Hartree-Fock approximation - a particular case of the variational method SJ = S f v|/*HH/dr = 0 in which the trial wave function is constructed from single-electron functions ► density functional theory (DFT) - application of variational method to the energy as functional of electron density (does not need to find a trial wave function) ► simple models: nearly-free electrons, tight-binding model F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 13/50 Hartree-Fock Method of Self-Consistent Field The wave function in the ground state is constructed from a set of orthonormal single-electron functions ^,{x), each a product of a spatial orbital function and a spin function cr(Sz). Single-electron functions are combined into multi-electron antisymmetric function by forming products in the form of N x N Slater determinant: ) The Hartree-Fock method needs to solve a set of coupled integro-differential equations that was obtained from the variational principle [#/+ I # Wrv -e'#/ = 0 in which the potential energy X^y/) / 0* ^ij1 (*1 ) 1 (*2) ^2(^2) >/ÄŽ! =>* used in quantum chemistry but is quite awkward for use in extended systems like solid surfaces F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces ka Zajíčková 14/50 E ensity hunctiona The electron density is the central quantity in DFT It is defined as the integral over the spin coordinates of all electrons and over all but one of the spatial variables {x = r, S) n(T) = N J \ty(x<[, x2,..., xN, )|2dS-| dS2 ... dS/vdx2dx3 ... dxN ► The first Hohenberg-Kohn theorem: For any system of interacting particles in an external potential Vext{r), n(r) uniquely determines the Hamiltonian operator (Vext(r) is a unique functional of n(r)) and thus all the properties of the system. ► The second Hohenberg-Kohn theorem: The energy of the many-body problem can be written as functional of the density, E[n]. The exact ground state is the global minimum value of this functional. A popular form of DFT functional was introduced by Nobel laureate W. Kohn and L. Sham £[n(r)] = T[n(r)] + ^Ze J dT-jj@- + \ J J drdr'^^l + Exc[n(r)] r ► T[n(r)] is the kinetic energy of a non-interacting inhomogeneous electron gas. ► The second term is ion-electron interaction. ► The third term is the average electrostatic interactions between the electrons. ► Exc[n(r)] is exchange-correlation term that represents all quantum mechanics many-body effects Exc[n(r)] = AEee + A 7. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková ensity The great advantage of this formulation is that the density that minimizes the energy is found by solution of a set of ordinary differential equations (Kohn-Sham) V + \/eff(r) l/eff(r) = Ze*J2 p1- + í df^L + i/xc(?) - \r — R\ J V — r \ r 1 1 where n(r) = l^/l Many-body interactions are hidden in the exchange-correlation potential V*c(r) = 5EyLC[n{f)]/8n(7) and practical implementation requires a good approximation for this quantity. The electron density appears in the effective potential which means that the Kohn-Sham equations needs to be solved self-consistently. The parameters e, and -0/(r) that enter the Schrodinger-like equation formally have no physical meaning. Nevertheless, they are frequently interpreted as one-particle excitation energies and eigenfunctions, respectively. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 16/50 Density Functional Theory - Local Density Approximation It is necessary to introduce an approximation of Vxc(r) - local density approximation (LDA): ► The exchange-correlation energy density of each infinitesimal region of the imhomogeneous electron distribution, n(r), is taken to be precisely equal to the exchange-correlation energy density of a homogeneous electron gas (HEG) with the same density as the corresponding infinitesimal volume element. n{xj The LDA is easy to apply because Vxc(r) is known very precisely for the homogeneous electron gas at all densities of physical interest {Ceperley, Alder, Phys. Rev. Lett. 45 (1980) 566). F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces ka Zajíčková 17/50 E ensity hunctiona ► The discrete ion cores are replaced by a uniform, positive background charge with density equal to the spatial average of the ion charge distribution. ► For the analogous surface problem, the semi-infinite ion lattice is smeared out as 1.2 1.0 • E 0.8 ■ > -0.6. tn \ \ \ \ 1 -1.0 -0.5 0.0 0.5 Distance z (Fermi wavelengths) n+(r) = n 0 z < 0, z>0 where n is expressed in terms of an inverse sphere volume 1 4 3 n 3 s Typical rs values are 2-5a0 (a0 being Bohr radius). The density variation perpendicular to the surface, n(z), reveals two features: 1. electrons spill out into vacuum region (z > 0) electrostatic dipole layer at the surface. 