Digital Circuits Based on Quantum Transistors

Abstract

It is proposed to use transistor-transistor logic (TTL) for future fast, low-power digital electronic circuits. A relaxation quantum transistor can be the basic element of these circuits. This approach allows us to circumvent the difficulties of using field effect transistors based on nanowires.

APPENDIX

MODELING THE CHARACTERISTICS OF A RELAXATION QUANTUM TRANSISTOR

For calculating the characteristics of an RQT, the model described in detail in the book [11] was used. A one-dimensional stationary charge transfer in the 0X direction was considered (see Fig. 1). It was assumed that the nanowires shown in Fig. 1, are parallelepipeds of a square cross section with the sides of the square equal to L(x), where L(x) is a piecewise constant function.

The electron energy was calculated using the approximate formula

$$E\left( {{{k}_{y}},{{k}_{z}},{{p}_{x}}} \right) = {{U}_{{{\text{min}}}}} + \frac{1}{{2{{m}_{e}}}}{{\left( {\frac{{\hbar \pi }}{L}} \right)}^{2}}\left( {k_{y}^{2} + k_{z}^{2}} \right) + \frac{{p_{x}^{2}}}{{2{{m}_{e}}}},$$

(A1.1)

where numbers ${{k}_{y}},{{k}_{z}}~$ = 1, 2, 3… number the levels of dimensional quantization; ${{p}_{x}}$ is the momentum of the electron in the direction 0X; ${{U}_{{{\text{min}}}}}$ is the minimum value of the one-dimensional potential for electrons

$$U\left( x \right) = {{U}_{0}}\left( x \right) - e\varphi \left( x \right),$$

(A1.2)

${{U}_{0}}\left( x \right)$ is the built-in potential determined by band discontinuities at heterointerfaces, and $\varphi \left( x \right)$ is the electrostatic potential.

The self-consistent system of Schrödinger, Poisson, and transport equations was solved numerically:

$$\frac{{{{\hbar }^{2}}}}{2}\frac{d}{{dx}}\left( {\frac{1}{{{{m}_{e}}}}\frac{{d{{\Psi }}\left( {x,{{E}_{x}}} \right)}}{{dx}}} \right) + \left( {{{E}_{x}} - {{U}^{{{\text{eff}}}}}\left( x \right)} \right){{\Psi }}\left( {x,{{E}_{x}}} \right) = 0,$$

(A2.1)

$$\left\{ \begin{gathered} \frac{d}{{dx}}\left( {\varepsilon \frac{{d\varphi }}{{dx}}} \right) = e\left( {{{n}_{e}} + {{n}_{c}} + {{n}_{b}} - N} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,({\text{A}}2.2) \hfill \\ \frac{{d{{J}_{{ec}}}}}{{dx}} = - R,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,({\text{A}}2.3)~ \hfill \\ \frac{{d{{J}_{b}}}}{{dx}} = R.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(A}}2.4) \hfill \\ \end{gathered} \right.$$

Here ${{\Psi }}\left( {x,{{E}_{x}}} \right)$ are wave functions of electrons with energy ${{E}_{x}}$ calculated from the minimum effective potential

$${{U}^{{{\text{eff}}}}}\left( x \right) = U\left( x \right) + \frac{1}{{{{m}_{e}}}}{{\left( {\frac{{\hbar \pi }}{{L\left( x \right)}}} \right)}^{2}},$$

(A3.1)

$\varepsilon$ is the permittivity of the material, expressed in units of the dielectric constant of the vacuum; ${{n}_{e}},~{{n}_{{c~}}},{{n}_{b}}$ is the concentration of electrons entering the structure from the emitter, collector, and base contacts, respectively; $N$ is the concentration of the dopant and holes, which was assumed in the semiconductor to be equal to the intrinsic concentration ${{n}_{i}}$; and in metal, the total charge of electrons; ${{J}_{{ec}}}$ is the total current density of electrons entering the structure from the emitter and collector contacts; and ${{J}_{b}}$, from the base contact; value $R$ describes the rate of relaxation of nonequilibrium electrons to the state of chemical equilibrium, for which the expression is adopted [11]

$$R = \frac{{e\gamma }}{\hbar }\left( {F - {{F}_{b}}} \right)\gamma \left( {{{n}_{e}} + {{n}_{c}} + {{n}_{b}}} \right),$$

(A3.2)

where γ is a dimensionless constant, which in the calculations was considered equal to unity.

