View source for Transport properties of ballistic and diffusive nanowires: A nonequilibrium Green function approach

The project explores phase-coherent transport, as well as influence of dephasing, in quasi-one-dimensional nanowires patterned using 2DEG. The wires are modeled using a simple square tight-binding lattice. This is achieved by using nonequilibrium Green function techniques to implement Landauer-Buttiker scattering formuals for steady state transport quantities.

== Phase-coherent transport and resonant transmission in 1D nanowires ==

[[Image:two_terminal.png||thumb|left|300px|FIGURE 1: Graphical illustration of the tight-binding model of a two-terminal 1D nanowire device.]]

A nanowire is modeled as tight-binding chain of 40 atoms (with single s-orbital per atom). The chain is attached to two semi-infinite ideal electrodes modeled by the same chain and with the hopping parameter <math> t </math> between the atoms in the lead and in the chain, as well as between the lead and the wire, being identical. This setup is illustrated in Fig. 1.


a) Compute conductance of this device, local density of states on atom 3 of the nanowire, and total density of states as a function of the Fermi energy <math> E_F=[-3t,3t] </math>.


b) Introduce two potential barriers of strength <math> V_0 = 1t </math> on atoms 5 and 36. Compute conductance of this device, local density of states on atom 3 of the nanowire, and total density of states as a function of the Fermi energy <math> E_F=[-3t,3t] </math>.

== Dephasing effects in 1D nanowires via Buttiker voltage probe method ==

[[Image:three_terminal.png||thumb|left|300px| FIGURE 2: Graphical illustration of the tight-binding model of a three-terminal 1D nanowire device.]]

The device in 1(b) exhibits resonant oscillatory transmission with resonances (<math> T(E_n) \approx 1 </math>) at discrete energies <math> E_n </math> as the hallmark of quantum interference effects in transport through phase-coherent nanostructures amenable to Landauer-Buttiker scattering approach. Buttiker as also introduced a phenomenological model of dephasing which should wash out such oscillations (as it happens in devices larger than the phase-coherent length) without having to model microscopic processes responsible for dephasing (such as electron-electron or electron-phonon interactions). 

In the so-called Buttiker voltage probe dephasing model, a third electrode (one can also additional such electrodes) is attached on atom 20 in the middle of the nanowire, as illustrated in Fig. 2, so that electron now have two possibilities to propagate from left electrode 1 to right electrode 2:

*electron propagates from 1 to 2 along direct "Feynman path" in phase-coherent fashion
*electron propagates from 1 to 3 and then from 3 to 2. In this case, it is assumed that electrode 3 will guide electron into the macroscopic reservoirs which will destroy memory of its quantum-mechanical phase. Thus, the "Feynman path" <math> 1 \rightarrow 3 \rightarrow 2 </math> is incoherent. 

The third-electrode is assumed to be a voltage probe, which means that its voltage <math> V_3 </math> is adjusted so that it does not draw any current, <math> I_3=0 </math>. That is, its voltage has to be adjusted so that current from 1 to 2 is exactly the same as current from 2 to 3. 

The multiprobe version of the Landauer-Buttiker formulae:

<math> I_p = \sum_q (G_{qp} V_p - G_{pq} V_q) </math> 

expresses current in lead <math> p </math> as the difference between current <math> \sum_q G_{qp} V_p </math> flowing out of this lead and current  <math> \sum_q - G_{pq} V_q </math> flowing from each lead <math> q </math> into lead <math> p </math>. Here <math> G_{pq} </math> are the conductance coefficients which can be expressed through the transmission function <math> T_{pq} (E) </math> describing probability of electrons to propagate from lead <math> q </math> to lead <math> p </math>:

<math> G_{pq} = \frac{2e^2}{h} T_{pq}(E) = \mathrm{Tr} \, [\mathbf{\Gamma}_p \mathbf{G}^r_{pq} \mathbf{\Gamma}_q (\mathbf{G}^r_{pq})^\dagger] </math>

