From c2d49e5f66d2980c67fad2c0ca9b69458acecef0 Mon Sep 17 00:00:00 2001 From: dm1sh Date: Wed, 15 Mar 2023 11:32:52 +0300 Subject: [PATCH] readme --- README.md | 253 ++++++++++++++++++ ...000001000000DC00000100F9B19430CB57B85C.png | Bin 0 -> 17614 bytes ...00000100000102000000EC388FD10724EFA862.png | Bin 0 -> 38432 bytes ...0000010000015E00000147D7C1BAAC07F91159.png | Bin 0 -> 19557 bytes ...000001000001C20000011D0C7FC81F7FC396D5.png | Bin 0 -> 52424 bytes ...000001000001CC00000131EEA170A8799B8110.png | Bin 0 -> 8502 bytes ...000001000002D0000003619A2670E62D0CE93C.png | Bin 0 -> 99621 bytes 7 files changed, 253 insertions(+) create mode 100644 README.md create mode 100644 pictures/Pictures/10000001000000DC00000100F9B19430CB57B85C.png create mode 100644 pictures/Pictures/1000000100000102000000EC388FD10724EFA862.png create mode 100644 pictures/Pictures/100000010000015E00000147D7C1BAAC07F91159.png create mode 100644 pictures/Pictures/10000001000001C20000011D0C7FC81F7FC396D5.png create mode 100644 pictures/Pictures/10000001000001CC00000131EEA170A8799B8110.png create mode 100644 pictures/Pictures/10000001000002D0000003619A2670E62D0CE93C.png diff --git a/README.md b/README.md new file mode 100644 index 0000000..9f389b5 --- /dev/null +++ b/README.md @@ -0,0 +1,253 @@ +МИНОБРНАУКИ РОССИИ + +Санкт-Петербургский государственный + +электротехнический университет + +«ЛЭТИ» им. В.И. Ульянова (Ленина) + +Кафедра Микрорадиоэлектроники и технологии радиоаппаратуры + +(МИТ) + +ОТЧЕТ + +**по ИДЗ** + +**по дисциплине «ОЭиР»** + +****Тема: Исследование контактных явлений в структуре +металл-полупроводник**** + +****Вариант 14,6,3**** + +https://www.meme-arsenal.com/memes/4598e00877a721c55a46dc4aafb78719.jpg + + ------------------ -- ------------- + Студент гр. 1181 Шишков Д.А. + Преподаватель Филипюк И.А + ------------------ -- ------------- + +Санкт-Петербург + +2023 + +Задание: + +Для заданной пары металл-полупроводник оценить кинетические свойства +заданных материалов, рассчитать и построить энергетическую диаграмму и +вольт-амперную характеристику контакта в заданном диапазоне температур, +дать рекомендации по применению исследуемого контакта. + +Таблица 1. Некоторые свойства металлов + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

No

+

ВАР.