For the wave functions of electrons ${{\Psi }}\left( {x,{{E}_{x}}} \right)$, the boundary conditions corresponding to the incident waves of unit amplitude were taken; only the states of the continuous spectrum were taken into account. At points ${{x}_{e}}$ and ${{x}_{c}}$, the solutions were stitched together from the conditions of continuity of the wave functions and their first derivatives.

As a result, at the point ${{x}_{e}}$ at$~{{E}_{x}} > {{U}^{{eff}}}\left( {{{x}_{e}}} \right)$

$$i{{p}_{x}}\left( {{{x}_{e}}} \right){{{{\Psi }}}_{e}}\left( {{{x}_{e}},{{E}_{x}}} \right) + \frac{{\hbar d{{{{\Psi }}}_{e}}\left( {{{x}_{e}},{{E}_{x}}} \right)}}{{dx}} = 2i{{p}_{x}}\left( {{{x}_{e}}} \right){{e}^{{ - i{{p}_{x}}\left( {{{x}_{e}}} \right){{x}_{e}}/\hbar }}},$$

$$i{{p}_{x}}\left( {{{x}_{e}}} \right){{{{\Psi }}}_{c}}\left( {{{x}_{e}},{{E}_{x}}} \right) + \frac{{\hbar d{{{{\Psi }}}_{c}}\left( {{{x}_{e}},{{E}_{x}}} \right)}}{{dx}} = 0$$

(A4.1)

and at ${{E}_{x}} < {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right)$ and ${{E}_{x}} > {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right)$

$${{{{\Psi }}}_{e}}\left( {{{x}_{e}},{{E}_{x}}} \right) = 0$$

$$\frac{{\hbar d{{{{\Psi }}}_{c}}\left( {{{x}_{e}},{{E}_{x}}} \right)}}{{dx}} - {{q}_{x}}\left( {{{x}_{e}}} \right){{{{\Psi }}}_{c}}\left( {{{x}_{e}},{{E}_{x}}} \right) = 0.$$

(A4.2)

At the point ${{x}_{c}}$ at ${{E}_{x}} > {{U}^{{eff}}}\left( {{{x}_{c}}} \right)$

$$i{{p}_{x}}\left( {{{x}_{c}}} \right){{{{\Psi }}}_{c}}\left( {{{x}_{c}},{{E}_{x}}} \right) + \frac{{\hbar d{{{{\Psi }}}_{c}}\left( {{{x}_{c}},{{E}_{x}}} \right)}}{{dx}} = 2i{{p}_{x}}\left( {{{x}_{c}}} \right){{e}^{{ - i{{p}_{x}}\left( {{{x}_{c}}} \right){{x}_{c}}/\hbar }}}$$

$$ - i{{p}_{x}}\left( {{{x}_{c}}} \right){{{{\Psi }}}_{e}}\left( {{{x}_{c}},{{E}_{x}}} \right) + \frac{{\hbar d{{{{\Psi }}}_{e}}\left( {{{x}_{c}},{{E}_{x}}} \right)}}{{dx}} = 0$$

(A4.3)

and at ${{E}_{x}} < {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right)$ and ${{E}_{x}} > {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right)$

$${{{{\Psi }}}_{c}}\left( {{{x}_{c}},{{E}_{x}}} \right) = 0$$

$$\frac{{\hbar d{{{{\Psi }}}_{e}}\left( {{{x}_{c}},{{E}_{x}}} \right)}}{{dx}} + {{q}_{x}}\left( {{{x}_{c}}} \right){{{{\Psi }}}_{e}}\left( {{{x}_{c}},{{E}_{x}}} \right) = 0.$$

