Matrix elements and symmetry

An important contribution to the photoemission intensity (eq. 3.7 and eq. 3.8) is the one-electron transition matrix elementMf,i. After factoring

out the(N−1)-electron system, the matrix element term has three parts: the one-electron final state wavefunction; the interaction Hamiltonian, which for photoemission is the dipole operator; and the one-electron wavefunction of the initial state, overall giving:

Mf,i=hφkf|A·p|φkii (3.19)

Since this term is an integral of the two wavefunctions over all space, if the integrand is an odd function (or has an odd parity eigenvalue given the definition in the previous chapter), the integral will be zero. The matrix ele- ment will therefore vanish, and hence too the photoemission intensity. This provides a powerful tool for selectively probing given initial states by mea- suring in a geometry which necessarily greatly reduces the intensity from initial states with a vanishing matrix element.

To illustrate this, consider the following highly idealised example (de- picted in fig. 3.7). Consider photoemission fromp-orbitals by linearly po- larised light. The propagation vector and polarisation vector of the light are governed by theA·pterm. In the geometry in this idealised example (fig. 3.7), light is incident exactly along thex-zdirection, which is along the mir- ror plane. The polarisation vector for s-polarised light is defined as having

44 Chapter 3. Methods p-pol s-pol hv e x y z mirror plane sample plane + + - -

FIGURE 3.7: Simplified geometry of light polarisation and initial state orbitals to explain matrix element effects. De- pending on the parity of the light polarisation and initial state orbital under reflection through the scattering plane (the mirror plane shown in the figure), the matrix element

may integrate to zero.

its polarisation vector parallel to the samples surface, whereas p-polarised light is defined as having a polarisation vector perpendicular to the sample surface. An important mirror plane, the scattering plane, is defined such that p-polarised light is even under parity transformation with respect to it, and s-polarised light has an odd-parity eigenvalue.

The final state is assumed to be that of a free-electron, as previously dis- cussed. We can further simplify this example by considering photoelectrons travelling along the mirror plane direction to an analyser which is itself in the mirror plane. Final states which have an odd parity eigenvalue will have a node in their wavefunction at zero and so the amplitude of the wavefunc- tion at the detector will also be zero. This means that the final states must be even parity. In real experiments however this idealised situation is never practically realised but through our work on BiTeI, it will become clear that this assumption about the final state is still practically useful in many cases. The parity of the matrix element as a whole must be even since an odd in- tegrand would integrate to exactly zero. Given the approximation of an

3.1. Angle-resolved photoemission spectroscopy 45

even parity final state, this imposes that the parity of the dipole operator (governed by the light polarisation vector) and the initial states must be the same (both even (+), or both odd (-), under parity inversion through the mirror plane) summarised as:

hφk_f|A·p|φk_ii=        h+|+|+i 6= 0 for p−pol

h+| − |−i 6= 0 for s−pol

where all other combinations integrate to zero. From this, p-polarised light will excite from even parity initial state orbitals, whereas s-polarised light will excite from odd parity initial state orbitals, with a non-vanishing one- electron transition matrix element.

Considering thep-orbitals (fig. 3.7),px andpz orbitals are mirror sym-

metric in the mirror plane, lying along the x andz axes. These therefore have even parity eigenvalues and are excited in this situation solely by p- polarised light. On the other hand, py orbitals have a contribution from

the phase of the wavefunction which is positive on one side of the mirror plane, and negative on the other. Taking this into account,py orbitals have

odd parity eigenvalue and so are excited solely by s-polarised light, in this example.

However in reality, beyond the so far idealised example considered, these strict conditions are more relaxed. For example the propagation vector of the light is at an angle to the mirror plane, which means the light polarisa- tion vector may not be purely p-polarised or purely s-polarised. This means the one-electron transition matrix element and therefore the photoemission intensity will not be strictly vanishing. Light which is p- or s-polarised will therefore only predominantly excite from the symmetry expected ini- tial state orbitals, with a non-vanishing matrix element for the other orbitals. This condition can also be relaxed by the analyser being at an angle to the mirror plane / sample surface and the final state therefore no-longer being

46 Chapter 3. Methods

strictly required to be even parity. Additionally, real orbitals could be mod- ified from the hydrogen-like simplified cases through bonding / hybridi- sation, which then makes the symmetry argument harder to make. In all these cases, instead of leading to a strictly vanishing matrix element, it will instead lead to a quantifiable reduction in the photoemission intensity. De- spite these convoluting factors, this still provides a means to study the initial state orbital composition through polarisation-dependent photoemission.