The last expression shows how to compute <math> T_{pq}(E) </math> using NEGF technique where the level broadening matrix <math> \mathbf{\Gamma}_p = i(\mathbf{\Sigma}_p - \mathbf{\Sigma}_p^\dagger) </math> is computed from the self-energies <math> \mathbf{\Sigma}_p </math> attached to atom <math> p </math> and <math> \mathbf{G}^r_{pq}</math> is the Green function matrix element connecting atoms <math> q </math> and <math> p </math>. 


a) Using these equations, evaluate Buttiker's expression of the transmission function <math> \tilde{T}_{21} </math> between electrodes 1 and 2 in the presence of the dephasing third electrode:

<math> \tilde{T}_{21}(E) = T_\mathrm{coherent}(E) + T_\mathrm{incoherent}(E) = T_{21}(E) + \frac{T_{23}(E) T_{31}(E)}{T_{31}(E)+T_{23}(E)} </math>

for the same nanowires with two barriers studied in 1(b). 


b) Comment on the difference between <math> \tilde{T}_{21}(E) </math> and <math>  T_{21}(E) </math> where the latter is computed in 1(b) in the absence of dephasing.

NOTE: For derivation of Buttiker's expression for <math> \tilde{T}_{21}(E) </math> see Datta's textbook Chapter 9, equation (9.5.7) and illustration in Fig. 9.5.8.

== Distribution of transmission eigenvalus in diffusive nanowires ==

To model the  diffusive transport regime through quasi-1D nanowires, we adopt Anderson Hamiltonian:


<math> \hat{H} = \sum_m \varepsilon_m |m \rangle \langle m| -  t \sum_{\langle m n \rangle}  |m \rangle \langle n | </math>


as a tight-binding model defined on a square lattice <math> 50 \times 50 </math> with nearest neighbor hopping <math> t </math> and disorder introduced as the on-site random potential  <math> \varepsilon_{m} \in [-W/2, W/2] </math>. This sample is attached to two semi-infinite ideal (i.e., disorder and interaction free) electrodes modeled on the same square lattice. Set the Fermi energy to <math> E_F=-0.5 t </math> and disorder strength to <math> W=1.5t</math> to ensure the diffusive transport regime.


a) To test the correctness of you m-file, plot the conductance <math> G(E_F) = G_Q T(E_E) </math> vs. <math> E_F </math> for a <math> 3 \times 3 </math> lattice in the interval <math> E_F \in [-4t,4t] </math>. You should see confirm that <math> G(E_F) </math> is quantized since two semi-infinite electrodes and the central region together form an infinite ideal wire with maximum of 3 open conducting channels. 


b) Using the NEGF expression <math> \mathbf{t}(E_F) \mathbf{t}^\dagger(E_F) = \mathbf{\Gamma}_2 \mathbf{G}^r_{21} \mathbf{\Gamma}_1 (\mathbf{G}^r_{21})^\dagger </math> for the product of the Landauer-Buttiker transmission matrix <math> \mathbf{t}(E_F)</math> and its Hermitian conjugate, compute transmission eigenvalues (in the chosen examples there are 50 of them) for a single realization of disorder and plot their histogram.


c) Repeat b) for at least 10 samples with different realizations of disorder and plot the disorder-averaged histogram of <math> T_n </math> as the estimate for the distribution of 
transmission eigenvalues <math> P(T) </math>.


d) Add analytical expressions (the Dorokhov formula)

<math> P(T) = \frac{\langle G \rangle}{2G_Q} \frac{1}{T\sqrt{1-T}} </math>.

on the same graph as in c) and compare numerical and analytical distributions.

== References ==

* [[Computing]] contains the following m-files that will assist you: 
** '''qt_1d.m''' is a program which computes the conductance and density of states for 1D nanowire
** '''self_energy.m''' is a function which returns the self-energy for a semi-infinite electrode modeled on the square lattice. 
* Datta Chapters 9,10.