### Theoretical model and propagating states

The schematic representation of the system under study with an arbitrary nanojunction region is presented in Figure 1. With reference to the Landauer-Büttiker theory for the analysis of the electronic scattering processes [47, 48], this system is divided into three main parts, namely the finite silicon-doped carbon wire nanojunction region, made up of a given composition of carbon (black) and silicon (orange) atoms, and two other regions to the right and left of the nanojunction which are semi-infinite quasi one-dimensional carbon leads. Moreover, for the purpose of quantum conductance calculations, the so-called irreducible region and the matching domains are depicted (see the ‘Phase field matching theory’ subsection for more details). Figure 1 is used throughout the ‘Methods’ section as a graphical reference for analytical discussion.

The system presented in Figure

1 is described by the general tight-binding Hamiltonian block matrix:

$H=\left[\begin{array}{cccc}\ddots & \cdots & 0& 0\\ \vdots & {E}_{N-1,N-1}& {H}_{N,N-1}^{\u2020}& 0& 0\\ 0& {H}_{N,N-1}& {E}_{N,N}& {H}_{N+1,N}^{\u2020}& 0\\ 0& 0& {H}_{N+1,N}& {E}_{N+1,N+1}& \vdots \\ 0& 0& \cdots & \ddots \end{array}\right].$

(1)

This is defined in general for a system of *N*_{
x
}inequivalent atoms per unit cell, where *N*_{
l
}denotes the number of basis orbitals per atomic site, assuming spin degeneracy. In Equation 1, E_{i,j} denotes on-diagonal matrices composed of both diagonal ${\epsilon}_{l}^{n,\alpha}$ and off-diagonal ${h}_{l,{l}^{\prime},m}^{n,{n}^{\prime},\beta}$ elements for a selected unit cell. In contrast, the H_{i,j}matrices contain only off-diagonal elements for interactions between different unit cells. The index *α* identifies the atom type, C or Si, on the *n* th site in a unit cell. Each diagonal element is characterized by the lower index *l* for the angular momentum state. The off-diagonal elements ${h}_{l,{l}^{\prime},m}^{n,{n}^{\prime},\beta}$ describe the *m*-type bond, (*m*=*σ*,*Π*), between *l* and *l*^{
′
} nearest-neighbor states. The index *β* identifies the types of interacting neighbors, C-C, Si-Si, or Si-C.

The

${h}_{l,{l}^{\prime},m}^{n,{n}^{\prime},\beta}$ elements are consistent with the Slater-Koster convention [

66] and may be expressed in the framework of the HTBT [

67] by the following:

${h}_{l,{l}^{\prime},m}^{n,{n}^{\prime},\beta}={\eta}_{l,{l}^{\prime},m}\frac{{\hslash}^{2}}{{m}_{e}{d}_{\beta}^{2}},$

(2)

where ${\eta}_{l,{l}^{\prime},m}$ values are the dimensionless Harrison coefficients; *m*_{
e
}, the electron mass in vacuum; and *d*_{
β
}, the interatomic distance for interacting neighbors. Explicit forms of the E_{i,j} and H_{i,j} matrices are given in Appendix Appendix 1. The tight-binding parameter schemes are illustrated in Figure 1; however, it is noteworthy that the *n* and *n*^{
′
}indices for coupling parameters are dropped for simplicity in this figure.

In our calculations, the single-particle electronic wave functions are expanded in the orthonormal basis of local atomic wave functions

*ϕ*_{
l
}(

**r**) as follows:

$\Psi (r,k)=\sum _{l,n,N}{c}_{l}({r}_{n}-{R}_{N},k){\varphi}_{l}(r-{R}_{N},k).$

(3)

In Equation

3,

k is the real wave vector;

R_{
N
}, the position vector of the selected unit cell; and

R_{
N
}, the position vector of the

*n* th atom in the selected unit cell. For ideal leads, the wave function coefficients

*c*_{
l
}(

r_{
n
}−

R_{
N
},

k) are characterized under the Bloch-Floquet theorem in consecutive unit cells by the following phase relation:

${c}_{l}({r}_{n}-{R}_{N+1},k)=z{c}_{l}({r}_{n}-{R}_{N},k),$

(4)

where

*z* is the phase factor

${z}_{\pm}={e}^{\pm ik{R}_{N}},$

(5)

which corresponds here to waves propagating to the right (+) or to the left (−).

