{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 霍尔效应实验数据处理 (Hall Effect)\n", "\n", "本 Notebook 用于对霍尔效应实验中直接测得的霍尔电压 $V_H$ 及零磁场电导电压 $V_\\sigma$ 进行线性拟合、导电类型判别、物理参数计算（电导率 $\\sigma$、霍尔灵敏度 $K_H$、霍尔系数 $R_H$、载流子浓度 $n$ 与迁移率 $\\mu$）及完整的不确定度传递分析。\n", "\n", "---\n", "### 实验原理与数学公式\n", "\n", "#### 1. 霍尔效应基本关系与载流子类型判别\n", "将通有控制工作电流 $I_s$ 的半导体薄片置于垂直磁场 $B$ 中，受洛伦兹力偏转产生横向霍尔电压 $V_H$：\n", "$$V_H = K_H \\cdot I_s \\cdot B$$\n", "* 若 $V_H > 0$：空穴导电，判定为 **P 型半导体**；\n", "* 若 $V_H < 0$：电子导电，判定为 **N 型半导体**。\n", "\n", "#### 2. 磁场计算（螺线管）\n", "励磁电流为 $I_m$，螺线管中心磁场强度满足：\n", "$$B = k_{\\mathrm{sol}} \\cdot I_m = \\left(\\mu_0 \\frac{N}{L_{\\mathrm{sol}}}\\right) I_m$$\n", "\n", "#### 3. 电导率 $\\sigma$ 计算\n", "在零磁场 ($I_m = 0$) 条件下测定不同工作电流 $I_s$ 下沿长度方向的电位差 $V_\\sigma$。线性回归斜率即为霍尔片电阻 $R = V_\\sigma / I_s$，则材料电导率为：\n", "$$\\sigma = \\frac{L}{R \\cdot b \\cdot d}$$\n", "* $L$：电极间距（长度）；$b$：霍尔片宽度；$d$：霍尔片厚度。\n", "\n", "#### 4. 霍尔系数 $R_H$、载流子浓度 $n$ 与迁移率 $\\mu$\n", "* 霍尔系数：$R_H = |K_H| \\cdot d$\n", "* 载流子浓度：$n = \\frac{1}{R_H \\cdot e} = \\frac{1}{|K_H| \\cdot e \\cdot d}$（$e = 1.602 \\times 10^{-19}\\,\\mathrm{C}$）\n", "* 载流子迁移率：$\\mu = \\sigma \\cdot R_H = \\sigma \\cdot |K_H| \\cdot d$\n", "\n", "#### 5. 回归斜率不确定度与相对误差合成\n", "* 线性回归斜率 $k$ 的不确定度：$u(k) = |k| \\sqrt{\\frac{1/r^2 - 1}{N - 2}}$\n", "* 各物理量相对不确定度传递：\n", "$$u_r(\\sigma) \\approx \\frac{u(R)}{R}, \\quad u_r(K_H) \\approx \\frac{u(k_1)}{k_1}, \\quad u_r(n) = u_r(K_H), \\quad u_r(\\mu) = \\sqrt{u_r^2(\\sigma) + u_r^2(K_H)}$$" ], "id": "c151d3bfb3ac3774" }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import math\n", "from python.utils import scientific_round, linear_regression\n", "\n", "print(\"霍尔效应计算与回归分析模块加载完成。\")" ], "id": "ebdba69fe30f7d93" }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1. 物理常数与霍尔元件几何尺寸" ], "id": "eb6332ab0ffbedaf" }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# 基本物理常量\n", "e_charge = 1.602e-19 # 元电荷 (C)\n", "mu_0 = 4.0 * math.pi * 1e-7 # 真空磁导率 (T·m/A)\n", "\n", "# 霍尔元件几何尺寸 (mm -> m)\n", "d_mm = 0.095 # 厚度 d (mm)\n", "b_mm = 0.235 # 宽度 b (mm)\n", "L_mm = 0.270 # 电极间距 L (mm)\n", "\n", "d = d_mm * 1e-3\n", "b = b_mm * 1e-3\n", "L = L_mm * 1e-3\n", "\n", "# 螺线管磁场比值常数 k_sol = B / Im (T/A)\n", "k_sol = 0.00209\n", "\n", "print(f\"霍尔元件尺寸: L={L_mm} mm, b={b_mm} mm, d={d_mm} mm\")\n", "print(f\"螺线管比值: k_sol = {k_sol} T/A\")" ], "id": "964b32703abf7887" }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2. 实验测量数据输入\n", "> **包含三组数据**：\n", "> 1. 固定 $I_m$，测量 $V_H$ 随 $I_s$ 的变化；\n", "> 2. 固定 $I_s$，测量 $V_H$ 随 $I_m$ 的变化；\n", "> 3. 