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<h2 class="hd hd-2 unit-title">Question 2.1</h2>
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Question 2.1
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<p>Si and Ge can be mixed together to form a Si<sub>x</sub>Ge<sub>1-x</sub> semiconductor material. The band gap increases linearly with \(x\), from the value for pure Ge at \(x = 0\) to the value for pure Si at \(x = 1\). You have a bar of intrinsic Si<sub>x</sub>Ge<sub>1-x</sub> material, with \(x = 0.2\) at one end and \(x = 0.8\) at the other. The composition varies linearly with distance. The equilibrium band structure is shown below. Use this band diagram to answer the following questions: </p>
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<b>Part A </b>
<p>What is the bandgap on the Ge-rich side of the bar?</p>
<p>\(E_{g1}\) (in \(eV\)):</p>
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<p>What is the bandgap on the Si-rich side of the bar?</p>
<p>\(E_{g2}\) (in \(eV\)):</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_62037e5dd8b74b80894cb7f13b6dbdf1_3_1" class="inputtype formulaequationinput">
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problem
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<b> Part B</b>
<p>Characterize the drift and diffusion of carriers taking place in the material:</p>
<p>Electron Drift:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a4d1d659fdc54594a0c066b455952764_2_1">
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_2_1" id="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a4d1d659fdc54594a0c066b455952764_2_1-choice_0-label" for="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_2_1"> Electrons drift from the Ge-rich region to the Si-rich region (left to right)
</label>
</div>
<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_2_1" id="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="a4d1d659fdc54594a0c066b455952764_2_1-choice_1-label" for="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_2_1"> Electrons drift from the Si-rich region to the Ge-rich region (right to left)
</label>
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<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_2_1" id="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a4d1d659fdc54594a0c066b455952764_2_1-choice_2-label" for="input_a4d1d659fdc54594a0c066b455952764_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_2_1"> There is no drift of electrons
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<p>Hole Drift:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a4d1d659fdc54594a0c066b455952764_3_1">
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_3_1" id="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a4d1d659fdc54594a0c066b455952764_3_1-choice_0-label" for="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_3_1"> Holes drift from the Ge-rich region to the Si-rich region (left to right)
</label>
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_3_1" id="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="a4d1d659fdc54594a0c066b455952764_3_1-choice_1-label" for="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_3_1"> Hole drift from the Si-rich region to the Ge-rich region (right to left)
</label>
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<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_3_1" id="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a4d1d659fdc54594a0c066b455952764_3_1-choice_2-label" for="input_a4d1d659fdc54594a0c066b455952764_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_3_1"> There is no drift of holes
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<p>Electron Diffusion:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a4d1d659fdc54594a0c066b455952764_4_1">
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_4_1" id="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a4d1d659fdc54594a0c066b455952764_4_1-choice_0-label" for="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_4_1"> Electrons diffuse from the Ge-rich region to the Si-rich region (left to right)
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_4_1" id="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="a4d1d659fdc54594a0c066b455952764_4_1-choice_1-label" for="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_4_1"> Electrons diffuse from the Si-rich region to the Ge-rich region (right to left)
</label>
</div>
<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_4_1" id="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a4d1d659fdc54594a0c066b455952764_4_1-choice_2-label" for="input_a4d1d659fdc54594a0c066b455952764_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_4_1"> There is no net diffusion of electrons
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<p>Hole Diffusion:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a4d1d659fdc54594a0c066b455952764_5_1">
<fieldset aria-describedby="status_a4d1d659fdc54594a0c066b455952764_5_1">