ЭлементСтруктураАтомная массаПараметр решетки, ÅПлотность, г/см3Удельное сопротивление, мкОм·смТемпература, КРабота выхода φ, эВ
Дебая (TD)Ферми (TF·10-4)плавления (Tпл)
14AuГЦК196.94.0819.282.21656.3913374.58
+ +1\) Определить класс симметрии заданных материалов, построить прямую и +обратную элементарные ячейки заданных материалов. Определить размеры +Зоны Бриллюэна в направлениях X, L, К. + +Гранецентрированная Кубическая решётка + +Формула симметрии: 3L~4~4L~3~6L~2~9PC + +Класс симметрии: m3m + +Так как формулы симметрии ГЦК и простой кубической решётки совпадают, +приведём на рис. 1-3 изображения осей, плоскостей и центра симметрии для +последнего + +![](pictures/Pictures/100000010000015E00000147D7C1BAAC07F91159.png){width="3.4504in" +height="2.9512in"} + +Рис. 1 Изображение осей симметрии кубической решётки + +![](pictures/Pictures/10000001000001CC00000131EEA170A8799B8110.png){width="4.5437in" +height="3.0126in"} + +Рис. 2 Изображение плоскостей симметрии куба + +![](pictures/Pictures/10000001000000DC00000100F9B19430CB57B85C.png){width="2.0882in" +height="2.4307in"} + +Рис. 3 Изображение центра симметрии куба + +Базисные вектора: + +$$a_{1} = \begin{bmatrix} +\frac{1}{\sqrt{2}} \\ +\frac{1}{\sqrt{2}} \\ +0 \\ +\end{bmatrix}$$, $$a_{2} = \begin{bmatrix} +\frac{1}{\sqrt{2}} \\ +0 \\ +\frac{1}{\sqrt{2}} \\ +\end{bmatrix}$$, $$a_{3} = \begin{bmatrix} +0 \\ +\frac{1}{\sqrt{2}} \\ +\frac{1}{\sqrt{2}} \\ +\end{bmatrix}$$, считая, что постоянная решётки = 1 + +Построим кристаллическую решётку по заданным векторам (рис. 4) + +![](pictures/Pictures/1000000100000102000000EC388FD10724EFA862.png){width="2.6874in" +height="2.4583in"} + +Рис. 4 ГЦК + +Объём элементарной ячейки: + +$$V = {|{\overrightarrow{a_{1}} \cdot \left\lbrack {\overrightarrow{a_{2}} \times a_{3}} \right\rbrack}|} = \frac{1}{\sqrt{2}}$$ + +Базисные вектора обратной решётки: + +$${a_{i}^{\ast} = \frac{2\pi}{V}}\left\lbrack {a_{j} \times a_{k}} \right\rbrack,{i \neq j \neq k}$$ + +$$a_{1}^{\ast} = \begin{bmatrix} +{\sqrt{2}\pi} \\ +{{- \sqrt{2}}\pi} \\ +{{- \sqrt{2}}\pi} \\ +\end{bmatrix}$$, $$a_{2}^{\ast} = \begin{bmatrix} +{{- \sqrt{2}}\pi} \\ +{\sqrt{2}\pi} \\ +{{- \sqrt{2}}\pi} \\ +\end{bmatrix}$$, $$a_{3}^{\ast} = \begin{bmatrix} +{{- \sqrt{2}}\pi} \\ +{{- \sqrt{2}}\pi} \\ +{\sqrt{2}\pi} \\ +\end{bmatrix}$$ + +Изобразим её на рис. 5 + +![](pictures/Pictures/10000001000002D0000003619A2670E62D0CE93C.png){width="4.128in" +height="3.6374in"} + +Рис. 5 Обратная решётка ГЦК + +Первая зона Бриллюэна (рис. 6): + +![](pictures/Pictures/10000001000001C20000011D0C7FC81F7FC396D5.png){width="4.6874in" +height="2.9689in"} + +Рис. 6 Первая зона Бриллюэна + +Размеры зоны Бриллюэна по направлениям X, L, K: + +$$X = \begin{bmatrix} +0 \\ +0 \\ +1 \\ +\end{bmatrix}$$ - центр верхнего квадрата + +$$L = \begin{bmatrix} +0.5 \\ +0.5 \\ +0.5 \\ +\end{bmatrix}$$ - центр шестиугольника + +$$K = \begin{bmatrix} +0.75 \\ +0.75 \\ +0 \\ +\end{bmatrix}$$ - середина грани соединяющей два шестиугольника + +2\) Определить концентрацию электронов для заданного металла из условия +касания зоны Бриллюэна и сферы Ферми и сделать суждение о применимости +теории свободных электронов. + +В момент касания волновой вектор kф, соответствующий радиусу сферы +Ферми, равен волновому вектору kз, при котором в направлении, +перпендикулярном отражающим плоскостям, выполняется уравнение +Вульфа-Брэггов. Условия касания для двумерной модели можно записать в +виде $$k_{ф} = k_{з}$$ или, поскольку $${k = 2}\frac{\pi}{\lambda}$$, то +$$\lambda_{ф} = \lambda_{з}$$. + +Объем сферы Ферми в пространстве импульсов равен +$$\frac{4}{3}\pi p_{ф}^{3}$$ , где $$p_{ф}$$ -- импульс электронов на +поверхности Ферми. С другой стороны, этот же объем равен +$$\frac{N}{2}\frac{h^{3}}{V}$$, где N -- число электронов в объеме V. +Таким образом: + +$$\frac{4}{3}\pi{p_{ф}^{3} = \frac{N}{2}}\frac{h^{3}}{V}$$ + +Откуда находим: + +$${p_{ф} = h}\left( \frac{3N}{8\pi V} \right)^{1/3}$$ + +Поскольку по соотношению де-Бройля $$\lambda = \frac{h}{p}$$, то +$$\lambda_{ф} = \left( \frac{8\pi V}{3N} \right)^{1/3}$$ + +Найдем теперь λз. Из всех граней первой зоны Бриллюэна для +гранецентрированной решетки ближе всего к началу координат находятся +грани, обусловленные отражением электронов от плоскостей {111}. Поэтому +сфера Ферми впервые коснется именно этих граней. Таким образом, для +определения условий касания сферы с первой зоной Бриллюэна необходимо +найти длину волны, при которой электроны взаимодействуют с плоскостями +{111}. Из уравнения Вульфа -- Бреггов +$${\mathit{n\lambda} = 2}d\mathit{sin\theta}$$ находим: + +3 + +$${\lambda_{з} = 2}{d = 2}{\frac{a}{\sqrt{H^{2} + K^{2} + L^{2}}} = 2}\frac{a}{\sqrt{3}}$$ + +(здесь θ = 90°, так как в точке касания волновой вектор перпендикулярен +к плоскостям {111}; п = 1, так как λ--наибольшее) + +Если в объеме V число атомов Na, то число элементарных ячеек будет +$$\frac{N_{a}}{4}$$, так как на одну ячейку гранецентрированной решетки +приходится четыре атома, тогда $${V = a^{3}}\frac{N_{a}}{4}$$ + +Откуда $$a = \left( \frac{4V}{N_{a}} \right)^{1/3}$$, следовательно +$${\lambda_{з} = \frac{2}{\sqrt{3}}}\left( \frac{4V}{N_{a}} \right)^{1/3}$$ + +Возводя обе части этого тождества в куб и произведя необходимые +сокращения, получаем: + +$$\frac{N}{N_{a}} = \frac{\pi\sqrt{3}}{4} = 1.36$$ + +Список литературы: + +Астанин В.В. 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