Auger Line Shapes in Solids Pierre Victor Auger (1899-1993) who discovered the law of the Auger effect in 1925 who discovered the Auger effect in 1923 1 Core-levels observed in photoemission are resonances and width is large for high binding energies Core-levels observed in photoemission are resonances and width is large for high binding energies Fe1s binding energy=7112 eV Width > 5 eV Decay mechanisms: X-ray emission, Auger electron emission. Auger electrons are seen by the ESCA apparatus along with photoelectrons. Their energies are independent of the photon energy. X ray notation level Ks1 1/2 Ls12 1/2 Lp22 1/2 Lp32 3/2 Ms13 1/2 Mp23 1/2 Mp33 3/2 Md43 3/2 Md53 5/2 Ns14 1/2 Np24 1/2 Np34 3/2 Nd44 1/2 Nd54 3/2 Nf64 5/2 Nf74 7/2 Meitner-Auger Effect and Spectroscopy An important spectroscopy is based on the Auger Effect. Actually, this effect was first reported in 1923 in Zeitschrift für Physik by the Austrian Physicist Lise Meitner (1878-1968). In 1925, the great French physicist Pierre Auger (1899- 1993) rediscovered the effect while investigating in a bubble chamber the emission of an electron from an atom that absorbs a X-ray photon, that is, photoemission. The photoelectrons have kinetic energy Ek = hν − EB, where EB is one of the binding energies of the atom inner levels. However the Auger electrons have ν-indepedent energies given approximately by the empirical Auger law. BBB EAuger ( )=E ( ) - E ( ) -E ( ) 7 Pierre Victor Auger (1899-1993) X-rays of adequate energy produce a primary hole in the state of binding energy of EB() and a photoelectron; since the ion with a core hole is unstable, the primary holes is filled up by an electron in a less bound level β, and the energy gained in the process is taken by an electron in level γ, than is emitted as the Auger electron. For particular transition to happen, it is necessary that the primary hole is deep enough (its binding energy must exceed the sum of those of and c). BBB EAuger ( )=E ( ) - E ( ) -E ( ) 8 The Auger transitions are denoted by the spectroscopic symbols of the shells (or, more precisely, sub-shells) involved: thus a KL1M2 transition is due to a primary hole in the K shell that decays leaving in the final state L1 and M2 holes and an Auger electron. M2 L1 K One speaks about Core-Core-Core, Core-Valence-Valence, etc.,transitions depending on the levels that are involved. The Auger effect is caused by the Coulomb interaction: two electrons of the system collide and while one fills up the primary hole, the other is shot out as the Auger electron. The energy of the primary-hole state is shared between the Auger electron and the ion left behind. 9 AES=Auger Electron Spectrum of Cd vapor (Z=48) X ray notation level Ks1 1/2 Ls12 1/2 M4 initial hole Lp22 1/2 Lp32 3/2 N4,5 final holes Ms13 1/2 Mp23 1/2 Mp33 3/2 Md43 3/2 Md53 5/2 Ns14 1/2 Np24 1/2 Np34 3/2 Nd44 1/2 Nd54 3/2 Nf64 5/2 Nf74 7/2 11 KL11 M Auger transition in the electron picture: L11 M K k the other electrons are (approximately) spectators 12 KL11 M Auger transition in the hole picture is simplest: K k L11 M hole k is in continuum 13 X decay is faster when the involved levels are distant in energy, because of the n3 factor in the transition probability due to the density of photon final states. The Auger decay prevails if the states , and are near in energy Auger is the dominant for inner shell holes of light atoms, while in heavy ones X decay is much more likely Wentzel (1927 ) theory: transition Primary holes final holes Primary holes: deep level α and free state k final holes : the spin-orbitals β and γ Two-step model: the photoemission process produces ion in its meta-stable ground state- Deep hole creation and Auger decay are disjoint processes . 15 Wentzel (1927 ) theory: transition Mechanism: the smoking gun The Auger transition is a Coulomb collision between two holes and is an energy conserving process. (in reality the other electrons are also involved in some way) In the hole picture, the initial state |Φi > and the final state |Φf > of the atom are represented by 2 × 2 Slater determinants; they have the same energy and are coupled by the Coulomb interaction. In a KLM transition, involving s states, |Φi >=Det|y1syk| --> |Φf >=Det|y2sy3s| 2 2 Rate by Fermi golden rule PHif| i C f | . 