The electronic equations of motion for a leads unit cell, independent of

*N*, may be expressed in a square matrix form, with an orthonormal minimal basis set of local wave functions as follows:

$(EI-{M}_{d})\times c(k,E)=0.$

(6)

*E* stands for the electron eigenvalues, and

I is the identity matrix, while the dynamical matrix

M_{
d
}contains the Hamiltonian matrix elements and the

*z* phase factors;

c(

k,

*E*) is the

*N*_{
x
}×

*N*_{
l
} size vector defined as follows:

$c(k,E)=\left[\begin{array}{l}{c}_{s}({r}_{1},k,E)\\ {c}_{{p}_{x}}({r}_{1},k,E)\\ \vdots \\ {c}_{{p}_{y}}({r}_{n},k,E)\\ {c}_{{p}_{z}}({r}_{n},k,E)\end{array}\right]\equiv \left[\begin{array}{l}{c}_{l}({r}_{1},k,E)\\ \vdots \\ {c}_{l}({r}_{n},k,E)\end{array}\right].$

(7)

Equation 6 gives the *N*_{
x
}×*N*_{
l
}eigenvalues with corresponding eigenvectors which determine the electronic structure of the lead system, where *l* under the vector c_{
l
}corresponds to *N*_{
l
}=4 orbitals *s*,*p*_{
x
},*p*_{
y
},*p*_{
z
}. Note that the choice of an orthonormal minimal basis set of local wavefunctions may result initially in an inadequate description of the considered electronic eigenvalues. However, as can be seen later, the proper choice of the TB on-site energies and coupling terms allows us to to obtain agreement with the DFT results. This is a systematic procedure in our calculations.

### Evanescent states

The complete description of electronic states on the ideal leads requires a full understanding of the propagating and evanescent electronic states on the leads. This arises because the silicon-doped nanojunction breaks the perfect periodicity of the infinite leads and forbids a formulation of the problem only in terms of the pure Bloch states as given in Equation

5. Depending on the complexity of a given electronic state, it follows that the evanescent waves may be defined by the phase factors for a purely imaginary wave vectors

k=

*i* κ such that

$z={z}_{\pm}={e}^{\mp \kappa {r}_{n}},$

(8)

or for complex wave vectors

k=

κ_{1} +

*i* κ_{2}such that

$z={z}_{\pm}={e}^{\mp (i{\kappa}_{1}-{\kappa}_{2}){r}_{n}}.$

(9)

The phase factors of Equations 8 and 9 correspond to pairs of hermitian evanescent and divergent solutions on the leads. Only the evanescent states are physically considered where spatial evanescence occurs to the right and left, away from the nanojunction localized states. It is important to note that the *l*-type evanescent state corresponds to energies beyond the propagating band structure for this state.

The functional behavior of

*z*(

*E*) for the propagating and evanescent states on the leads may be obtained by various techniques. An elegant method presented previously for phonon and magnon excitations [

59] is adapted here for the electrons. It is described on the basis of Equations

4 and

6 by the generalized eigenvalue problem for

*z*:

$\begin{array}{l}\left[\left[\begin{array}{ll}EI-{E}_{N,N}& {H}_{N,N-1}\\ I& 0\end{array}\right]-z\left[\begin{array}{ll}-{H}_{N,N-1}^{\u2020}& 0\\ 0& I\end{array}\right]\right]\phantom{\rule{2em}{0ex}}\\ \times \left[\begin{array}{l}c({R}_{N},z,E)\\ c({R}_{N-1},z,E)\end{array}\right]=0.\phantom{\rule{2em}{0ex}}\end{array}$

(10)

