Abstract
Thin, highdensity layers of dopants in semiconductors, known as δlayer systems, have recently attracted attention as a platform for exploration of the future quantum and classical computing when patterned in plane with atomic precision. However, there are many aspects of the conductive properties of these systems that are still unknown. Here we present an opensystem quantum transport treatment to investigate the local density of electron states and the conductive properties of the δlayer systems. A successful application of this treatment to phosphorous δlayer in silicon both explains the origin of recentlyobserved shallow subbands and reproduces the sheet resistance values measured by different experimental groups. Further analysis reveals two main quantummechanical effects: 1) the existence of spatially distinct layers of free electrons with different average energies; 2) significant dependence of sheet resistance on the δlayer thickness for a fixed sheet charge density.
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Introduction
Technological^{1} and fundamental^{2} difficulties with the continued downscaling of fieldeffect transistors have motivated the development of alternative beyondMoore and beyondCMOS (complementary metal–oxide–semiconductor) energyefficient computing systems. At the device level, both conventional CMOS and many new transistor technologies require a high drive current, low off current, and the ability to scale device sizes even smaller. To the extent that these demands can be passed down to the material level, they translate into the search for a highly conductive, highly confined material system that still allows precise control over the electric current. The high system’s conductivity is necessary for fast transistor charging/discharging, while the high degree of the electrostatic confinement in nanoswitches^{3} is necessitated by both the high density of nanodevices and the need to control each nanodevice channel. With the emergence of atomically precise techniques to produce planar dopantbased structures in semiconductors, a particular interest has been paid to the δlayer structures, when the dopant atoms form a mono or several atomic layers with the dopant densities much higher than the solid solubility limit^{4}. Such structures have been shown to possess very high current densities^{5,6} and thus have a strong potential for beyondMoore applications.
Electronic structure and conductive properties of Si:P δlayer systems, which consists of a thin, highly phosphorous (P) doped 2D sheet layer embedded in lightly doped silicon, as shown in Fig. 1a, have been a subject of active studies for the last several decades^{7,8,9,10,11,12,13,14,15,16,17,18} leading to applications in quantum computing^{19,20} and advanced microelectronic devices^{6,21,22}. Recent angleresolved photoemission spectroscopy (ARPES) measurements^{23,24,25,26,27} revealed the existence of shallow conductive states that determines the conductive properties of these systems. Mazzola et al.^{25} revealed the quantization of the conduction band and the valence band near the Fermi level. Most recently, based on highresolution ARPES experiments, the presence of three subbands has been observed: 1Γ, 2Γ and the shallow 3Γ for the δlayer donor density of 9.0 × 10^{13} cm^{−2} in Holt et al.^{26}, and for a donor density of 1.7 × 10^{14} cm^{−2} in Mazzola et al.^{27}, that could not be explained by the existing theoretical models without the need to adjust the value of Si dielectric constant to ϵ ≈ 40 for highdensity phosphorus regions. However, it is easy to see that a charge selfconsistent solution of the PoissonSchrodinger equation already takes into account the increase of the effective dielectric constant, by directly accounting for the free electrons, and thus the adjustment of ϵ in the Poisson equation itself can lead to doublecounting of the screening effect. The existence of Δband was also observed by Holt et al.^{26} for thicker δlayer systems.
Naturally, particular attention has also been given to numerical studies of the electronic structure of δlayer systems. Previous computational studies of these systems based on either effective mass^{16}, tightbinding^{15,18,28} or density functional theory^{13,14,17} formalisms can be classified into two categories: truly closedsystem approaches, with the Dirichlettype boundary conditions for the wavefunction, and approaches with periodic boundary conditions along the propagation direction. However, in both cases, if the conductive properties of the system need to be extracted, this principally cannot be done directly from the quantummechanical flux. Indeed, the current density, j, is proportional to Ψ ∇ Ψ^{*} − Ψ^{*} ∇ Ψ, which is zero for any closedsystem wavefunction Ψ or is simply proportional to the wavevector for the case of periodic boundary condition, which cannot be used to compute the flux in most nontrivial cases. Thus, most typically, a classical or semiclassical Drude–Sommerfeld^{29,30} approximation assuming that the current density is proportional to the electron density, j ~ n, has to be additionally employed to extract the conductive properties from the closedsystem solution. Periodic boundary conditions allow conductivity extraction in the case of idealized metallic nanowires in semiconductor^{28}, however, the corresponding assumption that the transmission coefficient equals unity for each propagating mode ignores the influence of discrete charged impurities and interface roughness. The use of truly opensystem boundary conditions allows direct computation of conductive properties from the quantummechanical flux and can be readily extended to simulate more complex structures such as tunnel junctions, gated devices, and transistors.
