Quantum Theory of Functionally Graded Materials

Abstract

Functionally graded materials (FGMs) are composites whose composition or microstructure varies continuously in space, producing position-dependent mechanical and functional properties. In recent years, FGMs have gained significant attention due to advances in additive manufacturing, which enable precise spatial control of composition and orientation. However, their graded, aperiodic structure breaks the assumptions of Bloch's theorem, making first-principles electronic and electromagnetic calculations challenging. Here we develop an ab initio quantum theoretical framework for the electromagnetic properties of FGMs. Using a non-interacting electron model, we formulate a theory of modulated Bloch states, derive effective field equations, and solve them by proposing a generalized WKB (GWKB) method, an effective mass approximation, the Boltzmann equation, and numerical approaches. Our GWKB solution is not semiclassical but remains valid in the fully quantum regime. We show that effective observables such as conductivity, magnetic permeability, and electric permittivity generally do not admit a tensorial description in graded media, and that engineered orientational gradients enable precise control of Landau quantization. As a device example, we further develop a theory of graded p-n junctions with enhanced electronic tunability. This framework lays the quantum foundation for predictive design of graded composite materials, enabling AI-accelerated discovery of next-generation functional architectures.

Figures (12)

• Figure 1: Illustration of the wave function ansatz used in the FGM theoretical framework. (a) The schematic depicts a spatially graded material where the periodicity and lattice parameters vary along the horizontal direction. (b) The wave function is decomposed into a fast-varying modulated Bloch component $u_{\lambda(\bm r)nK(\bm r)}(\bm r)$, and a slow-varying envelope function $\phi(\bm r)$, enabling the description of strongly aperiodic systems while retaining analytic control. Perturbative expansion in ratio $\kappa$ of local lattice spacing to lattice-modulation scale allows for fully quantum mechanical effective description valid for strong aperiodicity. Leading-order effective equations for slow modes are given by Eq. \ref{['eom']}.
• Figure 2: Calculated energy spectrum, linear density of state, and $k_z$ from the heuristic models of different FGM system. From left to right, the energy spectrum, the linear density of state, real part and imaginary part of $k_z$ are presented from the first column to the fourth column. (a)-(d) represent the cases for four different heuristic FGM systems. The system is discretized along the $z$-direction with a spatial step size of $dz = 1\times 10^{-8}\,\mathrm{m}$, resulting in a total of $100$ grid points over a length of $1\,\mu\mathrm{m}$. For (a), we have $U(z) = (0.5 + 0.01z)$eV, $m^*=(0.5 + 0.01z)$m$_{\mathrm{e}}$, $b=10^{12}$m$^{-2}$. For (b), we have $U(z) = (0.5 + 0.1z)$eV, $m^*=(0.5 + 0.01z)$m$_{\mathrm{e}}$, $b=10^{12}$m$^{-2}$. For (c), we have $U(z) = (0.5 + 0.01z)$eV, $m^*=(0.5 + 0.1z)$m$_{\mathrm{e}}$, $b=10^{12}$m$^{-2}$. For (d), we have $U(z) = (0.5 + 0.01z)$eV, $m^*=(0.5 + 0.05z)$m$_{\mathrm{e}}$, $b=10^{16}$m$^{-2}$. Here m$_{\mathrm{e}}$ reresents the electron mass.
• Figure 3: Workflow of the theoretical framework for functional graded materials (FGMs). The flowchart outlines the modeling procedure from inputting physical parameters (from heuristic models or ab initio DFT) to solving for wavefunctions and observables. Depending on the available data and desired precision, one may proceed through different branches, either quantum (via DFT and response theory) or semiclassical (via the Boltzmann equation), to obtain effective transport coefficients such as conductivity and susceptibility.
• Figure 4: Local conductivity calculated for heuristic model. The assumed heuristic system is discretized along the $z$-direction with a spatial step size of $dz = 1\times 10^{-8}\,\mathrm{m}$, resulting in a total of $100$ grid points over a length of $1\,\mu\mathrm{m}$. For (a), we have $U(z) = (0.5 + 0.02z)$eV, $m^*=(0.5 + 0.02z)$m$_{\mathrm{e}}$, $b=10^{12}$m$^{-2}$. For (a), the real part and imaginary part of the local conductivity at different positions are presented. The detailed frequency and z dependent conductivity are also presented in (b) and (c) with its real part and imaginary part.
• Figure 5: Directional dependent effective conductivity. The calculation is conducted with the corresponding heuristic FGM system with $U(z) = \left[0.5 + 0.1(z\mathrm{cos}\theta +x \mathrm{sin}\theta)\right]$eV, $m^*=\left[0.5 + 0.1(z\mathrm{cos}\theta+x\mathrm{sin}\theta)\right]$m$_{\mathrm{e}}$, $b=10^{12}$m$^{-2}$. As the integer part is identical across all angles, we present only the fractional part to better illustrate the variations. The voltage is applied as 0.5V only through the z direction. A total of $100$ grid points over a length of $1\,\mu\mathrm{m}$ through both x and z directions. In this relative formulation, the directional dependence of the effective conductivity is effectively captured by fixing the direction of the applied voltage while varying the material gradient direction, which is equivalent to the original setup where the gradient is fixed and the voltage direction is varied.