(A4.4)

Here impulses

$$p\left( {{{x}_{e}}} \right) = 2{{m}_{e}}\left( {{{x}_{e}}} \right)\sqrt {{{E}_{x}} - {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right),~} ~{{E}_{x}} > {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right),$$

$$p\left( {{{x}_{c}}} \right) = 2{{m}_{e}}\left( {{{x}_{c}}} \right)\sqrt {{{E}_{x}} - {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right),~} ~{{E}_{x}} > {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right)$$

and damping decrements

$${{q}_{x}}\left( {{{x}_{e}}} \right) = 2{{m}_{e}}\left( {{{x}_{e}}} \right)\sqrt {{{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right) - {{E}_{x}},~} ~{{E}_{x}} < {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right),$$

$${{q}_{x}}\left( {{{x}_{c}}} \right) = 2m{\text{*}}\left( {{{x}_{c}}} \right)\sqrt {{{U}^{{{\text{ext}}}}}\left( {{{x}_{c}}} \right) - {{E}_{x}},~} ~{{E}_{x}} < {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right).$$

As a wave function ${{{{\Psi }}}_{b}}\left( {{{x}_{b}}} \right)$, the following function was taken:

$${{{{\Psi }}}_{b}}\left( {{{x}_{b}},E} \right) = {{e}^{{ - ipx/\hbar }}}\sin \frac{{\hbar \pi {{k}_{y}}y}}{{{{L}_{1}}}}\sin \frac{{\hbar \pi {{k}_{z}}z}}{{{{L}_{1}}}},$$

(A4.5)

$$p = 2{{m}_{e}}\left( {{{x}_{b}}} \right)\sqrt {E - {{U}^{{eff}}}\left( {{{x}_{b}}} \right),~} E > {{U}^{{eff}}}\left( {{{x}_{b}}} \right),$$

(A4.6)

and ${{L}_{1}}$ is the transverse dimension of the base nanowire.

As boundary conditions for the electrostatic potential $\varphi \left( x \right)~$at points ${{x}_{e}}$ and ${{x}_{c}}$, the requirement of electroneutrality was accepted:

$${{\left. {\left( {{{n}_{e}} + {{n}_{c}} + {{n}_{b}} - N} \right)} \right|}_{{x = {{x}_{e}},{{x}_{c}}}}} = 0.$$

(A5.1)

The electron concentrations were calculated by the formulas

$${{n}_{e}}\left( x \right) = \frac{1}{{2\pi \hbar {{L}^{2}}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \int {d{{E}_{x}}} {{\sqrt {{{E}_{x}} - {{U}^{{{\text{eff}}}}}\left( {{{x}_{e}}} \right)} }^{{ - 1}}}\sum\limits_{{{k}_{y}},{{k}_{z}}} {\sin } \frac{{\hbar \pi {{k}_{y}}y}}{L}\sin \frac{{\hbar \pi {{k}_{z}}z}}{L}{{\left| {{{{{\Psi }}}_{e}}\left( {x,{{E}_{x}}} \right)} \right|}^{2}}f\left( {{{s}_{e}}\left( {F,T} \right)} \right),$$

(A5.2)

$${{n}_{c}}\left( x \right) = \frac{1}{{2\pi \hbar {{L}^{2}}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \int {d{{E}_{x}}} {{\sqrt {{{E}_{x}} - {{U}^{{{\text{eff}}}}}\left( {{{x}_{c}}} \right)} }^{{ - 1}}}\mathop \sum \limits_{{{k}_{y}},{{k}_{z}}} \sin \frac{{\hbar \pi {{k}_{y}}y}}{L}\sin \frac{{\hbar \pi {{k}_{z}}z}}{L}{{\left| {{{{{\Psi }}}_{c}}\left( {x,{{E}_{x}}} \right)} \right|}^{2}}f\left( {{{s}_{c}}\left( {F,T} \right)} \right),$$