In the following, we present an application of an opensystem nonequilibrium green function (NEGF) Keldysh formalism^{31} that enables a systematic quantummechanical study of conductive properties of semiconductor δlayer systems in the lowtemperature regime^{32}. We demonstrate that this opensystem treatment allows us to explain the origin of shallow conducting subbands recently observed in highresolution ARPES experiments and accurately reproduces the sheet resistance values obtained by different experimental groups for a wide range of the δlayer donor densities. Further analysis reveals two main effects: (1) the existence of spatially separated layers of free electrons with different average energies that are formed around the δlayer and (2) significant quantummechanical dependence of conductivity on the δlayer thickness for a fixed charge sheet density.
Results and discussion
Our device consists of a semiinfinite source and drains (represented by the NEGF open boundary conditions), in contact with the channel of length L, which is composed of a lightly doped Si cap, a very highly Pdoped layer, and a lightly doped Si body (see Fig. 1a). By using the NEGF open boundary conditions, the source and drain represent a way to extend the channel into infinity along the xaxis. Therefore, the source and drain have the same properties as the channel. The length of the channel along the xaxis is assumed to be L = 50 nm to avoid the boundary effect (between the source and drain contacts). The length along the ydirection is assumed to be infinite with a flat electrostatic potential, corresponding to planewave solutions of the Schrödinger equation along the yaxis. This ansatz allows us to seek the numerical solution of the PoissonopenSchrödinger equation in 2D. It is achieved by analytically integrating the Schrödinger equation over the yaxis momentum, which results in the effective 2D Fermi–Dirac distribution functions^{33}. The resulting 2D computational model for our device is shown in Fig. 1b. In all simulations, we assume a temperature of 4 K.
First, we investigate the density of states (DOS) induced by an embedded P δlayer in Si and its dependence on the layer depth, D, from the Si cap/body surfaces. For conceptual simplicity, we analyze symmetric configurations, where D in the Si cap/body is the same. We start by considering the concentration of donors of 1.0 × 10^{14} cm^{−2} in the δlayer and the δlayer thickness is set to 0.2 nm to approximate a monolayer of phosphorous atoms. We note that the nearestneighbor distance between atoms in Silicon is approximately 0.23 nm. The doping density in the δlayer is given by units of cm^{−2} to be consistent with the experiments’ nomenclature, i.e., \({N}_{D}^{2D}=t\times {N}_{D}^{3D}\), where t is the δlayer thickness, \({N}_{D}^{2D}\) is the doping density of the δlayer in cm^{−2} and \({N}_{D}^{3D}\) is the doping density in cm^{−3}. Throughout this work, the doping concentration of acceptors of 1.0 × 10^{17} cm^{−3} in the Si body/cap is assumed. The results of the corresponding opensystem quantummechanical treatment are shown in Fig. 2a, where the DOS are computed for D = 10, 20, and 40 nm. As expected in the opensystem quantummechanical formulation, the DOS is a continuous function of energy. The sharp peaks in the DOS correspond to the propagation modes (of energies E_{m}) that exist due to the confinement along the zaxis. The assumed source/drain geometry produces a onedimensional flux, therefore these modes have only one degree of freedom which explains their (\(1/\sqrt{E{E}_{m}}\))like shape. Below the Fermi level, there are two distinct occupied subbands referred to in the traditional band structure formulations as 1Γ and 2Γ, corresponding to the two peaks in the DOS approximately at the energies of −190 and −20 meV. Importantly, Fig. 2a demonstrates that these occupied states are independent of the depth distance D of the δlayer from the surfaces. Our calculations show that these states change only once they are shallow enough that D approaches the effective thickness of the electron cloud. This result is in agreement with the ARPES measurements of Mazzola et al.^{25} that show that the quantized conduction band states are largely independent of the dopant depth D, while strongly depending on the dopant density. As we will see later the number of occupied states and their respective energies are strongly dependent on both the δlayer thickness t and doping level N_{D}. On the other hand, the unoccupied states above the Fermi level are strongly dependent on the δlayer depth D, since they come from the conventional geometric device confinement. The number of these modes increases with the encapsulation distance of the δlayer, transitioning into the continuum mode in the limit of large D.
