How Franz-Keldysh effect, band tailing and low dimensional conductivity in Semiconductors are related?
Conductivity as measured and calculated is a response of the system in the weak-field limit, where the linear-response theory applies; as such, conductivity (conductivity tensor) is independent of the applied electric field. This is not the case with the Franz-Keldysh effect as observed under strong applied electric field (one can explicitly observe the non-linear effects of the applied electric field in the relevant expressions in the context of the Franz-Keldysh effect). Having said that, in considering systems with charged impurities/defects, one observes the Franz-Keldysh effect in the form of the Urbach tail [1,2], however in this case the Franz-Keldysh effect is not due an applied (i.e. external) electric field, but due to the electric field of the charged impurities/defects internal to the system. In such case, the electric field by which one measures the conductivity is not the one responsible for the Franz-Keldysh effect; this effect counts as an inherent property of the system and thus directly influences the conductivity - amongst others, it sets the energy at which photoconductivity takes place. Naturally, when measuring or calculating the differential conductivity under a constant applied electric field, the Franz-Keldysh effect due to the latter constant electric field influences the differential conductivity in the same way that the electric field due to charged impurities/defects does. In other words, the constant electric field around which one measures the differential conductivity is to be counted as internal to the system.
Lastly, and briefly, the dimensionality of the system is of direct influence on various power laws specific to the Franz-Keldysh effect. The extant relevant expressions are generally specific to three-dimensional systems.
[1] F Urbach, Phys. Rev. 92, 1324 (1953).
Article The Long-Wavelength Edge of Photographic Sensitivity and of ...
[2] D Redfield, Phys. Rev. 130, 916 (1963).
Article Effect of Defect Fields on the Optical Absorption Edge
Dr. Behnam Farid
I was working on a system of Te excess CdTe and it showed 1D conductivity.
Absorption spectroscopy showed Franz Keldysh effect.
Can you specify on the power law Franz Keldysh effect specific to ID conductivity?
If suggesting specific reference would make my approach very easy!
Regards
Roshan Mathew
Roshan Mathew : If you consider the original papers by Franz [1] and Keldysh [2], in the relevant expressions you will see momentum integrals over integrands containing Dirac δ-functions whose arguments are differences of one-particle energy dispersions. It is a matter of elementary calculus of integrals that the resulting power laws are directly relateded to the dimension of the underlying momentum space (this is exactly why in 1D interacting electrons cannot be Fermi liquids, and why in 2D there are all over the place logarithimic corrections -- reason why the straightforward Sommerfeld expansion fails in 2D [3], etc.). So, I would recommend you to do the calculations yourself. For the Franz-Keldysh effect in a slab geometry, where the quantum states are confined in the lateral direction, consult [4] (see in particular Figs 1, 2, and 4 herein and notice how for the width of the slab increasing, step-wise constant functions approach towards smooth functions).
[1] W Franz, Z. Naturforsch. 13a, 484 (1958).
[2] LV Keldysh, Sov. Phys. JETP 34, 788 (1958).
[3] NW Ashcroft, and ND Mermin, Solid State Physics (Harcourt, Philadelphia, 1976). Prob. 1, p. 53.
[4] DAB Miller, DS Chemla, and S Schmitt-Rink, Phys. Rev. B 33, 6976 (1986).
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