Gunn-Hilsum Effect in Mechanically Strained Silicon Nanowires: Tunable Negative Differential Resistance

Gunn (or Gunn-Hilsum) Effect and its associated negative differential resistivity (NDR) emanates from transfer of electrons between two different energy subbands. This effect was observed in semiconductors like GaAs which has a direct bandgap of very low effective mass and an indirect subband of high effective mass which lies ~300 meV above the former. In contrast to GaAs, bulk silicon has a very high energy spacing (~1 eV) which renders the initiation of transfer-induced NDR unobservable. Using Density Functional Theory (DFT), semi-empirical 10 orbital (sp3d5s*) Tight Binding and Ensemble Monte Carlo (EMC) methods we show for the first time that (a) Gunn Effect can be induced in silicon nanowires (SiNW) with diameters of 3.1 nm under +3% strain and an electric field of 5000 V/cm, (b) the onset of NDR in the I-V characteristics is reversibly adjustable by strain and (c) strain modulates the resistivity by a factor 2.3 for SiNWs of normal I-V characteristics i.e. those without NDR. These observations are promising for applications of SiNWs in electromechanical sensors and adjustable microwave oscillators. It is noteworthy that the observed NDC is different in principle from Esaki-Diode and Resonant Tunneling Diodes (RTD) in which NDR originates from tunneling effect.

a. Electron-LA phonon scattering Figure A1 shows how an electron at the bottom of the indirect conduction band can scatter into many available secondary states within E Debye window. If the rate of each scattering event is called , then the total scattering rate of the electron at k z is found by summation over all available secondary states (k ' z ) and phonon wave vectors ( ) i.e.: (A1) Figure A1. Inter-sub band electron-LA phonon scattering events starting from k z .
Total momentum relaxation rate is found by the same equation in which individual rates, W, are weighted by a factor of (1 -k z /k ' z ). Derivations of momentum relaxation rates have been explained in detail in references [1,2]. Here we explain how the total scattering rate (W kz ) as well as each individual scattering rate [ ] are calculated for EMC simulation. Using Fermi's golden rule, the rate of a single scattering event can be written as follows where both momentum and energy are conserved and ψ corresponds to the mixed (electron and phonon) states. (A2) The electron-phonon interaction Hamiltonian for phonons of LA type is given as: Since is the only term which depends on , therefore the integration over can be replaced by which is .
The precise calculation of scattering rate mandates large number of points in [0,2π] interval (N kz >1000). Now the Dirac delta function can be manipulated as: where we have used δ(ax)=δ(x)/a. Replacing with and using Krönecker's delta which we get, The integration over can be simplified more using , where x o is the single can be simplified further: Recalling that is only a function of . Using equation A14 a relation between and can be found as follows: Replacing according to equation A17 and using the sifting property of Dirac's delta function, equation A15 can be reduced to: If we write the integration as a discrete summation over grid points along the 1D BZ, then rewriting equation A17 reveals how it is possible to single out individual rate ( ) between a pair of given states. Recalling that we can write: (A18)

b. Electron-LO phonon scattering
Similar to equation A1, the total electron-LO phonon scattering rate can be written as: The only difference is that individual scattering rate between (at indirect conduction sub-band)

and (at direct conduction sub-band), includes LO phonon with wave vector and it is given by
Fermi's golden rule similar to equation A2: (A20) The electron-LO optical phonon interaction Hamiltonian matrix element is given as: Similar to the case of LA phonons, , and are structure factor, Bose-Einstein factor and frequency of dispersion-less LO phonon, respectively. Following the same procedure given for acoustic phonons, the summations in equation A19 can be simplified as follows: where . To single out the individual scattering rate i.e. , the integration over q z is written in its discrete form and noting that we have: By interchanging summation over k ' z and q p we have: k p are those values of secondary state wave vectors (k ' z ) which satisfy the F(k ' z ) =0. Hence corresponding to each k z (initial state) there are a few secondary states (k p ) to which electron can scatter by absorbing or emitting a LO phonon. And the total rate is simply found by adding each individual term according to equation A26. Albeit the derivative of can be calculated either numerically using finite difference scheme or using analytic derivation assuming effective-mass theory i.e. assuming that each sub band is of parabolic type with a defined effective mass. Both methods return the same values of scattering rates.

c. Post processing of scattering data for EMC simulation
The electron-LA phonon and LO phonon scattering rates (both inter-and intra-sub band) are implemented in MATLAB according to equation A18 and equation A26, respectively. Figure A2 shows the algorithm or pseudo code of saving individual electron-LA phonon scattering rates for initial states within the first sub band (S 1 ). The same method can be generalized to the cases where initial states are in second sub band (S 2 ), 3 rd sub band (S 3 ), and 4 th sub band (S 4 ). Figure A2. Flowchart of saving individual rates for e-LA phonon scattering events.
The MATLAB code which implements this algorithm groups the individual emission/absorption rates and adds them together. Decision is made based on the number of k j (secondary state indices) which 8 determines if the secondary state belongs to band1 (intra-sub band) or band2, 3 and 4 (inter sub band). The concept of the grouping of individual absorption and emission rates and sorting them according to inter-or intra-sub band scattering, were shown in Figure 2 of the manuscript. The algorithm for this decision making is shown in Figure A3. The same algorithm of Figure A3 is applicable to the case of electron-LO phonon scattering (using equation A26) but special care is required when secondary states are saved in the first loop of Figure A2. Figure A3. Algorithm of grouping and sorting individual LA phonon absorption and emission rates.
In contrast to LA phonons which have a continuum of energies from 0 to E Debye , the LO phonons have all a constant energy of E LO =63 meV. Thus if the criteria to choose the secondary state is set to be E i -63meV ≤ E i ≤ E i + 63meV, there may be cases with no phonon due to coarse grid of k z axis. On the other hand by choosing these criteria of E i -63meV±Tol ≤ E i ≤ E i + 63meV±Tol, it is possible to find many secondary states close to each other within the tolerance window. Resolving this problem is done by selecting a unique index among many closely spaced indices. For example if the indices of