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<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_5_1" id="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a4d1d659fdc54594a0c066b455952764_5_1-choice_0-label" for="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_5_1"> Holes diffuse from the Ge-rich region to the Si-rich region (left to right)
</label>
</div>
<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_5_1" id="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="a4d1d659fdc54594a0c066b455952764_5_1-choice_1-label" for="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_5_1"> Holes diffuse from the Si-rich region to the Ge-rich region (right to left)
</label>
</div>
<div class="field">
<input type="radio" name="input_a4d1d659fdc54594a0c066b455952764_5_1" id="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a4d1d659fdc54594a0c066b455952764_5_1-choice_2-label" for="input_a4d1d659fdc54594a0c066b455952764_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a4d1d659fdc54594a0c066b455952764_5_1"> There is no net diffusion of electrons
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<h2 class="hd hd-2 unit-title">Question 2.2</h2>
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<h3>Question 2.2</h3>
<p>A thick slab of Si (p-type, \(N_A = 10^{18}cm^{-3}\)) is illuminated on one side with light. The light creates an extra \(10^{10}cm^{-2}s^{-1}\) electron-hole pairs in the top \(1 \mu m\) of the Si. The lifetime of the carriers is \(10^{-5}s\), and their diffusivity can be taken as \(40cm^2s^{-1}\) (neglect the difference between electrons and holes).</p>
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<p>What does \(L\) represent in the figure above? (Please check all that apply.)</p>
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<input type="checkbox" name="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1[]" id="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="a921b6293b834adb8aa8b4dfd2bac13c_2_1-choice_1-label" for="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a921b6293b834adb8aa8b4dfd2bac13c_2_1"> The minority carrier diffusion length
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<input type="checkbox" name="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1[]" id="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="a921b6293b834adb8aa8b4dfd2bac13c_2_1-choice_3-label" for="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_a921b6293b834adb8aa8b4dfd2bac13c_2_1"> The average distance that excess minority carriers diffuse before recombination
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<input type="checkbox" name="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1[]" id="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="a921b6293b834adb8aa8b4dfd2bac13c_2_1-choice_4-label" for="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_a921b6293b834adb8aa8b4dfd2bac13c_2_1"> The average distance that excess electrons diffuse before recombination
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<input type="checkbox" name="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1[]" id="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="a921b6293b834adb8aa8b4dfd2bac13c_2_1-choice_5-label" for="input_a921b6293b834adb8aa8b4dfd2bac13c_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_a921b6293b834adb8aa8b4dfd2bac13c_2_1"> The location where the excess carrier concentration reaches 1/e of its value at the surface
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<p>What is the length \(L\) in the above plot?</p>
<p>\(L\) (in \(\mu m\)):</p>
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<b>Part B </b>
<p>What is the concentration of photogenerated electrons at the surface of the Si?</p>
<p>\(n\) (in \(cm^{-3}\)):</p>
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<p>For electrons, derive a steady-state expression that shows how their concentration varies with distance into the Si. Use this equation to predict the excess electron concentration in the material \(50 \mu m\) from the surface.</p>
<p>Excess minority carrier concentration \((n - n_p)\) (in \(cm^{-3}\)):</p>
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<b>Part C</b>
<p>What is the mobility of the carriers at room temperature? </p>
<p>Carrier mobility \(\mu\) (in \(cm^2/V \cdot s\)):</p>
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<p>Now assume that our silicon is only \(100 \mu m\) thick. What is the conductivity of the silicon when no light is shining on our material? </p>
<p>Conductivity \(\sigma\) (in \(S/cm\)):</p>
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<p>What is the conductivity when the light is turned on?</p>
<p>Conductivity \(\sigma\) (in \(S/cm\)):</p>
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<h2 class="hd hd-2 unit-title">Question 2.3</h2>
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<p>Consider a piece of Si with a band structure as shown below. At each end of the sample, the energy difference between the Fermi level and the band edge is \(0.1eV\). Take \(E_g = 1.1eV\) and neglect the mobility and mass difference between electrons and holes.</p>