2. n(z) oscillates as it approaches an asymptotic value that exactly compensates the uniform (bulk) background charge. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 18/50 Density Functional Theory - Friedel Oscillations 1.2- 1.0 • 0 8 ■ r 0.6 ™ 0.4 0.2 0.0 ■ /\ p"forrs = 5 \ \ Positive S background, i + p \\ P~ for \\rs = 2 a) -1.0 -0.5 0.0 0.5 Distance z (Fermi wavelengths) Electron oscillation arise because the electrons (with standing wave vectors between zero and kF, radius of Fermi sphere) try to screen out the positive background charge distribution which includes step at z = 0. Screening in metals is so effective that there are ripples in the response, corresponding to overscreening =^ Friedel oscillations with wavelength 7r//cF, where kF = (3tt277)1/3 In 1993, electron density oscillations were observed in STM images of individual adsorbed atoms on surfaces (Eigler's IBM group). By assembling adatoms at low T into particular shapes, these 'Quantum Corrals' produce stationary waves of electron density on the surface. Left corral is created from 48 iron atoms (the sharp peaks) on a copper surface. The wave patterns are formed by copper electrons confined by the iron atoms. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces kaZajickovä 19/50 p m ensity hunctiona e > S o >- E? £-1 V(z) ■ k//2 No sharp edge of the electron density - effective surface at '°° , dn(z) dzz The formation of a dipole layer means that the electrostatic potential far into vacuum is greater than the mean electrostatic potential deep in the crystal. Potential step AV = V(oo) - V(-oo) -1.5 -1.0 -0.5 0.0 0.5 1.0 Distance z (Fermi wavelengths) keeps the electrons within the crystal. It is surface property. The remainder of the surface barrier comes from short range Coulomb interactions -exchange and correlation. It is bulk effect. The work function is, therefore, divided into the part related to the bulk properties (band structure) and surface contribution 0 = AV - EF where V and chemical potentail n are referenced to the mean electrostatic potential deep in the bulk. The surface part is responsible e. g. for different work function from different crystal planes. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces ka Zajíčková 20/50 Nearly-Free Electron Mode ► The jellium description of a metal surface -1D model that neglects the details of electron-ion interaction and emphasizes the nature of the smooth surface barrier. ► The band structure approaches emphasize the lattice aspects and simplify the form of the surface barrier. 1D nearly-free electron model (appropriate to a metal surface): neglects e-e interaction and self-consistency effects present in LDA Schrodinger-like equation, i.e. effective potential includes only the ion cores and a crude surface barrier: dz2 + V(z) v|/(z) = Ev|/(z). The effect of the screened ion cores is modelled with a weak periodic pseudopotential V(z) = - V0 + 2Vgcosgz where g = 2n/a is the shortes reciprocal lattice vector of the chain. vacuum For solution see e. g. Kittel, 1966. A two-plane-wave trial function is sufficient: Vk{z) = aeikz + /3ei(*-fl0* F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 21 / 50 Nearly-Free Electron Model (contin.) Substituing the trial function into Schrodinger equation leads to the secular equation k2-V0-E 9 Vg (k-g)2-V0-E a = 0 which is readily solved for the wave functions and their energy eigenvalues: E=-V0 + (9/2f + k2 ± (g2k2 + V2)'/2 v|//f = ei/czCOS(^z/2 + 5) where e[26 = (E — k2)/Vg and the wave vector was written in term of its deviation from the Brillouin zone boundary k = g/2 + k. K imaginary The familiar energy gap appears at the Brillouin zone boundary k = 0. K>0 F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 22 / 50 