Equation 10 gives the 2*N*_{
x
}*N*_{
l
}eigenvalues as an ensemble of *N*_{
x
}*N*_{
l
}pairs of *z* and *z*^{−1}. Only solutions with |*z*|=1 (propagating waves) and |*z*|<1 (evanescent waves) are retained as physical ones. In Equation 10, k is then replaced by the appropriate energy *E* variable. Furthermore, for systems with more than one atom per unit cell, the matrices H_{N,N−1} and ${H}_{N,N-1}^{\u2020}$ in this procedure are singular. In order to obtain the physical solutions, the eigenvalue problem of Equation 10 is reduced from the 2*N*_{
x
}*N*_{
l
} size problem to the appropriate 2*N*_{
l
}one, using the partitioning technique (please see Appendix Appendix 2).

### Phase field matching theory

The scattering problem at the nanojunction is considered next. An electron incident along the leads has a given energy *E* and wave vector k, where *E*=*E*_{
γ
}(k) denotes the available dispersion curves for *γ* = 1, 2,.., *γ* propagating eigenmodes, where *γ* corresponds to the total number of allowed solutions for the eigenvalue problem of phase factors in Equation 10. In any given energy interval, however, these may be evanescent or propagating eigenmodes and together constitute a complete set of available channels necessary for the scattering analysis.

The irreducible domain of atomic sites for the scattering problem includes the nanojunction domain itself, (*N*∈[0,*D*−1]), and the atomic sites on the left and right leads which interact with the nanojunction, as in Figure 1. This constitutes a necessary and sufficient region for our considerations, i.e., any supplementary atoms from the leads included in the calculations do not change the final results. The scattering at the boundary yields then the coherent reflected and transmitted fields, and in order to calculate these, we establish the *system* of equations of motion for the atomic sites (*N*∈[−1,*D*]) of the irreducible nanojunction domain.

This procedure leads to the following general matrix equation:

${M}_{\mathrm{nano}}\times V=0.$

(11)

M_{
nano
} is a (

*D* + 2)×(

*D* + 4) matrix composed of the block matrices

$(EI-{E}_{N,N}-{H}_{N,N-1}-{H}_{N,N-1}^{\u2020})$, and the state vector

V of dimension

*D* + 4 is given as follows:

$V=\left[\begin{array}{l}{c}_{l}({r}_{1}-{R}_{-2},E)\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{-2},E)\\ \vdots \\ \vdots \\ {c}_{l}({r}_{1}-{R}_{D+1},E)\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{D+1},E)\end{array}\right].$

(12)

Since the number of unknown coefficients in Equation 11 is always greater than the number of equations, such a set of equations cannot be solved directly.

Assuming that the incoming electron wave propagates from left to right in the eigenmode

*γ* over the interval of energies

*E*=

*E*_{
γ
}, the field coefficients on the left and right sides of the irreducible nanojunction domain may be written as follows:

$\begin{array}{ll}{c}_{l}^{L}({r}_{n}-{R}_{N},{z}_{\gamma},{E}_{\gamma})& ={c}_{l}({r}_{n},{z}_{\gamma},{E}_{\gamma}){z}_{\gamma}^{-N}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+\sum _{{\gamma}^{\prime}}^{\Gamma}{c}_{l}({r}_{n},{z}_{{\gamma}^{\prime}},{E}_{\gamma}){z}_{{\gamma}^{\prime}}^{N}{r}_{\gamma ,{\gamma}^{\prime}}\left({E}_{\gamma}\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}N\le -1,\phantom{\rule{2em}{0ex}}\end{array}$

(13)

$\begin{array}{ll}{c}_{l}^{R}({r}_{n}-{R}_{N},{z}_{\gamma},{E}_{\gamma})& =\sum _{{\gamma}^{\prime}}^{\Gamma}{c}_{l}({r}_{n},{z}_{{\gamma}^{\prime}},{E}_{\gamma}){z}_{{\gamma}^{\prime}}^{N}{t}_{\gamma ,{\gamma}^{\prime}}\left({E}_{\gamma}\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}N\ge D,\phantom{\rule{2em}{0ex}}\end{array}$

