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<h2 class="hd hd-2 unit-title">Part 2.1</h2>
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<h2>Drift</h2>
<h3>Carrier Action</h3>
<p>There are three basic types of carrier action within semiconductors:</p>
<ol>
<li>Drift</li>
<li>Diffusion</li>
<li>Recombination and Generation (R & G)</li>
</ol>
<p>We are going to start by taking a closer look at <em>drift</em>.</p>
<p></p>
<h3></h3>
<h3>Drift Velocity</h3>
<p>Drift is the movement of charged particles in response to an electric field. </p>
<p>The drift velocity of a carrier is given as \(v_d = \mu \mathscr{E}\), where the variables are defined as follows:</p>
<ul>
<li>\(v_d\) is the drift velocity. Typical units are \(cm/s\)</li>
<li><span style="line-height: 25.6px;">\(\mu\) is the carrier mobility. Typical units are \(cm^2/V \cdot s\)</span></li>
<li><span style="line-height: 25.6px;"><span style="line-height: 25.6px;"><span style="line-height: 25.6px;">\( \mathscr{E}\)</span> </span><span style="font-size: 1em; line-height: 1.6em;">is the electric field. Typical units are \(V/cm\)</span></span></li>
</ul>
<p>Please note that this linear relationship between electric field and drift velocity applies only at low and moderate electric field strength. In the presence of a high electric field, the drift velocity is limited by the saturation velocity \(v_{sat}\) of the carrier.</p>
<h3>Current Density</h3>
<p>The current density \(J\) associated with electron and hole drift is given as follows:</p>
<p><span style="line-height: 25.6px;"><span style="line-height: 25.6px;">\(J = e(n \mu_n + p \mu_p) \mathscr{E}\)</span></span></p>
<p><span style="line-height: 25.6px;">where \(e\) is the charge of an electron, \(n\) and \(p\) are the carrier densities, and \(\mu_n\) and \(\mu_p\) are the mobilities of the electrons and holes, respectively. \(J\) is typically expressed in \(A/cm^2\). </span></p>
<p></p>
<h3>Conductivity</h3>
<p>Since the conductivity \(\sigma\) is defined as <span style="line-height: 25.6px;"><span style="line-height: 25.6px;"> \(J/ \mathscr{E}\)</span>, we can now easily write an expression for conductivity as a function of carrier mobility:</span></p>
<p><span style="line-height: 25.6px;">\(\sigma = e(n \mu_n + p \mu_p)\)</span></p>
<p><span style="line-height: 25.6px;">where \(\sigma\) is typically expressed in the units of \(S/ cm\).</span></p>
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Example 2.1A
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Assume that you have a piece of silicon doped with \(10^{16} /cm^3\) of phosphorus atoms. Assume that you apply an electric field of \(5 kV/cm\) across the silicon. At this doping level, \(\mu_n = 1248 cm^2/V \cdot s \) and \(\mu_p = 437 cm^2/V \cdot s \) in silicon at room temperature. The saturation velocity is approximately \(1 \times 10^7 cm/s\). </p>
<p>What is the velocity of the electrons in the silicon under these conditions?</p>
<p>\(v_d\) (in \( cm/s\))</p>
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Example 2.1B
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Example 2.1C
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Example 2.1D
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Now assume that you apply an electric field of \(30 kV/cm\) to the silicon described in Example 2.1. What is the velocity of the electrons under this condition?</p>
<p>\(v_d\) (in \(cm/s\)):</p>
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<h2 class="hd hd-2 unit-title">Part 2.2</h2>
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<h2>Diffusion</h2>
<h3>Diffusion Current</h3>
<p>A diffusion current arises from the response of carriers to a concentration gradient. The total diffusion current \(J_{diff}\) is given as follows:</p>
<p>\(J_{diff} = eD_n \triangledown n + e D_p \triangledown p\)</p>
<ul>
<li><span style="font-size: 1em; line-height: 1.6em;">where \(\triangledown n\) and \(\triangledown p\) are the concentration gradients of electrons and holes respectively.</span></li>
<li><span style="font-size: 1em; line-height: 1.6em;">and \(D_n\) and \(D_p\) are the diffusion coefficients of electrons and holes. The units of \(D\) are typically in \(cm^2/s\).</span></li>
</ul>
<h3>Einstein Relationship</h3>
<p>The Einstein relationship allows us to relate the mobility of a carrier to its diffusion coefficient. This relationship is given as follows:</p>
<p>\(\frac{D}{\mu} = \frac{k_B T}{q}\)</p>
<p>where \(D\) and \(\mu\) refer to the diffusion coefficient and mobility of either electrons or holes.</p>
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Example 2.2A
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<p>At room temperature, electrons in GaAs have a mobility of about \(7000 \; cm^2/V \cdot s\) at a doping level of \(10^{16}cm^{-3}\). Calculate the diffusion coefficient of electrons in GaAs at this doping concentration at room temperature.</p>
<p>\(D_e\) (in \(cm^2/s\)):</p>
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Example 2.2B
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<p>What happens to the diffusion coefficient if we increase the doping level to \(10^{18}cm^{-3}\)?</p>
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<h2 class="hd hd-2 unit-title">Part 2.3</h2>
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<h2>Interpreting Band Diagrams</h2>
<p>Here are some important ideas to remember when drawing and interpreting band diagrams:</p>
<div>
<ul>
<li>A diffusion current arises in response to a concentration gradient.</li>
<li style="margin-left: 25px;">Carriers diffuse from areas of high concentration to areas of low concentration.</li>
<li>This diffusion current leads to the generation of an electric field within the material as a space charge develops. </li>
<li>This electric field can be either externally applied or internally generated, and leads to a drift current.</li>
<li>Tilted bands indicate the presence of an electric field.</li>
<li style="margin-left: 25px;">Electrons drift down the band. (Think of them like marbles rolling down a hill.)</li>
<li style="margin-left: 25px;">Holes drift up the band. (Think of them like balloons floating to the sky.)</li>
<li><span style="line-height: 25.6px;">The Fermi Level \(E_F\) is flat at equilibrium</span></li>
<li style="margin-left: 25px;">If the Fermi Level isn't flat, the semiconductor isn't at equilibrium.</li>