(A5.3)

$${{n}_{b}}\left( x \right) = \frac{1}{{2\pi \hbar {{L}_{1}}^{2}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \int {dE} {{\sqrt {E - {{U}^{{{\text{eff}}}}}\left( {{{x}_{b}}} \right)} }^{{ - 1}}}\mathop \sum \limits_{{{k}_{y}},{{k}_{z}}} \sin \frac{{\hbar \pi {{k}_{y}}y}}{{{{L}_{1}}}}\sin \frac{{\hbar \pi {{k}_{z}}z}}{{{{L}_{1}}}}{{\left| {{{{{\Psi }}}_{b}}\left( {{{x}_{b}},E} \right)} \right|}^{2}}f\left( {{{s}_{b}}\left( {{{F}_{b}},T} \right)} \right),$$

(A5.4)

$$f\left( s \right) = \frac{1}{{{{e}^{s}} + 1}}$$

(A5.5)

– the Fermi–Dirac distribution function;

$${{s}_{e}}\left( {F,T} \right) = \frac{{E\left( {{{k}_{y}},{{k}_{z}},{{p}_{x}}} \right) - F + {{\tau }_{e}}{{u}_{e}}\frac{{dF}}{{dx}}}}{{{{k}_{{\text{B}}}}T}},$$

(A5.6)

$${{s}_{c}}\left( {F,T} \right) = \frac{{E\left( {{{k}_{y}},{{k}_{z}},{{p}_{x}}} \right) - F + {{\tau }_{e}}{{u}_{c}}\frac{{dF}}{{dx}}}}{{{{k}_{{\text{B}}}}T}},$$

(A5.7)

$${{s}_{b}}\left( {F,T} \right) = \frac{{E\left( {{{k}_{y}},{{k}_{z}},p} \right) - {{F}_{b}} + {{\tau }_{e}}{{u}_{e}}\frac{{dF}}{{dx}}}}{{{{k}_{{\text{B}}}}T}},$$

(A5.8)

$${{\tau }_{e}} = \frac{{{{m}_{e}}{{\mu }_{e}}}}{e}$$

(A5.9)

is the average relaxation time of the electron pulse;

$${{u}_{e}} = \frac{{{{j}_{e}}\left( {x,{{E}_{x}}} \right)}}{{{{{\left| {{{{{\Psi }}}_{e}}\left( {x,{{E}_{x}}} \right)} \right|}}^{2}}}}~,\,\,\,\,{{u}_{c}} = \frac{{{{j}_{c}}\left( {x,{{E}_{x}}} \right)}}{{{{{\left| {{{{{\Psi }}}_{c}}\left( {x,{{E}_{x}}} \right)} \right|}}^{2}}}},\,\,\,\,~{{u}_{b}} = \frac{{{{j}_{b}}\left( {{{x}_{b}},E} \right)}}{{{{{\left| {{{{{\Psi }}}_{b}}\left( {{{x}_{b}},E} \right)} \right|}}^{2}}}}$$

(A5.10)

– the microscopic electron velocities

$${{j}_{e}}\left( {x,{{E}_{x}}} \right) = \frac{{i\hbar }}{{2{{m}_{e}}}}\left( {{{\Psi }}_{e}^{*}\left( {x,{{E}_{x}}} \right)\frac{{d{{{{\Psi }}}_{e}}\left( {x,{{E}_{x}}} \right)}}{{dx}} - \frac{{{\text{d}\Psi }_{e}^{*}\left( {x,{{E}_{x}}} \right)}}{{dx}}{{{{\Psi }}}_{e}}\left( {x,{{E}_{x}}} \right)} \right),$$