As we will demonstrate in the following, in order to understand the electronic transport mechanism in highly conductive δlayer systems, it is essential to analyze the electron distribution along the confinement direction in real space (LDOS). Figure 2b shows the LDOS(E,z) computed using our opensystem treatment for the considered Si:P δsystem with D = 10 nm. In addition to the energy quantization of the occupied states, they are also separated in space and with a distinct structure: the lowest energy mode, around −190 meV, is centered around the δlayer (z = 0.0 nm); the highest energy mode, around −20 meV, is located offcenter around ±1 nm from the δlayer on both sides. These occupied states remain spatially invariant with the depth of the δlayer as well, as illustrated in Fig. 2d showing the LDOS for D = 20 nm.
In general, LDOS(E,z) in such semiconductor δlayer systems takes a peculiar shape that we term “quantum menorah” as shown in Fig. 2b. We note that a “binary” (occupied/unoccupied) version of such LDOS was previously predicted using the traditional closedsystem approximations for an abstract semiconductor^{9}. A particular balance of donor/acceptor concentrations and the thickness value of the δlayer determine the specific “menorah” shape and the exact location of the Fermi level, thus determining the system’s conducting properties. For instance, the increase of the acceptor cap/body doping results in vertical stretching of the structure, while the decrease of the donor concentration results in the compression. The corresponding transmission coefficient for each mode shown in Fig. 2c is a consequence of a quantummechanical relation T(E) = ∑_{m}Γ_{mm}(E)A_{mm}(E), where Γ_{mm}(E) and A_{mm}(E) are the Hermitian form of the system’s selfenergy and the spectral function respectively, with both matrices written in the mode representation^{34}.
Next, we evaluate the influence of the δdoping profile on the conduction subband structure. Our simulations reveal that the number of conduction subbands and their corresponding energy splittings are strongly influenced by both the δlayer thickness t and the doping density N_{D}. Figure 3 shows the free electron distribution (i.e., the DOS weighted with the Fermi–Dirac distribution) for a fixed 2 nm δlayer thickness with diverse doping densities, and for a constant sheet doping density of 2 × 10^{14} cm^{−2} with different δlayer thicknesses, respectively. For a fixed δlayer thickness, the increment of the sheet doping increases the number of conducting modes, as well as the splitting energy between them, passing from a sole subband (1Γ) to two subbands (1Γ and 2Γ), and from two subbands to three subbands (1Γ, 2Γ, and 3Γ), as shown in Fig. 3a. In contrast, for a fixed sheet doping, the increment of the δlayer thickness increases the number of modes, but decreases the energy splitting between them, particularly between 1Γ and 2Γ for t = 4 nm, as reflected in Fig. 3b. We note that this may be an indication that previously reported low (<50 meV) energy splitting between these subbands^{27} is only true for relatively thick (>5 nm) δlayers. The highly Pdoped δlayer creates an electrostatic potential well (see Fig. 2b, gray line), which becomes sharper for smaller δlayer thicknesses and higher doping densities. A sharper confinement potential leads to an increased energy splitting between subbands, causing a larger average energy difference between the electrons in the distinct parallel layers. The energy splitting between subbands is proportional to the sheet doping density, which is in agreement with previous tightbinding calculations^{15}.
The current spectrum is also shown in Fig. 3. Even though more carriers are allocated in the lowestenergy subbands for high confinement potentials, the electrical current is carried out equally among the conductive subbands. For example, for the 0.2 nm δlayer thickness with a doping density of 1.2 × 10^{14} cm^{−2} in Fig. 3b, the free electron distribution between subbands is approximately 90%(1Γ) and 10%(2Γ), but the current contribution of each subband is similar (about 50%). However, the contribution of higherenergy subbands on the current becomes more significant as the confinement potential becomes weaker, i.e. electrons at the outer layers will start to carry the majority of the current. For instance, the 4 nm δlayer thickness with a doping density of 1.2 × 10^{14} cm^{−2} in Fig. 3b, the electron distribution between subbands is approximately 35%(1Γ), 45%(2Γ), and 20%(3Γ), however, the current contribution of each subband is around 10%(1Γ), 25%(1Γ), and 65%(3Γ).
The effective electron cloud thickness as a function of the sheet doping density N_{D} and the δlayer thickness t is shown in Fig. 4. For low doping densities, below N_{D} = 10^{13} cm^{−2}, the electron cloud thickness is almost independent of the δlayer thickness and is only determined by the doping value (the weak confinement regime). However, for high doping densities, above N_{D} = 5 × 10^{13} cm^{−2}, the effective electron thickness becomes independent of the doping value itself and is determined only by the δlayer thickness value (the strong confinement regime).