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<b>Part A</b>
<p>Indicate which of the following statements with regards to the band structure shown are true: </p>
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<b>Part B</b>
<p>Charactarize the drift and diffusion of carriers taking place in the material:</p>
<p>Electron Drift:</p>
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<p>Hole Drift:</p>
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<p>Electron Diffusion:</p>
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<option value="There is no net electron diffusion"> There is no net electron diffusion</option>
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<p>Hole Diffusion:</p>
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<option value="Holes diffuse from left to right"> Holes diffuse from left to right</option>
<option value="Holes diffuse from right to left"> Holes diffuse from right to left</option>
<option value="There is no net hole diffusion"> There is no net hole diffusion</option>
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<p>Indicate whether the following statements are true or false:</p>
<p>When we heat the left edge of the material, the Fermi Level increases:</p>
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<p>When we heat the left edge of the material, the diffusion currents do not change:</p>
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<p>When we heat the left edge of the material, the drift currents do not change:</p>
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<p>When we heat the left edge of the material, we induce a net current flow in the material:</p>
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<p>When we heat the left edge of the material, the material is no longer at equillibrium:</p>
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<h2 class="hd hd-2 unit-title">Question 2.4</h2>
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Problem 2.4
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<p>Indium antimonide (InSb) is a semiconductor with a narrow band gap of \(0.17eV\) and an intrinsic carrier concentration of \(2 \times 10^{16} cm^{-3}\). The mobility of electrons in InSb is \(78000 cm^2/ V \cdot s\) and their effective mass is \(0.014m_0\), while the holes have a mobility of \(1000 cm^2/ V \cdot s\) and a mass of \(0.43m_0\) (all at \(300K\)).</p>
<b>Part A</b>
<p>What is the Fermi level with respect to mid-gap in intrinsic InSb?</p>
<p>\(\Delta E_F\) (in \(eV\)):</p>
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<p> </p>
<p>Suppose that we have a piece of InSb at \(300K\) and we apply an electric field of \(1000 V/ cm\) across it. What current density of electrons and holes would flow?</p>
<p>\(J\) (in \(A/cm^2\)):</p>
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<b>Part B </b>
<p>Now suppose that we disconnect the electric field, and instead heat up a small region of the InSb by \(10K\). What happens to the carrier concentration in this small region of the material? Express your answer as a ratio of \(n_i\) in the heated material to \(n_i\) in the material at room temperature.</p>
<p>\(\frac{n_i(310K)}{n_i(300K)}\):</p>
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<p>Consider the relative fluxes of electrons and holes diffusing away from the hot region. What is the ratio between the electron diffusivity and the hole diffusivity?</p>
<p>\(D_n/D_p\):</p>
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<h2 class="hd hd-2 unit-title">Question 2.5</h2>
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<p>A slab of intrinsic GaAs, \(3 cm\) long, \(2 cm\) wide, and \(0.3 cm\) thick, is exposed to light. The light is absorbed with an absorption coefficient \(\alpha = 500 cm^{-1}\) (this means that the light intensity decreases in the material exponentially with distance \(t\), and is proportional to \(exp(- \alpha t)\)). The light is monochromatic with a wavelength of \(750 nm\) and an intensity of \(5 \times 10^{-4} W/cm^2\). (Photon energy (\(eV\)) is given by \(1.24\)/wavelength (\(\mu m\)).)</p>
<p>What photon flux is incident on the slab?</p>
<p>photon flux (in photons \(cm^{-2}s^{-1}\)):</p>
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<p>At what depth does the intensity decrease to \(5 \%\) of its initial value</p>
<p>\(t\) (in \(\mu m\)):</p>
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<p>Calculate the number of electron-hole pairs generated per second in the slab. You may assume that since the photon energy is greater than the band gap, each photon produces an electron-hole pair.</p>
<p>pairs per second:</p>
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<p>What is the excess carrier concentration due to the light? You may assume a recombination time of \(2 \times 10^{-4}\) sec.</p>
<p>\(\Delta n\) (in \(cm^{-3}\)):</p>
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<p>By how much does the conductivity of the material change due to the light?</p>
<p>Conductivity increases by a factor of:</p>
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