(A5.11)

$${{j}_{c}}\left( {x,{{E}_{x}}} \right) = \frac{{i\hbar }}{{2{{m}_{e}}}}\left( {{{\Psi }}_{c}^{*}\left( {x,{{E}_{x}}} \right)\frac{{d{{{{\Psi }}}_{c}}\left( {x,{{E}_{x}}} \right)}}{{dx}} - \frac{{{\text{d}\Psi }_{c}^{*}\left( {x,{{E}_{x}}} \right)}}{{dx}}{{{{\Psi }}}_{c}}\left( {x,{{E}_{x}}} \right)} \right),$$

(A5.12)

$${{j}_{b}}\left( {x,E} \right) = \frac{p}{{{{m}_{e}}}}{{\left| {{{{{\Psi }}}_{b}}\left( {{{x}_{b}},E} \right)} \right|}^{2}}$$

(A5.13)

is the electron flux density.

Current densities ${{J}_{{ec}}}$ and ${{J}_{b}}$ are determined by expressions

$${{J}_{{ec}}}{{\;}} = {{J}_{e}} + {{J}_{c}},$$

(A6.1)

$${{J}_{e}} = \frac{e}{{2\pi \hbar {{L}^{2}}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \smallint d{{E}_{x}}{{\sqrt {{{E}_{x}} - {{U}^{{eff}}}\left( {{{x}_{e}}} \right)} }^{{ - 1}}}\mathop \sum \limits_{{{k}_{y}},{{k}_{z}}} {{j}_{e}}\left( {x,{{E}_{x}}} \right)f\left( {{{s}_{e}}\left( {F,T} \right)} \right),$$

(A6.2)

$${{J}_{c}} = \frac{e}{{2\pi \hbar {{L}^{2}}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \int {d{{E}_{x}}} {{\sqrt {{{E}_{x}} - {{U}^{{eff}}}\left( {{{x}_{c}}} \right)} }^{{ - 1}}}\mathop \sum \limits_{{{k}_{y}},{{k}_{z}}} {{j}_{c}}\left( {x,{{E}_{x}}} \right)f\left( {{{s}_{c}}\left( {F,T} \right)} \right),$$

(A6.3)

$${{J}_{b}} = \frac{e}{{2\pi \hbar L_{1}^{2}}}\sqrt {\frac{{{{m}_{e}}}}{2}} \int E {{\sqrt {E - {{U}^{{eff}}}\left( {{{x}_{b}}} \right)} }^{{ - 1}}}\mathop \sum \limits_{{{k}_{y}},{{k}_{z}}} {{j}_{b}}\left( {x,E} \right)f\left( {{{s}_{b}}\left( {{{F}_{b}},T} \right)} \right).$$

(A6.4)

For the chemical potentials $F$ and ${{F}_{b}}$, the following boundary conditions were set:

$$F\left( {{{x}_{e}}} \right) = - e{{V}_{e}},\,\,\,\,~F\left( {{{x}_{c}}} \right) = - e{{V}_{c}},$$

(A7.1)

$${{F}_{b}}\left( {{{x}_{b}}} \right) = - e{{V}_{b}},~\,\,\,\,~{{F}_{b}}\left( {{{x}_{e}}} \right) = - e{{V}_{e}},~~{{F}_{b}}\left( {{{x}_{c}}} \right) = - e{{V}_{c}}.$$

(A7.2)

where ${{V}_{e}}$, ${{V}_{c}}~$, and ${{V}_{b}}$ are the voltages applied to the emitter, collector, and base contacts, respectively. In terms of their physical meaning, these conditions are similar to those used in the one-dimensional modeling of bipolar transistors (see Engl, V.L., Dirks, H.K., and Meinerzhagen, B., Modeling semiconductor devices, Works of the Institute of Electrical and Electronics Engineers (IEEE). M., 1983, vol. 71. no. 1, pp. 14–42).