Nearly-Free Electron Model - Shockley Surface States ► In bulk, the solutions with imaginary k are discarded because of exponential growth at |z| ->> oo - do not satisfy the usual periodic boundary conditions ► For the semi-infinite problem, the solution that grows for positive z is acceptable since it will be matched at z = a/2 onto a function that describes the decay of the wave function in the vacuum: v|/(z) = eKZ cos(gz/2 + 5) z < a/2 v|/(z) = e~qz z > a/2 where q2 = V0 — E. If the logarithmic derivative of v|/(z) is continuous at z = a/2 =^ existence of electronic state localized at the surface of the lattice chain. The energy of this surface state lies in the bulk energy gap. ... here, bottom figure (for Vg > 0), curve 2 This solution is often called a Shockley state. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces ka Zajíčková 23/50 1D tight-binding model - wave functions are constructed from atomic-like orbitals (appropriate for semiconductor surface). The lattice potential is constructed from a superposition of N free atom potentials, Va(T), arranged on a chain with lattice constant a: N VUr) = J2V*(r-nä) n=1 where [-V2 + Va(r) - Ea]0(r) = 0 The non-self-consistent Schrodinger equation for the bands is {-V2 + Va(r) + [VL(r) - Va{T)]}V{r) = The simplest trial function is a superposition of s-like Wannier orbitals - one on each site: N = J2°nHr-na) n=1 Tamm surface states F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 24/50 Surface States - fast, slow Surface states can be ► fast - Equilibrium between surface and volume is established during relaxation time of « 10-8 s; density depends on the method of surface treatment, 1011 —1012 cm-2 for well-etched semiconductors. Example of fast states that can be found at clean ideal surfaces: ► Shockley surface states - periodic potential is symmetrical terminated at the surface and atoms are sufficiently close and interact strongly appropriate calculation method is NFE (nearly-free electrons) ► Tamm surface states - change in the potential in the outermost cell of crystall whose atoms are far appart (weak interaction between states) tight-binding model Demonstration for transition metals: nearly-free s and p electrons form Shockley surface states extending several layers into the solid; d electrons forming state with atomic-like wave functions localized on surface atoms. Example of fast states related to adsorb gases and surface deffects of crystal lattice. ► slow - Relaxation time ms till hours, density 1014-1015 cm-2 States related to thin oxide layers and presence of charges at its surface as well as its volume. F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 25/50 Surface States - donors, acceptors Surface states can be ► donors - they are occupied (neutral) for EDs < EF and unoccupied (positive) for EDs > EF ► acceptor - they are occupied (negative) for EAs < EF and unoccupied (neutral) for EAs > EF Repetition of terms donor (donating electrons)/acceptor (accepting electrons) from doping of semiconductors: Phosphorus Conduction band Conduction band Conduction band I*" New band gap 1 I L Donor level Band gap New band gap Acceptor level ■ 1 Gallium P-doped silicon Valence band Add Valence band Add Valence band group 15 I I group 13 ..„ atoms atoms n-Type semiconductor <- Pure silicon -► p-Type semiconductor (a) Doping with a group 15 element Ga-doped silicon (b) Doping with a group 13 element F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 26/50 ► Energy levels are straight up to the surface in case of no electrical field and no surface states. ► Appearance of acceptor surface state below EF =^ electrons from valence band will occupy it =^ surface will be negatively charged and positive space-charge sublayer occurs upwards bending of energy bands it -H E b) S(x) Obohocení Rovné posy Ochuzení Inverze m ±±+1 —J I e e© e e -U ^-Ec 0\ I d) In p, n n, r n Example for p-type semiconductor F7360 Characterization of thin films and surfaces: 2.2 Electronic Structure of Clean Surfaces Lenka Zajíčková 27/50 Surface States - example of ZnO nanowires ZnO is wide bandgap semiconductor. Nanowires have large surface - surface defects, near surface traps and surface adsorbed species play important role. Electron