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 BBB E()=EAuger  () - E () -E ()

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).

 

 

  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

M4 initial hole Ls12 1/2

Lp22 1/2

N4,5 final Lp32 3/2 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 061 22spS  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  32 4 1  S jPS0 from2s0 2p mixed parity 33 243  PjP1 fromeven and J=1 22spP    1  32 4 1  1  jPD2 from2s22 2p mixed parity  D

3 P1 even and J=1 requires Auger electron in even state with J=1  forbidden

22 1  324 1  S jPS0 from2s0 2p mixed parity 33 243  PjP1 fromeven and J=1 22spP    1  324 1  1  jPD2 from2s22 2p mixed parity  D

3 P1 even and J=1 requires Auger electron in even state with J=1  forbidden

3 3 P2 and P0 become more and more allowed as Z grows.

3 Instead, P1 remains purely P and forbidden (as long as one can neglect the mixing with higher configurations). Thus, the number of lines grows to 9.

0 6 1 Allowed Auger transitions 2s 2 p S , 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 , 3 P , 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 kBZ 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 IMdNN()||()()111 NE()()   k  kBZ M  Wentzel'smatrixelement 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. Phys Rev (1973)

Quasiatomic CVV spectrum

27 Milano 4 Luglio 2006 28 George A. Sawatzky valence atomic orbitals Closed band theory

closed-shell valence ? band

core-hole

Initial state Final state:do holes delocalize?

Model Hamiltonian H HHHHatomSolidatom Solidfree electrons

HnnUmmˆˆ m mcc c c (,,,) †† atom dh dhmm m m m  1 2 3 4 1 2 4 3 mm m m m 1 2 3 4

ˆ Hnfree electronsk k

HHSolid atom solid enter through LDOS; one may take Anderson model 30 Take Spin-up core hole in initial state

† icdeep hole 1 atomic Auger :(A ,', m )(1) m k (2)(1)(2) deep holemm k ' r12

neglect inter-atomic transitions i.e. transitions take to the set of states

†† mncc n m

with two holes on valence orbitals of the same atom.

31 Milano 4 Luglio 2006 Cross section: from Fermi Golden Rule * SAkAk()()()()  D mnpqmnpq mnpq

where the local density of states (LDOS) for the two final-state valence holes is

DpqHmnmnpq ()() 

†† mncc n m

H is all the interacting Hamiltonian except free Auger electron term

32 Milano 4 Luglio 2006 1 hole analogy :( )(mnnm ) mHn  iHt Fouriertransform( ) mn tm en

DpqHmnmnpq ( )()  iHt Fouriertransform( ) Dtpqemnmnpq  

The fact that the LDOS appears, rather than the band DOS, as in Lander theory, is one of the main results, and qualifies Auger spectroscopy as a local probe of valence states.

33 Milano 4 Luglio 2006 Simple limiting cases:

Atomic limit multiplets

Non-interacting case

Two-hole correlation function iHt †† Dmnpq(), t pq e mn  mn   c n cm  U=0 limit, taking primary hole spin 

,Dmnpq ( t )  mp t  nq  t spins 

,Dmnpq ( t )  mp t  nq t   mq t  np  t spins 

34 Milano 4 Luglio 2006 To calculate the interacting density of states use the identity

1 1 1 1 H1 HHHH  00    

One-body part without Repulsion term free electrons

†† Take matrix elements over mncc n m

The coupled equations can be easily solved in general.

However, if the solid does not significantly perturb the spherical symmetry of the atom, the the D matrices are diagonal in the

|LMLSMS > representation, where Hr is diagonal, and obey uncoupled equations. 35 Trivial example: s band: The two-holes Green’s function is defined as 0

0 1  i0 ()0000 0 D (') 0 0 0 Hi Id()' i()()()   I   i  D  0   '

D = two-holes DOS 2 holes on same site

1111 000000000000H1 HHH 00H

H U(,,,) m m m m c†† c c c 1 1 2 3 4 m1 m 2 m 4 m 3 m1 m 2 m 3 m 4 in closed s band reduces to

HU1  00 00

36 1111 000000000000H1 HHHH 00 1111 0000000000000000U HHH 0 0 H

(0)   Exactsolution( )( )   Ii D 1 iU(0)  

D0 () D( ) Re [1(UIU )](022 ) D 2 02

two-hole resonances

Hole binding by repulsion: a purely quantum effect 37 Example: rectangular band

1   (||) 2

  

 =

1  TriangularDOS : D0 ( )  (2    )(  ) 242 38 1  D0 ()(2)() Triangular dos 242

0  Dy() Idy0 ()     y

1 2   22 4  I 0 ( ) log | | log | | 2  2  4 22 

D0 () D( ) Re [1(UIU )](022 ) D 2 02 shows the distortion of band-like lineshapes as U grows.

For U>1.44  the intensity of the continuous line shape drops and tends to zero for large U.

The overall intensity cannot drop, however.

39 124 22 I 0 ()log  ||log || 224  22

Letting( )(,zi  , Izizw0(0)  e  ) h re

1z 2 z z22 4 iz0 ( )  log( )  log( ) as  0, 2zz 2  4  22 I00()() i  D 

D0 () D( ) Re [1UI0 (  )] 2  2 U 2 D 0 (  ) 2 D0 () D()Re  [1()]()UIUD022202

The missing intensity is into split-off the two-hole resonances, that however come into the theory as 0 over 0 singularities.

DUI000,10

This is equivalent to convolving the line shape with a Lorentzian of width  , which, incidentally, is always appropriate to describe lifetime broadenings which are present in reality even if they are not considered in the model.

41 U   istheparameter

 Applications to desorption theory (Knotek Feibelman mechanism) Feibelman (Knotek theory desorption to Applications

42 Hole binding by repulsion: a purely quantum effect

Band continuum

I called them resonances because they must eventually decay, since they are excited states Strong correlation means bound states, new frequencies, failure of mean field

W=0 pairing yields bound states from repulsion in the presence of high enough symmetry (Cini and Balzarotti 1995) but this is 43 another topic!