(14)

where *γ*^{
′
}∈*Γ* is an arbitrary channel into which the incident electron wave scatters, and c_{
l
}(r_{
n
},*z*_{
γ
},*E*_{
γ
}) denotes the the eigenvector of the lead dynamical matrix of Equation 6 for the inequivalent site *n* at *z*_{
γ
} and *E*_{
γ
}. The terms ${r}_{\gamma ,{\gamma}^{\prime}}$ and ${t}_{\gamma ,{\gamma}^{\prime}}$ denote the scattering amplitudes for backscattering and transmission, respectively, from the *γ* into the *γ*^{
′
}eigenmodes and constitute the basis of the Hilbert space which describes the reflection and transmission processes.

Equations 13 and 14 are next used to transform the (

*D* + 2)×(

*D* + 4) matrix of the system of equations of motion, Equation

11, into an inhomogeneous (

*D* + 2)×(

*D* + 2) matrix for the scattering problem. This procedure leads to the new form of the following vector:

$\begin{array}{ll}V& =\left[\begin{array}{lllllll}{z}^{2}& 0& \cdots & \cdots & \cdots & \cdots & 0\\ z& 0& \vdots \\ 0& 1& \vdots \\ \vdots & \ddots & \vdots \\ \vdots & 1& \vdots \\ \vdots & \ddots & \vdots \\ \vdots & 1& 0\\ \vdots & 0& z\\ 0& \cdots & \cdots & \cdots & \cdots & 0& {z}^{2}\end{array}\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.56865pt}{0ex}}\phantom{\rule{2.56865pt}{0ex}}\phantom{\rule{2.56865pt}{0ex}}\times \left[\begin{array}{l}{r}_{\gamma ,{\gamma}^{\prime}}\\ {c}_{l}({r}_{1}-{R}_{0},{E}_{\gamma})\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{0},{E}_{\gamma})\\ \vdots \\ \vdots \\ {c}_{l}({r}_{1}-{R}_{D-1},{E}_{\gamma})\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{D-1},{E}_{\gamma})\\ {t}_{\gamma ,{\gamma}^{\prime}}\end{array}\right]+\left[\begin{array}{l}{c}_{l}({r}_{1},{z}_{\gamma},{E}_{\gamma}){z}_{\gamma}^{-2}\\ \vdots \\ {c}_{l}({r}_{n},{z}_{\gamma},{E}_{\gamma}){z}_{\gamma}^{-2}\\ {c}_{l}({r}_{1},{z}_{\gamma},{E}_{\gamma}){z}_{\gamma}^{-1}\\ \vdots \\ {c}_{l}({r}_{n},{z}_{\gamma},{E}_{\gamma}){z}_{\gamma}^{-1}\\ 0\\ \vdots \\ \vdots \\ 0\end{array}\right].\phantom{\rule{2em}{0ex}}\end{array}$

(15)

The rectangular sparse matrix in Equation 15 has the (*D* + 4)×(*D* + 2) size. The vectors ${r}_{\gamma ,{\gamma}^{\prime}}$ and ${t}_{\gamma ,{\gamma}^{\prime}}$ are column vectors of the backscattering and transmission Hilbert basis.

Substituting Equation

15 into Equation

11 yields an inhomogeneous system of equations as follows:

$M\times \left[\begin{array}{l}{r}_{\gamma ,{\gamma}^{\prime}}\\ {c}_{l}({r}_{1}-{R}_{0},{E}_{\gamma})\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{0},{E}_{\gamma})\\ \vdots \\ \vdots \\ {c}_{l}({r}_{1}-{R}_{D-1},{E}_{\gamma})\\ \vdots \\ {c}_{l}({r}_{n}-{R}_{D-1},{E}_{\gamma})\\ {t}_{\gamma ,{\gamma}^{\prime}}\end{array}\right]=-\left[\begin{array}{l}{M}_{1}^{\mathrm{in}}\\ {M}_{2}^{\mathrm{in}}\\ 0\\ \vdots \\ 0\end{array}\right].$

(16)

In Equation 16, M is the *matched*(*D* + 2)×(*D* + 2) square matrix, and the vector of dimension (*D* + 2) which incorporates the ${M}_{1}^{\mathrm{in}}$ and ${M}_{2}^{\mathrm{in}}$ elements, regroups the inhomogeneous terms of the incident wave. The explicit forms of the M matrix elements and and ${M}_{N}^{\mathrm{in}}$ vectors are presented in Appendix Appendix 3.