Depletion Electron AeeumuJalUm Ground Stalt (c) Ground Slate {<]) F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 28/50 Difference between Fermi level and chemical potential is neglected (temperature below 1000 K) ► for metals: X = Eaf - Ei The electron affinity Eaf is the difference between the vacuum level E0, and the bottom of the conduction band Ec. ► for semiconductors: thermoelectric work function - difference between E0 and EF photoelectric work function (ionization potential) -difference between E0 and Ev conductive band ■St vacuum level Xf Xt ■af eu F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 29/50 Contact Potential - Metal/Metal Before Contact t Metal A After Contact t Metal A MA) .-----------VacuĽiii Level EF(B) Metal B Metal B metals A and B are electrically isolated (xa < xb) =>* an arbitrary potential difference may exist metals A and B are brought into contact =^ electrons flow from the metal B to the metal A until the electrochemical potentials (Fermi energies) are equal. The actual numbers of electrons that passes between the two phases is small, and the occupancy of the Fermi levels is practically unaffected. Metal A is charged positively and metal B negatively, i. e. work functions does not change but contact potential appears. eVcont. = xb ~ xa F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 30/50 Contact Potential - Semiconductor/Metal © © © ■o J x m T—r Xts ■af ■af .1_______ ■Fm ■Fs EF -E„ equilibrium between metal and semiconductor not yet established equilibrium metal/semiconductor distance is large close contact contact potential: eV^cont. — Xm — XTs Development of space-charge region in the semiconductor in case of close contact => band bending F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 31 / 50 Measurement of Work Function from Contact Potential Experimental methods for determination of work function measurement of contact potential difference eVcont. = xb - xa in which the work function of one material has to be known; ► measurement of characteristics of various electron emission processes Measurement of contact potential difference by the condensator method (Kelvin probe): between two surfaces creating a condensator with capacity C. If C is changed, a current / will flow dt where U is the voltage difference between the condensator plates. This voltage is equal to the contact potential difference if there is no external voltage applied. If we apply external voltage compensating the contact potential the field between the plates can be reduced to zero external voltage is equal to the contact potential difference. i" *' Cl. ÍM The changes of capacity are realized by vibration of one electrode vibrating capacitor method. SKP470 Scanning Kelvin Probe http://www.bio-logic.info/instruments/skp470-2/ F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 32/50 Change of Work Function with Temperature ► For metals the work function has a linear relation with the temperature change: x(T) = x(T0) + a(T-T0)i where a has values between 10-4-10-5 eV/K. ► For semiconductors and insulators the chemical potential varies strongly with temperature: Eg- kT , A/c 6(T) = Eaf + + — n —, ; at -r 2 -r 2 ^, Eg - band gap, Eaf - electron affinity, Nc and A/v - densities of the conduction and valence bands. ► For n-type semiconductors at lower ionizations: x(7) = Eaf + ^ + Tln-, A Ed - activation energy, A/D - density of the donor states. At high temperatures it is change to: ► In case of p-type semiconductors: kT , A/c x(r) = Eaf + -ln- D and for high T fT, r: , r: AEA kT A/v x(7) = Eaf + Eg-^-Tln- / í- í- A/v x(T) = £af+ £g- — In F7360 Characterization of thin films and surfaces: 2.3 Work Function Lenka Zajíčková 33/50 Change of Work Function in Electrical Field If we apply el. field E close to the surface of metal there will be two types of forces exerting to the electrons ► attractive image force (between the electron and its mirror image inside metal) ^0 = 1 6tt£oX2 ► el. force accelerating electrons out of the metal F(x) = F0(x) - eE. In certain distance xk from the surface, the final force F(x) will be equal to zero and for x > xk the electron will be accelerated from the surface. Work function will be then equal oo oo *k x = J (F0(x) - eE)dx = J F0dx- J F0dx- J eEdx o 0 Xk 0 X0 - eExk = xo - e> eE This dependence of work function on external el. field is called Schottky effect. F7360 Characterization of thin films and surfaces: 2.4 Thermoemission Len ka Zajickova 34 / 50 Thermoemission Addition of heat =^ increased energy of lattice vibration and energy of electrons =^ some electrons obtain energy required to pass surface potential barrier and are emitted from the surface Process of thermoemission can be described 1. by thermodynamics - electrons are the evaporated material, terms like heat of evaporation, Clausius-Clapeyron equation, equation of state 2. by statistical physics - known statistical distribution of electron velocities is taken to calculate those electrons that have enough energy to overcome work function. This approach was originally suggested by Richardson for metals (1901) but using classical statistics. Statistical description of thermoemission using Fermi-Dirac statistics For a system of identical fermions, the average number of fermions in a single-particle state /, is given by the Fermi-Dirac (F-D) distribution <"/> = —9J— exp^ + 1 where Q\ is the state degeneracy (the number of states with energy e,) F7360 Characterization of thin films and surfaces: 2.4 Thermoemission ka Zajíčková 35/50 Distribution of Energies Perpendicular to Surface In solids, the states are characterized by a quasi-continuum energies with defined density of states g(e) (the number of states per unit energy range per unit volume) and the average number of electrons per unit energy range per unit volume is for metals (n(e)) 9(e) exp ^ + 1 where from Heisenberg principle of uncertainity g(e) = 2/h3 and for reasonable T we assume fi = EF Number of electrons having momentum in the range from (px,py,pz) to (px + dpx,py + dpy,pz + dpz)\ N(p)dpxdpydpz = g0 dpx dpy dp. h3 exp(^2^) + 1 The axis z will be perpendicular to the surface and we look for the number of electrons with the energy from (pz,Pz + dpz). After integration in polar coordinates (/ ^x+r0^ = * — ^0 + ex)) and substitution e = p^/2m we have N(e)de 7rg0m J 2m kT In (l +e" e — fj, ~kT~ de. F7360 Characterization of thin films and surfaces: 2.4 Thermoem ission Lenka Zajíčková 36/50 Density of Thermoemission Current Number of electrons impinging on unit surface area per unit time with energy (e, e + de) is obtained by multiplication with vz = \/2e/m He)de = ^UlkT\n (l+e-^)de. Emitted electron have to fulfil the condition e > Eaf but integration of the flux term is not possible in general. Assuming (e — EF)/kT > 1 the flux can be simplified as u(e)de= -e~ kT h3 and density of emission current / is obtained by integration considering a certain probability of electron reflection at the surface barrier R(e) oo i=e J [1 - R(e)]v(e)de. R(e) is for simplicity approximated by an averaged value R = 1 - D and then / = -Anmek2 o Eaf-^ - n o ^/kT D--—T2e--vr- = DA0T2e~x/kT, h3 F7360 Characterization of thin f ilms and surfaces: 2.4 Thermoemission Lenka Zajíčková 37/50 Richardson-I Dushman equation From previous slide we have i=DA0T2exp (--^) in which the constant D should not differ for different metals but it was found out that DA0 is quite different for different metals =^ we need to consider temperature dependece of work function x(7~) = x(7"0) + a(T - T0) and obtain Richardson-Dushman equation i=DA0T2exp(-a/k)exP(-x{To)k-aTo)=AT2Bxp(-^), where Richardson constant A is not an universal constant but it is characteristics for given material and is reduced or Richardson work function. metal melting point (K) /A(Acm~2K2) x (eV) ~~W 3640 80 4fi Ta 3270 60 4.1 Pt 2050 170 5.6 d^/d7" is of the order of 10-4 to 10-3 eV/K, with both positive and negative signs. When data is taken over a limited range of T, this temperature dependence will not show up on such a plot, but will modify the pre-exponential constant. F7360 Characterization of thin films and surfaces: 2.4 Thermoemission Lenka Zajíčková 38 / 50 easurement of Wor k Function Using T hermoelectric Methods Using Richardson line: a plot of log(//7~2) versus 1/7" yields a straight line whose negative slope gives the work function 0. The constant, A, can be measured in principle, but is complicated in practice because we need to know the emitting area independently, since what is usually measured is the emission current / rather than the current density, /. Je třeba zajistit, aby se měření proudu uskutečňovalo v režimu nasyceného proudu, tj. aby v měřícím systému nehráli roli prostorové náboje. To znamená, že mezi emitující katodu a anodu musí být vloženo dostatečně velké napětí. Při větších napětích se pak ovšem uplatňuje Schottkyho jev, takže naměřené hodnoty by měly být správně extrapolovány na nulové vnější pole: i = AT2 exp(-^-) = AT2 exp(-^)exp f ® /_?