# Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening

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## Abstract

**:**

_{2}/SiO

_{2}/metal (ll-) and InSb/S(sulfur)/HfO

_{2}/metal (lh-κ type) multi-layer structures. A substantial enhancement of the binding energy is obtained with the non-degenerate Q2D EG for the ll-κ-type structure, reaching an almost fourfold value of the InSb bulk sample (~0.66 meV).

## 1. Introduction

_{2}O

_{3}[5] and HfO

_{2}[6], as well as HfO-AlO [7,8], ZrO-AlO [8], HfO-LaO [9] and HfO-chalcogenide [10] nanolaminate alloy forms, are currently of key interest. The aforementioned active channel materials possess a narrow energy band gap, resulting in the easy generation of defect states, for example, in the multi-layer quantum structure [7]. Certainly, such states can directly affect transport and optical properties, and clarifying the specific roles of defect states, impurity states in particular, in the noted system is fundamental [11].

## 2. Screened Coulomb Interaction Potential

_{0}> 0 near the semiconductor/oxide heterojunction. The screened Coulomb interaction potential φ(ρ,z) of the impurity center is related to Poisson’s equation as

_{i}is the induced charge density, which, according to the Tomas–Fermi method, has the form [14]:

_{0}≫ k

_{B}T is the chemical potential in the absence of the Coulomb perturbing field, k

_{B}T is the energy scale factor, $g\left(z\right)={\left|\psi \left(z\right)\right|}^{2}$ is the normalized charge distribution, ψ(z) is the self-consistent effective mass equation normalized solution of the electrons transverse motion, and ${\overline{\phi}}_{s}\left(\overrightarrow{\rho}\right)$ is averaged over the z coordinates by ψ(z) screened potential as

_{e}is the electron effective mass.

_{0}is the zero-th order Bessel function.

_{k}(z) = 0 while following Equation (11a). Among the coefficients C

_{i}, only C

_{1}corresponds to the channel region and has to be calculated. Applying the boundary conditions after Equation (10) at the i-th interface, we obtain C

_{1}as

_{0}determines a criterion of two-dimensionality of the problem, and for that, as shown in [26], an active channel plasma should be dense and, therefore, weakly nonideal. To the latter in size, quantum limit corresponds to the condition ${n}_{s}{a}_{0}^{2}$ ≫ 1, meaning that the number of surface electrons on the area of the Bohr orbit should be large enough.

_{0}≪ 1, which permits in the leading order of the parameter d/a

_{0}to obtain a solution of Equation (9a) for any arbitrary normalized charge distribution function g(z). Such a calculation gives the result as

_{S}(ρ) in the discussed case holds the same k and ρ dependences as the corresponding value in the multi-layer quantum structure with the bulk dielectric layer [14,15], but is enhanced twice (due to the factor ε

_{S}/2) in relation to the dielectrically homogeneous (${\epsilon}_{s}={\epsilon}_{D\left(L\right)}$) structure. A similar result is also obtained in the screening problems with the nano-scale thickness dielectric layer for any proportion of ${\epsilon}_{s}$ and ${\epsilon}_{D\left(L\right)}$ in the multi-layer quantum structure [18] and with the bulk low-κ-type dielectric layer in the three-layer quantum systems [22,26,27].

_{S}(ρ) is characterized by the more general (compared to previous cases) effective screening parameter ${q}_{S}^{\ast}$, which depends simultaneously on the key physical parameters of 2D EG and dielectric layers (${n}_{s}$, ${\epsilon}_{s}$, ${\epsilon}_{D},{\epsilon}_{L}$, L, D).

- (1)
- Enhancement of the carrier saturation effect specific up to this point at the pure 2D EG screening only;
- (2)
- Dependence of the screening mechanism nature (SS and ES) on the low-κ and high-κ types of the oxide layers.

_{S}/T parameter allowed for an enhancement of the carrier saturation effect become proper, and can be obtained from Equations (21), (4) and (6) as

_{S}/T values are also consistent with Equation (7b).