In practice, Equation 16 can be solved using standard numerical procedures, over the entire range of available electronic energies, yielding the coefficient c_{
l
} for atomic sites on the nanojunction domain itself as well as the *γ* reflection ${r}_{\gamma ,{\gamma}^{\prime}}\left(E\right)$ and the *γ* transmission ${t}_{\gamma ,{\gamma}^{\prime}}\left(E\right)$ coefficients.

The reflection and transmission coefficients give the reflection

${r}_{\gamma ,{\gamma}^{\prime}}\left(E\right)$ and transmission

${t}_{\gamma ,{\gamma}^{\prime}}\left(E\right)$ probabilities, respectively, by normalizing with respect to their group velocities

*v*_{
γ
} in order to obtain the unitarity of the scattering matrix as follows:

${R}_{\gamma ,{\gamma}^{\prime}}\left(E\right)=\frac{{v}_{{\gamma}^{\prime}}}{{v}_{\gamma}}{\left|{r}_{\gamma ,{\gamma}^{\prime}}\left(E\right)\right|}^{2},$

(17)

${T}_{\gamma ,{\gamma}^{\prime}}\left(E\right)=\frac{{v}_{{\gamma}^{\prime}}}{{v}_{\gamma}}{\left|{t}_{\gamma ,{\gamma}^{\prime}}\left(E\right)\right|}^{2},$

(18)

where *v*_{
γ
}≡*v*_{
γ
}(*E*) denotes the group velocity of the incident electron wave in the eigenmode *γ*. The group velocities are calculated by a straightforward procedure as in Appendix Appendix 4. For evanescent eigenmodes, ${v}_{{\gamma}^{\prime}}=0$. Although the evanescent eigenmodes do not contribute to the electronic transport, they are required for the complete description of the scattering processes.

Furthermore, using Equations

17 and

18, the overall reflection probability,

*R*_{
γ
}(

*E*), for an electron incident in the

*γ* eigenmode and the total electronic reflection probability,

*R*(

*E*), from all the eigenmodes may be expressed, respectively, as follows:

${R}_{\gamma}\left(E\right)=\sum _{{\gamma}^{\prime}}^{\Gamma}{R}_{\gamma ,{\gamma}^{\prime}}\left(E\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}R\left(E\right)=\sum _{\gamma}^{\Gamma}{R}_{\gamma}\left(E\right).$

(19)

Similarly, for transmission probabilities, we may write the equivalent equations as follows:

${T}_{\gamma}\left(E\right)=\sum _{{\gamma}^{\prime}}^{\Gamma}{T}_{\gamma ,{\gamma}^{\prime}}\left(E\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}T\left(E\right)=\sum _{\gamma}^{\Gamma}{T}_{\gamma}\left(E\right).$

(20)

The

*T*_{
γ
}(

*E*) and

*T*(

*E*) probabilities are very important for the electronic scattering processes since they correspond directly to the experimentally measurable observables. Likewise, the total transmission

*T*(

*E*_{
γ
}) allows to calculate the overall electronic conductance. In this work, we assume the zero-bias limit and write the total conductance in the following way:

$G\left({E}_{F}\right)={G}_{0}T\left({E}_{F}\right).$

(21)

In Equation 21, *G*_{0} is the conductance quantum and equals 2*e*^{2}/h. Due to the Fermi-Dirac distribution, *G*(*E*_{
F
}) is calculated at the Fermi level of the perfect lead band structure since electrons only at this level give the important contribution to the electronic conductance. The Fermi energy can be determined using various methods where, in the present work, *E*_{
F
} is calculated as the basis of the density of state calculations.