</ul>
<p><br /><br /></p>
<p><strong>Consider the band diagram shown below, and then answer the following questions:</strong></p>
<p><img src="/assets/courseware/v1/801646a91fd0f9cf205d952b6da71c5b/asset-v1:MITX+3.15.1x+2T2017+type@asset+block/315Tutorial3_3A.jpg" alt="n-type Si on the left, linearly graded to i-Ge on the right" type="saveimage" target="[object Object]" width="500" height="277" /></p>
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Exercise 2.3A
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<p>Is the semiconductor at equilibrium?</p>
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Exercise 2.3B
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<p>What is the direction of electron diffustion?</p>
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Exercise 2.3C
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<p>What is the direction of electron drift?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a858e4f37138482aa9aac377dea56aec_2_1">
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<input type="radio" name="input_a858e4f37138482aa9aac377dea56aec_2_1" id="input_a858e4f37138482aa9aac377dea56aec_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a858e4f37138482aa9aac377dea56aec_2_1-choice_0-label" for="input_a858e4f37138482aa9aac377dea56aec_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a858e4f37138482aa9aac377dea56aec_2_1"> The electrons drift from left to right
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<input type="radio" name="input_a858e4f37138482aa9aac377dea56aec_2_1" id="input_a858e4f37138482aa9aac377dea56aec_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a858e4f37138482aa9aac377dea56aec_2_1-choice_2-label" for="input_a858e4f37138482aa9aac377dea56aec_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a858e4f37138482aa9aac377dea56aec_2_1"> There is no net electron drift current
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Exercise 2.3D
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<p>What is the direction of hole diffusion?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_4ce32b57b67846b5a5979c4d35439913_2_1">
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<input type="radio" name="input_4ce32b57b67846b5a5979c4d35439913_2_1" id="input_4ce32b57b67846b5a5979c4d35439913_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="4ce32b57b67846b5a5979c4d35439913_2_1-choice_0-label" for="input_4ce32b57b67846b5a5979c4d35439913_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_4ce32b57b67846b5a5979c4d35439913_2_1"> The holes diffuse from left to right
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<input type="radio" name="input_4ce32b57b67846b5a5979c4d35439913_2_1" id="input_4ce32b57b67846b5a5979c4d35439913_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="4ce32b57b67846b5a5979c4d35439913_2_1-choice_1-label" for="input_4ce32b57b67846b5a5979c4d35439913_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4ce32b57b67846b5a5979c4d35439913_2_1"> The holes diffuse from right to left
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Exercise 2.3 E
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<h2 class="hd hd-2 unit-title">Part 2.4</h2>
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<h2>Recombination and Generation</h2>
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<li><span style="font-size: 1em; line-height: 1.6em;">Recombination and generation occurs:</span></li>
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<ol>
<li><span style="line-height: 1.6; font-size: 1em;"> By direct R & G</span></li>
<li><span style="line-height: 1.6; font-size: 1em;"> Via RG centers (traps)</span></li>
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<p></p>
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<li><span style="font-size: 1em; line-height: 1.6em;">The recombination rate \(R\) is proportional to the concentration of electrons and holes. </span></li>
</ul>
<p>\(R = rnp\)</p>
<p>where \(r\) is a rate constant.</p>
<p></p>
<p></p>
<ul>
<li><span style="font-size: 1em; line-height: 1.6em;">At equilibrium, the rate of recombination \(R\) and the rate of generation \(G\) are equal:</span></li>
</ul>
<p></p>
<p></p>
<p>\(R = G = rnp = rn_i^2\)</p>
<p></p>
<ul>
<li><span style="font-size: 1em; line-height: 1.6em;">Shining light on a semiconductor causes additional generation of carriers. These excess carriers \(n_l\) and \(p_l\) decay once the light is turned off.</span></li>
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Example 2.4
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Consider an illuminated p-type semiconductor. The illumination is shut off at time \(t = 0\).
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<p>
Write an expression for the concentration of electrons and holes as a function of \(N_A\), \(n_i\), and the excess photogenerated electron and hole concentrations \(n_l\) and \(p_l\).
</p>
<p>\(p\)</p>
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<p>\(n\)</p>
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Example 2.4, continued
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Assume the the number of photogenerated carriers is small compared to \(N_A\), but is large compared to the minority carrier concentration in the unilluminated semiconductor. Write simplified expressions for \(n\) and \(p\).
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<p>\(p\)</p>
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<p>\(n\)</p>
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<p>Write an expression for the net rate of change of carriers \(R -G\) as a function of \(n\), \(p\), \(n_i\) and the rate constant \(r\).</p>
<p>\((R - G)\)</p>
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<p>
Now express the rate of recombination of minority carriers \(-dn/dt\) in terms of \(r\), \(N_A\), and \(n_l\) only.
</p>
<p>\(-dn/dt\):</p>
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Problem 2.4, Continued
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<p>
The above equation has a solution of \(n_l = n_{l, t=0}exp(-t/\tau_n)\), where \(\tau_n\) is the minority carrier lifetime.
</p>
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Express \(\tau_n\) in terms of \(r\) and \(N_A\).
</p>
<p>\(\tau_n\):</p>
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