Next, we validate our calculations against experimental data by several groups^{5,35,36,37} by comparing the macroscopic conductive properties of the system. To carry out this, we first need to introduce a heuristic elastic defect scattering model for meso and macroscopic scale into our quantum transport framework (see Supplementary Method 2 for further details). In this work, we neglect the effects of inelastic scattering, since in Si:Pδ systems the phaserelaxation length l_{ψ} is larger than the mean free path l_{m} at low temperatures^{5}. There are two kinds of elastic scattering events that we consider in our opensystem treatment: (1) scattering due to channel geometry and confinement due to a specific doping profile, which we refer to as geometry scattering, and (2) defect scattering (e.g., vacancies, dislocations, and impurities). The geometry scattering is already taken into account by our charge selfconsistent quantum transport framework that also accounts for electronelectron interaction (via local density approximation exchangecorrelation potentials). We can simulate defect scattering via abstract coherencebreaking scatterers represented as a linear density ν per transmission mode m along the channel of length L. The linear defect density is defined as ν = N/L, where N is the total number of elastic scatterers per transmission mode. Then the effective transmission function per mode along the channel can be expressed as^{32}
where T_{mm}(E) is the transmission function per mode without defect scatterers across the channel, and \({t}_{mm}^{{\prime} (i)}(E)\) is the transmission probability per mode due to a scatterer i. In other words, the first term of the right side of Eq. (1) takes into account the geometry scattering, whereas the second term encompasses the defects. Assuming that all scatterers have identical transmission probabilities \({t}_{mm}^{{\prime} (i)}={t}_{mm}^{{\prime}}\), the effective transmission function results as
The term \(\nu \frac{1{t}_{mm}^{\prime}}{{t}_{mm}^{\prime}}\) can either be assumed to be constant for all modes and energies or as being inversely proportional to the characteristic mean free path l_{m}^{32}. To account for the reported increase of l_{m} for very high δlayer densities^{5} we assume that \(\nu \frac{1{t}_{mm}^{\prime}}{{t}_{mm}^{\prime}} \approx \frac{{\alpha}}{{l}_{m}(N_{2D})^{\prime}}\), where α is proportional to the linear defect density in the system. We note the conductance obtained from the effective transmission in Eq. (2) is analogous (equivalent in the limit of low voltages) to the conductance calculated using previous elastic scattering models^{28,38,39}.
By applying the described scattering model in our quantum transport simulations, we compute the sheet resistance for a wide range of δlayer thickness (from 0.2 to 5 nm) and doping densities (from 4 × 10^{11} to 7.5 × 10^{14} cm^{−2}). The computed sheet resistances are shown in Fig. 5 for α = 1.0 and 2.0 and compared against experimental data^{5,35,36,37}. The opensystem quantummechanical simulations generally reproduce the experimental data for the entire doping range and from diverse sources by only adjusting the defect linear density parameter α. It is important to point out that all systematic electrical measurements^{5,36,37} performed for a wide range of densities can be reproduced very well with our model with a particular value of the linear defect density α, as reflected in Fig. 5. Moreover, the shown experimental results can be classified into the distinct groups based on their samples impurity density: as the data in Fig. 5 suggest, the earlier samples^{5} had twice higher linear impurity density α = 2, while more recent samples^{36,37} had the lower impurity density as α = 1 for them. Figure 6 shows the computed sheet conductance for the system as a function of the sheet doping density N_{D} and the δlayer thickness t. It is evident that the complex subband structure has a significant effect on the system’s conductive properties that manifests itself in the conduction decrease for the sharper confining potentials. Indeed, semiclassically the conductivity can only depend on the sheet δlayer density, but not on the δlayer thickness, unless it is assumed that the mobility strongly depends on the distance from the δlayer and properly reflects the difference in average energies of electrons in different occupied subbands. Thus, the predicted strong dependence of the conductivity on the δlayer thickness presented in Fig. 6 is a direct consequence of the fully quantummechanical opensystem treatment.