^_ ) KV kTJ KV kTJ K ^AT y 47T£0/ Musíme měřit dostatečně přesně teplotu katody (pyrometrická metoda nebo pomocí změn odporu žhaveného vlákna). F7360 Characterization of thin films and surfaces: 2.4 Thermoem ission Lenka Zajíčková 39/50 Measurement of Work Function Using Thermoelectric Methods Metoda kalorimetrická: emitované elektrony s sebou odnášejí určitou energii, tj. katoda se ochlazuje a chceme-li, aby její teplota zůstala konstantní, musíme zvětšit příkon. Energie spotřebovaná na jeden elektron je e = % + 2kT = e 1000 K), hodí se tato metoda pouze pro látky s dostatečně vysokým bodem tání a pro látky, které při použitých teplotách nedissociují. ► Při pomalém zahřívání a udržování vzorku na vysoké teplotě nastává difúze nečistot z objemu. ► Díky tepelné desorpci může dojít k porušení stechiometrie a naleptávání povrchu krystalu. ► Nelze odstranit libovolnou nečistotu. F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces Lenka Zajíčková 45 / 50 E esorpce v silném ei . poli Kov je kladným pólem. Je-li el. pole dost silné (řádově 108 V/cm), může se hladina valenčního elektronu adsorbované látky vyrovnat s Fermiho hladinou kovu, resp. se dostat těsně nad ni. V tomto případě je umožněno protunelování elektronu do kovu. Z atomu se stává kladný iont, který je elektrostatickými silami odmrštěn od kladného povrchu kovu. Nejsnadněji lze realizovat desorpci elektropozitivnívh adsorbátů, je však možné desorbovat i látky elektronegativní, ovšem potřebná pole jsou větší a může dojít i k vytrhávání vlastních atomů (tzv. vypařování v poli). Tento způsob čištění je usnadněn při zahřátí (větší migrace). Nevýhody: omezeno na materiály, ze kterých umíme a chceme vyrobit velmi ostrý hrot (pod 1 /iim), a na kovy. F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces Lenka Zajíčková 46/50 Desorpce elektronovým bombardovaním Přímé ostřelování zkoumaného povrchu elektrony relativně nízkých energií (50-200 e V), takže zahřátí je nepatrné. Jedná se pravděpodobně o přechod adsorbované částice do excitovaného stavu, který může být k povrchu vázán slaběji nebo vůbec. F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces ka Zajíčková 47/50 Používají se těžší ionty, většinou Ar nebo Xe. Díky své hmotnosti předávají ionty účinně energii povrchové částici. Je důležitá čistota pracovního plynu a správné soustředění svazku (pozor na rozprašování okolních materiálů!). Výhody: ► univerzální metoda pro libovolnou látku ► umožňuje postupné odprašovaní. Nevýhody: ► u dielektrika se musí neutralizovat náboj iontů, ► jsou vytvářeny poruchy v bombardovaném materiálu =^ kombinace bombardu a vyhřátí, F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces Lenka Zajíčková 48/50 Čištění pomocí laserového paprsku Moderní modifikace čištění tepelnou desorpcí. Laserový svazek dopadá na čištěný povrch skrz okénko. Výhody: ► vakuum, žádné cizí částice ► pro krátké pulzy dojde k ohřevu jen povrchu a nikoliv objemu Nevýhody: ► lokální tavení materiálu ► (jako pro ostatní tepelné metody) nelze odstranit libovolnou nečistotu ► vysoká cena a prostorové nároky vhodných laserů F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces ka Zajíčková 49/50 Vhodné pro některé monokrystaly. Čistota povrchu je dokonalá. štípání břitem zatlačovaným do vrypu na povrchu krystalu =^ povrch má většinou hodně nepravidelností ► lámání krystalu ohybem =^ povrch lepší F7360 Characterization of thin films and surfaces: 2.7 Methods for Preparation of Clean Surfaces Lenka Zajíčková 50 / 50 Vhodné v určitých speciálních případech. Organické nečistoty lze odoxidovat v kyslíku, některé nečistoty lze převést na plynné sloučeniny zahřátím ve vodíku atd. Většinou se vzorek v dané atmosféře žíhá.