Dressel.Gruner.Electrodynamics of Solids.2003

formula (6.5.3) for isolated atoms; the energy levels are close to the top of the valence band for p-type impurities or below the bottom of the conduction band for n-doped material, as in the case of Si:P. Because of valley–orbit splitting, the optical absorption shows a complicated set of narrow lines located in the spectral range of a few millielectronvolts, in good accord with our estimations in Section 6.5.2. The results [Tho81] are displayed in Fig. 13.13b. For small impurity concentrations, the sharp lines correspond to transitions between atomic levels; the broad maxima above 400 cm−1 correspond to transitions to the conduction band. With increasing impurity density the lines become broadened due to the interaction between neighboring donor states; because of the random donor positions this is most likely an inhomogeneous broadening. Further increase in the phosphorus concentration leads to further broadening; and the appropriate model is that of donor clusters, with progressively more and more impurity wavefunctions overlapping as the average distance between the donors becomes smaller. Because of the large spatial extension of the phosphorus impurity states, all this occurs at low impurity concentrations. A detailed account of these effects for Si:P is given by Thomas and coworkers [Tho81], but studies on other dopants or other materials can also be interpreted using this description.

For increasing impurity concentration N , the dc conductivity and also various thermodynamic and transport measurements provide evidence for a sharp zero temperature transition to a conducting state. In contrast to classical phase transitions with thermal fluctuations, this is regarded as a quantum phase transition. The delocalization of the donor states increases, and at a critical concentration of Nc = 3.7 × 1018 cm−3 in the case of Si:P an impurity band is formed. The optical conductivity [Gay93] for samples which span this transition is displayed in Fig. 13.13a. On the insulating side the conductivity is typical of that of localized states: zero conductivity at low frequencies with a smooth increase to a maximum above which the conductivity rolls off at high frequency according to ω−2 (the frequency of this Drude roll-off is nevertheless small because of the small donor concentration). The observed peaks are associated with the transitions already seen for lower donor concentrations (Fig. 13.13b). On the metallic side, the conductivity is of Drude type, with all its ramifications, such as the conductivity which extrapolates to a finite dc value, and absorption which is proportional to the square of the frequency (both also seen in a variety of highly doped semiconductors). Transitions between the various donor states lead to additional complications which can nevertheless be accounted for. While the insulator–metal transition is clearly seen by the sudden change of the conductivity extrapolated to zero frequency σ1(ω → 0), the optical sum rule is obeyed both below and above the transition; the integral

Fig. 13.14. Reflectance spectra R(ω) of insulating and metallic Si:P samples determined at T = 10 K (after [Gay93]). The transition from the insulating state to a metallic conduction state occurs at the impurity concentration of Nc = 3.7 × 1018 cm−3.

is proportional to the donor concentration N at both sides of the critical density Nc. The reflectivity R(ω) of samples near to the critical concentration, displayed in Fig. 13.14, provides evidence for the metal–insulator transition. While there is no plasma edge for samples below Nc [Gay93], in the conducting state a well defined plasma edge appears in the optical reflectivity; the impurity concentration where this appears is in full agreement with the sum rule given above. Increasing the impurity concentration further, we find that the impurity states merge with the conduction band. This is evidenced, for instance, by the plasma edge which can be analyzed to give the full number of states in the silicon band in contrast to the number of impurity states. Optical studies of the so-called critical region, very close to the critical concentration (studies such as those conducted on the system Nbx Si1−x and discussed in Section 12.2.3) have not been performed to date.

13.2.2 Electron–phonon and electron–electron interactions

Interactions between electrons and also electron–lattice interactions may lead to a non-conducting state even for a partially filled electron band which, in the absence of these interactions, would be a metal. Several routes to such non-conducting states – which we call a semiconducting state – have been explored; and the emergence of these states depends, broadly speaking, on the relative importance of the kinetic energy of the electron gas, and the interaction energy.

The former is usually cast in the form of

H = ti j ci+,σ c j,σ + c+j,σ ci,σ , (13.2.2) i, j

where ti j is the transition matrix element between electron states i and j, and the terms in the parentheses describe electron transitions between sites i and j; the spin is indicated by σ . In general this Hamiltonian is treated in the tight binding approximation. Often only nearest neighbor interactions are included, and in this case the transfer integral is t0. Electron–electron interactions are described, as a rule, by the Hamiltonian

H = U ni,σ ni,−σ , (13.2.3)

i

where U represents the Coulomb interaction between electrons residing at the same site. This term would favor electron localization by virtue of the tendency for electrons to avoid each other. When electron–lattice interactions are thought to be important, they are accounted for by the Hamiltonian

where u refers to the lattice position. Here the spin of the electrons can, at first sight, be neglected; α is the electron–lattice coupling constant. The consequences of these interactions have been explored in detail for one-dimensional lattices, with nearest neighbor electron transfer included. In this case the Hamiltonian reads

For a half filled band both interactions lead – if the interactions are of sufficient strength – to a non-conducting state, the essential features of which are indicated in Fig. 13.15. Coulomb interactions lead – if U > W (the bandwidth given by Eq. (13.2.2)) – to a state with one electron localized at each lattice site, with an antiferromagnetic ground state in the presence of spin interactions. The state is usually referred to as a Mott–Hubbard insulator. The lattice period, which is defined as the separation of spins with identical orientation, is doubled. The

(c)

(d)

Fig. 13.15. Simple representation of correlation driven semiconducting states for a half filled band. (a) Coulomb correlations lead to the magnetic ground state and (b) electron– lattice interactions to the state with a bond alternation, or (c) to a lattice with a period doubling. In (d) a soliton state is shown, in the usual representation appropriate for the polymer trans-(CH)x .

period of this broken symmetry ground state is 2a; thus it is commensurate with the lattice period a. The broken symmetry state which arises as a consequence of electron–lattice interactions represents either a displacement of the ionic positions or an alternating band structure, such as shown in Fig. 13.15; such states are referred to as Peierls insulators. Again the structure is commensurate with the underlying lattice period. In contrast to incommensurate density waves which develop in a partially filled band (see Chapter 7), here the order parameter is real. Consequently phase oscillations of the ground state do not occur; instead, non-linear excitations of the broken symmetry states – domain walls or solitons – are of importance.

The single particle gap in the insulating state – for strong interactions – is

U − W ,

where U is the strength of the electron–electron or electron–lattice interaction and W is the bandwidth; for weaker interactions is a non-analytic function of the parameters U and W . The optical properties of these states are – in the absence of non-linear excitations – those of a semiconductor with σ1(ω) depending on the dimensionality of the electron structure, as discussed before. This is fairly straightforward and also comes out of calculations.