_{s}(q)│

_{z >0}in Equation (22) from the responsible parameter ${{q}_{S}^{\ast}|}_{ES}$ (i.e., n

_{S}/T) of the ES mechanism indicates that the joint effect of the image charges located in the dielectric and metal gate layers of the discussed multi-layer quantum structure would also lead to the enhanced saturation effect taking place for the low n

_{S}/T parameter values, i.e., even for the non-degenerate 2D EG statistics.

_{S}/T parameter (corresponding to the degenerate 2D EG statistics) only are responsible for the saturation effect [14]. Based on Equations (22) and (23), here we have a completely different form (${{q}_{S}^{\ast}|}_{SS}$) and meaning for the screening parameter, which does not depend on the 2D carrier concentration at all. The spatial and dielectric characteristics of the quantum structure entirely determine this form. It directly indicates a change in the nature of the saturation effect concerning the purely 2D EG screening case.

## 3. Screened Background Impurity State Properties

_{0}, where m

_{0}is the free electron mass)) and a macroscopically significant impurity effective Bohr radius (a

_{0}≈ 64.1 nm) since, under these parameters, the criterion of two-dimensionality of the problem is fully satisfied.

_{2}/SiO

_{2}/metal [28] as the ll-κ type, InSb/S(sulfur)/HfO

_{2}/metal [10] as the lh-κ type. In particular, the numerical results of the effective screening parameter ${q}_{S}^{\ast}$, the screened impurity ground state 2D effective radius a

_{sc}≈ λ

^{−1}, the 2D effective screening length ρ

_{s}≈ ${q}_{S}^{\ast}{}^{-1}$ and the impurity binding energy ${E}_{b}$ as functions of the oxide nano-scale thickness values D ϵ (1.5–8 nm), with the two characteristic n

_{S}/T parameter values n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K as case (a) and n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K as case (b), are demonstrated.

_{s}/T|

_{c}≈ 6.6 × 10

^{9}cm

^{−2}/

^{0}K) as case (c) are also clarified. The theoretical calculations developed here do not impose restrictions on the selection of semiconductor materials in terms of the energy band gap. However, in connection with device applications, here we are interested in materials with a high mobility, most of which have a small value band gap.

_{2}≈ 3.9. Based on the real situation [28], the thickness of the passivation layer is taken as L = 0, and the values of the dielectric constants are ${\epsilon}_{S}$|InSb ≈ 16.9 and ${\epsilon}_{D}$|SiO

_{2}≈ 3.9. As seen from Table 1, case (a), in the interval D ϵ (1.75–7 nm), the results of the required parameters are related to the non-degenerate type of 2D EG following Equation (23). Herewith, the ${{q}_{S}^{\ast}|}_{SS}$ contribution in the ${q}_{S}^{\ast}$ is more than 80%, and the effective screening parameter magnitude forms dominantly due to the SS mechanism contribution.

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K and from the values D > 3.75 nm at n

_{s}/T|

_{b}≈ 6.6 × 10

^{9}cm

^{−2}/

^{0}K, respectively. Comparing these results, we may conclude that the SS and ES mechanism’s correlation relationship is more sensitive to changes for the SS parameter D compared to the ES parameter n

_{s}/T.

_{S}≈ 1 and ${\epsilon}_{D}$|

_{HfO2}≈ 25.

_{s}/T parameter.

_{sc}and 2D screening length ρ

_{s}parameters. Figure 3 and Figure 4 show the dependences of these quantities as a function of the oxide thickness D for the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K and n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K parameter fixed values for the InSb/SiO

_{2}/metal (ll-κ type) and InSb/S/HfO

_{2}/metal (lh-κ type) multi-layer quantum structures, respectively.

_{sc}= ρ

_{s}= 23.74 nm (marked by a horizontal dashed-dotted line in the figure). This fact is entirely absent in the other presented cases in both figures, due to the joint correlation between the 2D EG statistic types on the one hand, and the dielectric types of the oxide layer on the other, reflected already in Table 1 and Table 2. In this connection, note that in the investigated before cases with the QW structure where the metal media is absent, the effective radii support such specified intersecting behavior regardless of the dielectric type of the oxide (see Figures 1 and 3 in [21]).

