Electron Field Emission from Diamond-Like Carbon
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CB
------.------ -Evac
(-1
-
FL.
Zn
VBM
X
S Fig. 1. Schematic5:diagram of the development of atomic orbitals, bonds and bands in a covalent solid, showing S band edge energies with respect to the vacuum level.
(D"
~
c8 *~'
~A~b 6 7 -
n~
-,
Gdsb 4 4
5
Damon
5i Ge
Si
I
-
-
Z~e
~ 6
W/
7 E . theory (eV)
8
9
Fig. 2. Comparison of the calculated valence band maximum energy and the measured photoelectric threshold for diamond and zincblende semicnoductors. 0o -------------------------------------
Vacuum level
-2 -Ec oEp-U/2
0-
> 4 gap -6 a W-8 -10-
Fig. 3
E
C
Si
Ge
Sn
Variation of band energies for group IV elements. 778
Here, the 2s and 2p valence orbitals of carbon hybridise to produce the sp3 hybrids on each atom which interact to form the bonding (a) and antibonding (a*) states. The or state broadens into the valence band while the o* state broadens into the conduction band of the solid. Now, the valence states must lie below the vacuum level Evc. However, there is no such requirement on the o* or conduction states. For example, the or* states of methane lie above Evc. The conducton band minimum (CBM) E. can lie below or above Ev.. A simple analysis of the EA can be given in terms of atomic orbitals using tightbinding theory [13,141. We express Ec as lying at the band gap energy E. above the
valence band maximum (CBM) Ev. This valence state is a bonding p state of r 25' symmetry in all diamond/zincblende semiconductor crystals. Its energy is given by E(F25) = Vo
+
(V22 + V32)11
where Vo = /2 (E(p,c) + E(p,a)) the average of the p orbital energy on the cation and anion sites for a zincblende semiconductor like c-BN, V3 = Y2(E(p,c)-E(p,a)) is 0the2 ionic energy. V2 is the covalent energy, which is given empirically by V2 = V 2 /d where d is the bond length and V20 = 18.6 eV/A&2 [141. The orbital energies E(p) etc are the orbital energies in the solid which are given in terms of the ionisation potentials of the free atom and the intra-atom correlation energy U by [13-1 5] E(p) = E-
Y2 U
This treatment has been found to give a reasonably successful accurate value of E, below Evac which experimentally is the photoelectric threshold (Fig. 2)[13,141. However, the model is still incomplete as the predictions are less successful for first row systems like diamond. We now apply this model to the case of diamond and amorphous C in particular. The low EA of diamond is actaully rather surprising, if one considers the
work function of metals varies linearly with the electronegativity [1 61, and that C is the most electronegative of the group IV elements. To understand this further, we plot in Fig. 3 the values of E, and Ec for the group IV elements. We see that E, follows the p orbital energy Ep which does decline from Sn to C. In contrast, Er rises sharply from Si to diamond because of diamond's very wide band gap. Thus, diamond has the lowest electron affinity of the group IV semiconductors, despite having the largest electronegativity, because it has the widest gap. A second ques
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