16 An alternative mechanism : internal photoemission M M2 2 hn L L1 1 K K One of electrons in the upper states β, γ could fill up the deep hole via a normal radiative process, emitting a X-ray photon; this photon could then cause the photoemission of the other electron. Would’nt the final state be the same? Indeed, this alternative process does exist, and has the name of internal photoemission; unlike the Auger effect, it obeys optical selection rules. The transition probability of the internal photoemission can be calculated by perturbation theory, but since it is a second-order process in A.P it turns out to be quite small compared to the Auger effect. 17 Atomic Auger Selection Rules 2 Conserved quantities in the transition: energy, J , Jz and parity . Actually parity conservation is violated but the effect is too small to be of any importance in Auger spectroscopy. 2 Selection rules arise in the two-step model from the conservation of J , Jz and parity between the initial, core-hole state and the final state including the Auger electron. 2 2 In the L-S approximation, L , Lz and S , Sz are also conserved, while in the jj scheme, the states are labeled by the j quantum numbers. 18 Atomic Auger Selection Rules KLL transitions If the spin-orbit interaction is more important than the Coulomb interaction, as in high Z elements, the jj scheme applies. If the Coulomb interaction dominates, the LSMLMS scheme is OK. The pure jj scheme one would predict 6 transitions, namely, KL1L1,KL1L2, KL1L3,KL2L2,KL2L3, and KL3L3. L3 L 3 L3 L L 3 L3 3 L2 L 2 L2 L L 2 L2 2 L1 L 1 L1 L L 1 L1 1 K K K K K K 19 In the pure LS scheme, the possible final states are 5: 2s22p4 3P is forbidden by 0 6 1 22s p S parity conservation; indeed, the primary hole has L=0 2 4 3 1 and is even, 2s 2p P is 15 P even and has L=1 ; by L 22sp 3 P allowed conservation the Auger electron must be in a p state, which is odd.Hence 1S the final state is odd. 2 4 3 22s p P The transition to the odd 1 D 2s12p5 3P atomic final state is allowed, and one predicts 5 lines. LS terms split by Coulomb interaction 20 Dependence of atomic levels on coupling scheme for 2 electrons of p type When the spin-orbit interaction is introduced, in intermediate coupling the forbidden final-state splits : 2 4 3 3 3 3 2s 2p P → P2, P1, P0; 3 then, states with the same j and different spin mix: P2 mixes with 1 3 1 2 4 D2 and P0 mixes with S0 from the same 2s 2p configuration. Now the decay can occur with d or s Auger electrons. 1 3 2 4 1 S j0 from P0 2s 2p S mixed parity 33 2 4 3 P j1 from P even and J=1 22s p P 1 3 2 4 1 1 j2 from P22 2s 2p D mixed parity D 3 P1 even and J=1 requires Auger electron in even state with J=1 forbidden 22 3 2 4 1 3 3 1 j0 from P 2s 2p S mixed parity P2 and P0 become S more and more allowed as Z0 grows. 33 2 4 3 P j1 from P even and J=1 22s p P 1 Instead, 3P remains purely P and forbidden (as3 long as 2 one 4 1can neglect 1 1 j2 from P22 2s 2p D mixed parity the mixing with higherD configurations). Thus, the number of lines grows to 9. 3 0 6 1 AllowedP1 evenAuger andtransitions J=1 requires Auger electron2s 2in p even S state, with J=1 forbidden in intermediate coupling 1 5 1 3 3 3 2s 2 p P2, P 0 , P 1, P 2 , 2s2 2 p 4 1 S, 3P , 3 P , 1 D 0 0 2 2 23 Core-Valence-Valence transitions in solids: Lander independent-particle theory Level scheme E k kinetic energy of Auger electron k vacuum level 1 =binding energyof level 1= electron jumps to free state energyof hole in level 1 NE()() k k BZ Auger law: 1 = ck12 2 core-hole 2 valence holes+Auger electrons electron jumps to core Auger spectrum: C ck(12 )= Auger energy 24 CVV transitions in solids: Lander theory (1953) Level scheme ck( 12 )= Auger electron energy E All hole pairs such that 12 k vacuum level contribute to spectrum at kc . electron jumps to free state Auger intensity 2 I( ) | M | d 1 N ( 1 ) N ( 1 ) NE()() k k BZ M Wentzel's matrix element 1 2 Density of final hole states electron jumps to core d N()() N 1 1 1 C self-convolution of N NB in Lander theory the holes are created in Bloch stetes 25 Powell, prl (1973) stressed that spectra can be atomic-like or band-like 26 Milano 4 Luglio 2006 S.P.Kowalczyk et al.