_{s}> >ε

_{D}(the low-κ oxide), the effective radii react quite differently to an increase in D. In case (a), the a

_{sc}graph starts to decrease noticeably, indicating a significant enhancement of the impurity Coulomb interaction at the required distances. This is due to the growing influence of dielectric polarization in the low-κ oxide layer, and parallel with that, a weakening of the metal layer induced polarization with increasing D. These two positive factors of impurity interaction enhancement outweigh the influence of the ES mechanism at these distances. In these conditions, the effective screening parameter q

_{s}* behaves as a

_{sc}, so the ρ

_{s}graph increases noticeably. This situation is also favored by the ES mechanism’s insignificance according to the condition following Equation (22). At the same time, the rate decrease of the a

_{sc}graph is much slower (≈22%) than the high growth rate of the ρ

_{s}≈ q

_{S}*

^{−1}graph (≈83%). In case (b), for which the condition following Equation (23) is no longer satisfied, the a

_{sc}values become almost saturated again, as for the D < 4 nm oxide thicknesses, and the ρ

_{s}growth rate also persists (≈85%). The reason for such results is the strong influence of the EC mechanism, the action of which now outweighs the contribution of dielectric enhancement to the impurity interaction.

_{sc}values decrease almost imperceptibly (≈1.5%) in case (a). In contrast, these values decrease noticeably (≈6.6%) for case (b), while the ρ

_{s}values maintain the growth rate for the (a) and (b) cases as ≈ 31% and ≈ 20%, respectively.

_{s}

^{ll-k}and E

_{s}

^{lh-k}, respectively). Figure 5 and Figure 6, respectively, demonstrate the E

_{s}

^{ll-k}(D) and E

_{s}

^{lh-k}(D) functional dependencies for the fixed n

_{s}/T values following Figure 3 and Figure 4 numerical data.

_{s}

^{ll-k}values increase in parallel with the oxide thickness D, but the growth rates in cases (a) and (b) are quite different. In the first case, the binding energy grows significantly, increasing for the InSb bulk sample from values E

_{s}≈0.66 meV (marked by a horizontal dashed-dotted line in the figure) at the oxide thickness D ≈ 2.84 × 10

^{−7}nm (marked by a vertical dashed-dotted line) to an almost fourfold increased value ≈ 2. 52 meV (marked by a horizontal dashed-dotted line) at thickness D ≈ 6.75 × 10

^{−7}nm (marked by a vertical dashed-dotted line), for which the effective radii have equal values (a

_{sc}=ρ

_{s}= 23.74 nm in accordance with Figure 3). With the oxide thickness range D ϵ (3–7 nm), the E

_{s}

^{ll-k}(D) functional dependence might be quite satisfactorily characterized as linear growth, as demonstrated in the graph by the dashed line. The latter, as can be clearly seen from the graph, should no longer be attributed to case (b), where the increase in the binding energy is obviously slower and, in terms of magnitude, differs slightly from the InSb bulk sample result E

_{s}≈ 0.66 meV starting from the D > 5.33 × 10

^{−7}nm value (marked by a vertical dashed-dotted line in Figure 5). This fully corresponds to those already given in the analysis of the data from Figure 2 regarding the behavior of effective radii related to the increasing influence of the SS mechanism at these oxide thicknesses, which outweighs the contribution of the low-κ dielectric mismatch in the impurity interaction.

^{lh-k}remains sizable starting from D > 3 × 10

^{−7}nm (marked by a vertical dashed-dotted line in the figure), while the second case takes tiny values for the entire discussed interval D ϵ (2–8 nm). Such a pattern is wholly consistent with the long-range nature restriction of the Q2D impurity interaction potential due to the high-κ dielectric mismatch effect. In this regard, it should be emphasized that the still sizable values of the binding energy in case (a) are also conditioned by the favorable parameters of the low-k type passivation layer.

## 4. Conclusions

_{S}or/and SS) on the low-κ and high-κ dielectric types of the barrier oxide layers.