Conclusion
We have presented an opensystem quantum transport treatment for semiconductor δlayer systems that revealed a quantized conduction subband structure that we termed “quantum menorah”. The structure predicts the existence of spatially separated layers of free electrons with significantly different average energies. With the rapidly advancing atomically precise manufacturing techniques, this property could be used for thermoelectric applications^{40}, when free electrons of particular energy would be “mechanically” blocked or allowed to pass through, e.g., by the means of the appropriately located, thin dielectric or acceptorδ layer(s). Together with the high δlayer systems conductivity, the ability to selectively filter charge carriers based on their kinetic energy could be used to design thermoelectric systems with a high figure of merit^{41}.
Our simulation results for Si:P δsystems suggest that the quantized subband structure manifests itself in two main effects: (1) the highly nonlinear dependence of the electron cloud on the confining δlayer doping profile and (2) the general quantummechanical resistivity increase for sharper δlayer doping profiles. Understanding the dependence of the electron cloud thickness on the δlayer doping profile is especially important for designing qubits, ultrascaled fieldeffect transistors and beyond CMOS devices, when the device critical dimensions are already shrunk to the nmscale.
Finally, we point out that the nontrivial results presented in this work, particularly the strong deviation from the semiclassical j ~ n dependence for high donor densities shown in Fig. 6, indicate the general necessity of the fully quantummechanical opensystem treatment for highly confined and conductive systems, which were previously studied only with the bandstructure approaches based on the closedsystem approximation(s).
Methods
We concentrate our analysis on the free electron properties in Si:P δlayer systems obtained through the charge selfconsistent solution in the “jellium approximation”^{16} of the Poisson equation and the singleband (Γvalley) effective mass Schrödinger equation with the opensystem boundary conditions (obtained using the nonequilibrium Green’s functions \(G(r,r^{\prime} ,E)\)^{32}). In this model, the influence of the Si nuclei, and the core and valence electrons is introduced only through the Si effective mass tensor, while the main effect is being determined by the process of establishing the global Fermi level for the open system, which we numerically emulate by finding the charge selfconsistent realspace solution using the Hartree potential augmented with the exchangecorrelation corrections^{42}. Then the resulting quantummechanical opensystem quantities of interest, such as the local DOS \(LDOS(r,E)=\frac{1}{\pi }\,{{\mbox{Im}}}\,[G(r,r,E)]\) and the transmission function T(E) = Tr[Γ_{L}(E)G(E)Γ_{R}(E)G^{†}(E)] are used to compute the current spectrum j(E) and the macroscopic conductive properties such as the sheet resistance. For an efficient implementation of this rather demanding computational scheme we utilized the Contact Block Reduction method^{34,43} and the opensystem predictor–corrector method^{44} augmented with Anderson mixing scheme^{45} (see Supplementary Method 1 for further details). The accuracy and reliability of the contact block reduction method have been established in numerous publications^{34,43,44,46,47,48}. We justify the use of the simple singleband approximation by (1) the desire to simplify the freeelectron Hamiltonians to reduce the computational burden for the first opensystem treatment of these systems and (2) the recent report^{26}, where the relative contribution of Γ and Δ bands have been studied using highresolution ARPES measurements, demonstrating that only the Γ band is occupied for the monoatomic δlayers (see Table I within^{26}). We note that no fitting parameters of any kind were used in the chargeself consistent calculations of this work. We also note that the sheet resistance/conductance computing model contains a single parameter, the linear defect density, as described in the discussion of Eq. (2). The standard values of electron effective masses m_{l} = 0.98 × m_{e}, m_{t} = 0.19 × m_{e}, and the dielectric constant ϵ_{Si} = 11.7 of Si were used. The use of the bulk effective masses has been proven to give very accurate results for Si:P δlayer systems when compared to both tightbinding and density functional theory calculations^{16}.
Data availability
The data that support the plots within this paper are available from the corresponding authors upon reasonable request.
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Acknowledgements
D.M. expresses his immeasurable and irredeemable gratitude to Galina Yakovlevna Mamaluy (1940–2020). The authors are thankful to TzuMing Lu (Sandia National Laboratories) for insightful discussions on meso vs. macroscopic conductivity. This work is funded under Laboratory Directed Research and Development Grand Challenge (LDRD GC) program, Project No. 213017, at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DENA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
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D.M. and J.M. performed the central calculations and analysis presented in this work. X.G. and S.M. conducted the analysis of the available experimental data and contributed to discussions framing the results, along with the methodology section. The paper was written by all the authors.
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Mamaluy, D., Mendez, J.P., Gao, X. et al. Revealing quantum effects in highly conductive δlayer systems. Commun Phys 4, 205 (2021). https://doi.org/10.1038/s42005021007051
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DOI: https://doi.org/10.1038/s42005021007051
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