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Instead of detecting photoelectrons at a single angle, or more realistically within a small solid angle, as in ARPES, the photoelectrons can be collected over a large solid angle in anangle-integratedmode. At relatively low photon energies, the transitions must still be direct, and so integrating over all wavevectors yields the joint density of states between initial and final states [71]. In reality, the acceptance angle of a spectrometer will be limited, which may restrict the portion of the Brillouin zone sampled in the measurement.

However, if x-ray photons are used, several other factors become important. First, for photoelectrons with high kinetic energies, small angles of emission still correspond to high values of wavevector, and so even for relatively modest accep- tance angles, it is possible to sample the entire Brillouin zone. Second, assuming the Debye factor is not too large, the influence of phonon scattering at higher energies can become very significant, meaning that direct transitions are no longer required. It should also be noted that the wavevector of the photon can no longer be consid- ered negligible for x-rays, resulting in the breakdown of the dipole approximation introduced in Eqn. 3.5, and the necessity to include a term for the photon wavevec- tor in the overall wavevector conservation term. Integrating Eqn. 3.9 over ki and kf independently yields the total photocurrent

I(E)∝¯¯M¯f i

¯ ¯2

N(Ei)N(Ef)D(E)T(E) (3.16)

where N(Ei) and N(Ef) denote the initial and final density of states, respectively,

3.1. Photoemission spectroscopy 35 8 6 4 2 0 In te n si ty (a rb . u n it s) VB-DOS QPC-DFT (unbroadened) VB-DOS QPC-DFT (broadened) (1120) VB photoemission Γ K H A Γ M L A PWZ I SWZ I PWZ II (a)

Binding Energy (eV) 5 3 3 ₅ 1 6 1 6 3 2 1 3 3 1,2 3 5,6 1,3 4 3 2 1 3 1 1,3 2,4 1,3 5,6 1,3 (b) Figure 3.2: Shirley-background-subtracted valence band x-ray photoemission spec- trum and QPC-DFT VB-DOS calculations shown without (shaded) and with lifetime and instrumental broadening for wurtzite InN(11¯20). The main features in the VB- DOS are identified, after Ley et al. [82]. The measured valence band photoemission is rigidly shifted to lower energies by 1.53 eV to align the VBM at 0 eV binding energy as for the calculations. The XPS and QPC- DFT spectra are normalized to the peak PWZ

I intensity. The corresponding QPC-

DFT valence band structure for wurtzite InN is shown in (b). High symmetry points are denoted using double group symmetry notation, although the symmetry point la- bel has been dropped for clarity of presen- tation. Therefore, for example, at the va- lence band maximum the label 6 denotes Γ6

symmetry. The XPS spectrum was mea- sured at the National Centre for Electron Spectroscopy and Surface Analysis, Dares- bury Laboratory, UK, using a photon en- ergyhν = 1486.6 eV. For more details, see Ref. [83].

the matrix element and the final density of states, as well as terms due to the second and third steps of the three-step model, can be treated as constant over the range of the measurement, and so valence-band x-ray photoemission spectroscopy (XPS) approximates well the initial density of states of the material. While in rare cases, direct transition effects can be observed in XPS measurements [80], these can normally be neglected, at least at room temperature and above. Indeed, Shirley [75] found excellent agreement between broadened valence band density of states (VB- DOS) calculations and valence band XPS measurements from gold, while Pollak et al.[81] and Leyet al.[82] showed that this holds for a range of group IV, III-V and II- VI semiconductors. Subsequent investigations have also shown excellent agreement between theoretical calculations and XPS measurements for the VB-DOS of many other materials. An example of this, for the case of InN, is shown in Fig. 3.2 [83].

The energy reference is generally taken to be the Fermi level in a PES mea- surement. It should be noted that, in a semiconductor where space-charge regions can exist, the reference Fermi level is that at the surface due to the surface speci-

3.1. Photoemission spectroscopy 36 600 400 200 0 In 4p In te n sity ( a r b . u n its)

Binding energy (eV)

XPS HXPS In 3p In 3d N 1s O 1s C 1s In 4s In 4d

Figure 3.3: XPS (hν= 1486.6 eV) and HXPS (hν= 7600 eV) wide energy scans from InN(11¯20), performed at the National Centre for Electron Spectroscopy and Surface Analysis, Daresbury Laboratory, UK, and beamline ID16, European Synchrotron Research Facility, Grenoble, France, respectively. The positions of several core-level peaks are shown. The Fermi level is defined as the zero of the binding energy scale, and the spectra are vertically offset for clarity.

ficity of the technique. Consequently, by either extrapolating the leading edge of the valence band photoemission to the background level, or by aligning it with VB- DOS calculations, the surface Fermi level of a semiconductor can be determined. This will be used extensively in this work. While XPS is not as surface sensitive as PES performed with lower photon energies, it is still dominated by the signal from the surface. Additionally, interpretation of valence band photoemission spectra ob- tained using lower photon energies is complicated by final state effects and also the presence of surface states. Consequently, XPS measurements are perhaps the most suitable for determining the surface Fermi level position in semiconductors.