_{2}/SiO

_{2}/metal (the ll-κ type) and InSb/S(sulfur)/HfO

_{2}/metal (the lh-κ type) nano-scale oxide layer quantum structures at the non-degenerate and near-degenerate 2D EG cases. An oxide type (low-κ or high-κ) plays a non-significant role in forming an impurity bound state up to D < 4 nm thickness values for the discussed structures. With D > 4 nm values, for the ll-κ-type structure, a significant enhancement of the impurity binding energy E

_{s}

^{ll-k}is obtained in the non-degenerate 2D EG case. The substantial enhancement of E

_{s}

^{ll-k}here is quite satisfactory and should be characterized by a linear-like growth (Figure 5), and it reaches, in particular, almost a fourfold value relative to the InSb bulk sample analog (E

_{s}≈0.66 meV). In the degenerate 2D EG case, a strong influence of the ES mechanism outweighs the positive contributions of the ll-κ-type dielectric mismatch effect on the interaction, and Es

^{ll-k}subtly differs from the results of a similar bulk sample. For the obtained lh-κ-type structure, the binding energy Es

^{lh-k}(D) linear-like growth is quite explicit (Figure 6). However, the high-κ dielectric mismatch leads to a sufficient decline in the intensity of the impurity interaction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Multi-layer quantum structure. The solid (dotted) lines mark the media (IL) boundaries. z

_{0}denotes the impurity position in the IL.

**Figure 3.**Two-dimensional effective radius of the screened impurity bound state a

_{sc}≈ λ

^{−1}(solid lines) and 2D screening length ρ

_{s}(dashed lines) as a function of the oxide thickness D with the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K parameter (bold lines) and n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K parameter (thin lines) for the InSb/SiO

_{2}/metal (ll-κ type) multi-layer quantum structure.

**Figure 4.**Two-dimensional effective radius of the screened impurity bound state a

_{sc}≈ λ

^{−1}(solid lines) and 2D screening length ρ

_{s}(dashed lines) as a function of the oxide thickness D with the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K parameter (bold lines) and n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K parameter (thin lines) for the InSb/S/HfO

_{2}/metal (lh-κ type) multi-layer quantum structure.

**Figure 5.**Binding energy E

_{s}

^{ll-k}of the screened impurity state as a function of the oxide thickness D with the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K parameter (bold line) and n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K parameter (thin line) for the InSb/SiO

_{2}/metal (ll-κ type) multi-layer quantum structure.

**Figure 6.**Binding energy E

_{s}

^{ll-k}of the screened impurity state as a function of the oxide thickness D with the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K parameter (bold line) and n

_{s}/T|

_{b}≈6.6 × 10

^{8}cm

^{−2}/

^{0}K parameter (thin line) for the InSb/SiO

_{2}/metal (ll-κ type) multi-layer quantum structure.

**Table 1.**Numerical results of the ${q}_{S}^{\ast}$, ${{q}_{S}^{\ast}|}_{SS}$ and ${{q}_{S}^{\ast}|}_{ES}$ parameter values, the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K, n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K, n

_{s}/T|

_{b}≈ 6.6 × 10

^{9}cm

^{−2}/

^{0}K parameter values for the InSb/SiO

_{2}/metal (ll-κ type) multi-layer structure.

$\frac{{\mathit{n}}_{\mathit{s}}}{\mathit{T}}$ (cm ^{−2}/K)
| D (nm) | ${\frac{{\mathit{n}}_{\mathit{s}}}{\mathit{T}}|}_{\mathit{S}\mathit{S}}$ (cm ^{−2}/K)
| ${\mathit{q}}_{\mathit{S}}^{\ast}$ (nm ^{−1})
| $\frac{{\text{}{\mathit{q}}_{\mathit{S}}^{\ast}|}_{\mathit{S}\mathit{S}}}{{\mathit{q}}_{\mathit{S}}^{\ast}}$ (%) | $\frac{{\text{}{\mathit{q}}_{\mathit{S}}^{\ast}|}_{\mathit{E}\mathit{S}}}{{\mathit{q}}_{\mathit{S}}^{\ast}}$ (%) |
---|---|---|---|---|---|

Case (a): n_{s} /T |_{a} ≈ 6.6 × 10^{7} | 1.75 | 1.1 × 10^{9} | 1.4 × 10^{6} | 94 | 6 |