零磁场下测量 $V_\\sigma$ 随 $I_s$ 的变化。" ], "id": "bae2c240cece3f2a" }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# --- 1. VH vs Is 数据 (固定 Im = 600 mA) ---\n", "Im_fixed_mA = 600.0\n", "B_fixed = k_sol * (Im_fixed_mA * 1e-3) # T\n", "Is_1 = [0.20 * (i + 1) for i in range(10)] # mA: 0.20 ~ 2.00\n", "# 实测 VH 带符号读数 (mV)\n", "VH_signed_1 = [-1.52, -3.05, -4.58, -6.11, -7.63, -9.16, -10.68, -12.21, -13.74, -15.26]\n", "VH_abs_1 = [abs(x) for x in VH_signed_1]\n", "\n", "# --- 2. VH vs Im 数据 (固定 Is = 1.50 mA) ---\n", "Is_fixed_mA = 1.50\n", "Im_2 = [100.0, 150.0, 200.0, 250.0, 300.0, 350.0, 400.0, 450.0, 500.0, 550.0, 600.0] # mA\n", "VH_signed_2 = [-1.91, -2.87, -3.82, -4.78, -5.73, -6.69, -7.64, -8.60, -9.55, -10.51, -11.46] # mV\n", "VH_abs_2 = [abs(x) for x in VH_signed_2]\n", "\n", "# --- 3. Vσ vs Is 数据 (零磁场 Im = 0) ---\n", "Is_3 = list(Is_1) # mA\n", "Vsigma = [45.2, 90.5, 135.8, 181.0, 226.3, 271.5, 316.8, 362.0, 407.3, 452.5] # mV\n", "\n", "print(\"实验数据载入完成。\")" ], "id": "b944163d5b6fa5f7" }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3. 线性回归分析与参数计算" ], "id": "913d723bf7836a79" }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# 1. 回归 1: VH(mV) vs Is(mA) -> 斜率单位 mV/mA = V/A\n", "k1, b1, r1, uk1 = map(float, linear_regression(Is_1, VH_abs_1))\n", "\n", "# 2. 回归 2: VH(mV) vs Im(mA)\n", "k2, b2, r2, uk2 = map(float, linear_regression(Im_2, VH_abs_2))\n", "\n", "# 3. 回归 3: Vσ(mV) vs Is(mA) -> 样品电阻 R (Ω)\n", "R_sample, b3, r3, u_R = map(float, linear_regression(Is_3, Vsigma))\n", "\n", "# 导电类型判定\n", "carrier_type = \"P 型 (空穴导电)\" if sum(VH_signed_1)/len(VH_signed_1) > 0 else \"N 型 (电子导电)\"\n", "\n", "# 电导率 σ = L / (R · b · d)\n", "sigma = L / (R_sample * b * d)\n", "u_r_sigma = abs(u_R / R_sample)\n", "u_sigma = sigma * u_r_sigma\n", "sigma_final, u_sigma_final = scientific_round(sigma, u_sigma)\n", "\n", "# 霍尔灵敏度 KH = k1 / B_fixed\n", "KH = k1 / B_fixed\n", "u_r_KH = abs(uk1 / k1)\n", "u_KH = KH * u_r_KH\n", "KH_final, u_KH_final = scientific_round(KH, u_KH)\n", "\n", "# 霍尔系数 RH = KH * d\n", "RH = KH * d\n", "\n", "# 载流子浓度 n = 1 / (RH · e)\n", "n_carrier = 1.0 / (RH * e_charge)\n", "u_n = n_carrier * u_r_KH\n", "n_final, u_n_final = scientific_round(n_carrier, u_n)\n", "\n", "# 迁移率 μ = σ · RH\n", "mu = sigma * RH\n", "u_r_mu = math.sqrt(u_r_sigma**2 + u_r_KH**2)\n", "u_mu = mu * u_r_mu\n", "mu_final, u_mu_final = scientific_round(mu, u_mu)\n", "\n", "print(\"=\" * 55)\n", "print(\" 霍 尔 效 应 实 验 结 果 汇 总 \")\n", "print(\"=\" * 55)\n", "print(f\"材料导电类型 : {carrier_type}\")\n", "print(f\"VH-Is 回归斜率 k1 : {k1:.4f} V/A (r={r1:.4f})\")\n", "print(f\"VH-Im 回归斜率 k2 : {k2:.4f} V/A (r={r2:.4f})\")\n", "print(f\"样品内阻 R (斜率k3): {R_sample:.4f} Ω (r={r3:.4f})\")\n", "print(\"-\" * 55)\n", "print(f\"电导率 σ : {sigma_final} ± {u_sigma_final} S/m\")\n", "print(f\"霍尔灵敏度 |KH| : {KH_final} ± {u_KH_final} V/(A·T)\")\n", "print(f\"霍尔系数 RH : {RH:.4e} m³/C\")\n", "print(f\"载流子浓度 n : {n_final} ± {u_n_final} m⁻³\")\n", "print(f\"载流子迁移率 μ : {mu_final} ± {u_mu_final} m²/(V·s)\")\n", "print(\"-\" * 55)\n", "print(f\"相对不确定度 u_r(σ) : {u_r_sigma*100:.2f}%\")\n", "print(f\"相对不确定度 u_r(KH): {u_r_KH*100:.2f}%\")\n", "print(f\"相对不确定度 u_r(μ) : {u_r_mu*100:.2f}%\")\n", "print(\"=\" * 55)" ], "id": "7f25b56c710f5a1f" } ], "metadata": { "language_info": { "name": "python" } }, "nbformat": 4, "nbformat_minor": 5 }