4.5 | 4.1 × 10^{8} | 5.9 × 10^{5} | 87 | 13 | |

7 | 2.6 × 10^{8} | 4.1 × 10^{5} | 81 | 19 | |

Case (b): n_{s} /T |_{b} ≈ 6.6 × 10^{8} | 2.5 | 7.4 × 10^{8} | 1.4 × 10^{6} | 67 | 33 |

5 | 3.7 × 10^{8} | 9.2 × 10^{5} | 50 | 50 | |

6.5 | 2.8 × 10^{8} | 8.1 × 10^{5} | 44 | 56 | |

8 | 2.3 × 10^{8} | 7.4 × 10^{5} | 39 | 61 | |

Case (c): n_{s} /T |_{c} ≈ 6.6 × 10^{9} | 2 | 9.3 × 10^{8} | 1.8 × 10^{6} | 64 | 36 |

3 | 6.2 × 10^{8} | 1.4 × 10^{6} | 55 | 48 | |

3.75 | 4.9 × 10^{8} | 1.24 × 10^{6} | 50 | 50 | |

8 | 2.3 × 10^{8} | 0.9 × 10^{6} | 31 | 69 |

**Table 2.**Numerical results of the ${q}_{S}^{\ast}$, ${{q}_{S}^{\ast}|}_{SS}$ and ${{q}_{S}^{\ast}|}_{ES}$ parameter values with the n

_{s}/T|

_{a}≈ 6.6 × 10

^{7}cm

^{−2}/

^{0}K, n

_{s}/T|

_{b}≈ 6.6 × 10

^{8}cm

^{−2}/

^{0}K, n

_{s}/T|

_{b}≈ 6.6 × 10

^{9}cm

^{−2}/

^{0}K parameter values for the InSb/SiO

_{2}/metal (ll-κ type) multi-layer structure.

$\frac{{\mathit{n}}_{\mathit{s}}}{\mathit{T}}$ (cm ^{−2}/K)
| D (nm) | ${\frac{{\mathit{n}}_{\mathit{s}}}{\mathit{T}}|}_{\mathit{SS}}\text{}$ (cm ^{−2}/K) | ${\mathit{q}}_{\mathit{S}}^{\ast}$ (nm ^{−1})
| $\frac{{{\text{}\mathit{q}}_{\mathit{S}}^{\ast}|}_{\mathbf{SS}}}{{\mathit{q}}_{\mathit{S}}^{\ast}}$ (%) | $\frac{{{\text{}\mathit{q}}_{\mathit{S}}^{\ast}|}_{\mathit{ES}}}{{\mathit{q}}_{\mathit{S}}^{\ast}}$ (%) |
---|---|---|---|---|---|

(a) n_{s} /T |_{a} ≈ 6.6 × 10^{7} | 2 | 6.9 × 10^{8} | 9.6 × 10^{5} | 92 | 8 |

8 | 5.1 × 10^{8} | 7.1 × 10^{5} | 89 | 11 | |

(b) n_{s} /T |_{b} ≈ 6.6 × 10^{8} | 2 | 6.9 × 10^{8} | 1.32 × 10^{6} | 65 | 35 |

8 | 5.1 × 10^{8} | 1.1 × 10^{6} | 58 | 42 | |

(c) n_{s} /T |_{c} ≈ 6.6 × 10^{9} | 2 | 6.9 × 10^{8} | 1.48 × 10^{6} | 58 | 42 |

8 | 5.1 × 10^{8} | 1.26 × 10^{6} | 50 | 50 |

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## Share and Cite

**MDPI and ACS Style**

Aharonyan, K.; Kokanyan, N.; Kokanyan, E.
Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening. *Crystals* **2023**, *13*, 83.
https://doi.org/10.3390/cryst13010083

**AMA Style**

Aharonyan K, Kokanyan N, Kokanyan E.
Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening. *Crystals*. 2023; 13(1):83.
https://doi.org/10.3390/cryst13010083

**Chicago/Turabian Style**

Aharonyan, Kamo, Ninel Kokanyan, and Edvard Kokanyan.
2023. "Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening" *Crystals* 13, no. 1: 83.
https://doi.org/10.3390/cryst13010083