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3.4.3: Threshold Voltage

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    89970
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    Diagram of MOS at threshold, with a graph of the charge density as a function of position x. The graph's y-axis aligns with the boundary between the oxide and the p-silicon. For the gate, which has negative values of x, the graph takes the form of a tall, narrow rectangle starting on the x-axis and rising to height Q_g. The region of the p-silicon occupied only by the fixed negative charges, as the holes have moved to the right side of the silicon, is graphed as a short, long rectangle with its top left corner at the origin and a height of q N_a.
    Figure \(\PageIndex{1}\): Charge distribution at threshold

    As before, we will do the integral graphically, starting at the left hand side of the picture. The field outside the structure must be zero, so we have no electric field until we get to the delta function of charge on the gate, at which time it jumps up to some value we will call \(E_{\text{ox}}\). There is no charge inside the oxide, so \(\frac{dE}{dx}\) is zero and thus \(E(x)\) must remain constant at \(E_{\text{ox}}\) until we reach the oxide/silicon interface.

    Graphical integral to find the electric field in the oxide layer. Below the graph from Figure 1 above, a second set of axes E(x) vs s is placed with the y-axes aligned. The graph of the electric field in the oxide takes the form of a tall rectangle with its bottom right corner at the origin, with its base on the x-axis and its left side aligned with the right side of the gate charge-distribution rectangle.
    Figure \(\PageIndex{2}\): Electric field in the oxide

    If we were to put our little "pill box" on the oxide-silicon interface, the integral of \(D\) over the face in the silicon would be \(\varepsilon_{\text{Si}} E_{\text{Si}} \Delta (S)\) where \(E_{\text{Si}}\) is the strength of the electric field inside the silicon. On the face inside the oxide it would be \(-\left( \varepsilon_{\text{Si}} E_{\text{Si}} \Delta (S) \right)\) where \(E_{\text{ox}}\) is the strength of the electric field in the oxide. The minus sign comes from the fact that the field on the oxide side is going into the pill box instead of out of it. There is no net charge contained within the pill box, so the sum of these two integrals must be zero. (The integral over the entire surface equals the enclosed charge, which is zero. \[\varepsilon_{\text{Si}} E_{\text{max}} \Delta (S) - \varepsilon_{\text{ox}} E_{\text{ox}} \Delta (S) = 0\]

    or \[\varepsilon_{\text{Si}} E_{\text{max}} = \varepsilon_{\text{ox}} E_{\text{max}}\]

    Within the oxide layer, which is on the left side of the silicon, an electric field of magnitude E_ox points towards the right. Starting at the boundary between the substances, an electric field of magnitude E_max points towards the right. A small rectangular region horizontally centered on the boundary represents the side view of the cylindrical "pillbox", which has a cross-sectional area of Delta S.
    Figure \(\PageIndex{3}\): Using Gauss's Law at the silicon/oxide interface

    This is just a statement that it is the normal component of displacement vector, \(D\), which must be continuous across a dielectric interface, not the electric field \(E\). Solving Equation \(\PageIndex{3}\) for the electric field in the silicon: \[E_{\text{Si}} = \frac{\varepsilon_{\text{ox}}}{\varepsilon_{\text{Si}}} E_{\text{ox}}\]

    The dielectric constant of oxides about one third that of the dielectric constant of silicon dioxide, so we see a "jump" down in the magnitude of the electric field as we go from oxide to silicon. The charge density in the depletion region of the silicon is just \(- \left(q N_{a}\right)\) and so the electric field now starts decreasing at a rate \(\frac{- \left(q N_{a}\right)}{\varepsilon_{\text{Si}}}\) and reaches zero at the end of the depletion region, \(x_{p}\).

    Graph of electric field in the oxide from Figure 2 above, also showing the electric field within the silicon. The area under the graph of E_ox is Delta V_ox. The graph of the silicon electric field takes the form of a line that slants down from the point E_Si on the positive y-axis to the point x_p on the positive x-axis. Slope of this line is -q N_a divided by the electric constant of silicon. The area beneath this line is Delta V_Si.

    Figure \(\PageIndex{4}\): Electric field and voltage drops across the entire structure

    Clearly, we have two different regions, each with their own voltage drop. (Remember the integral of electric field is voltage, so the area under each region of \(E(x)\) represents a voltage drop.) The drop in the little triangular region we will call \(\Delta \left(V_{\text{Si}} \right)\) and it represents the potential drop in going from the bulk, down to the bottom of the drooping conduction band at the silicon-oxide interface. Looking back at the earlier figure on threshold, you should be able to see that this is nearly one whole band-gap's worth of potential, and so we can safely say that \(\left( \Delta \left(V_{\text{Si}}\right) \simeq 0.8 \right) \rightarrow 1.0 \mathrm{~V}\).

    Just as with the single-sided diode, the width of the depletion region \(x_{p}\), is (which we saw in a previous equation): \[x_{p} = \sqrt{ \frac{2 \varepsilon_{\text{Si}} \Delta \left(V_{\text{Si}}\right)}{q N_{a}} }\]

    from which we can get an expression for \(E_{\text{Si}}\) \[\begin{array}{l} E_{\text{Si}} &= \frac{q N_{a}}{\varepsilon_{\text{Si}}} x_{p} \\ &= \sqrt{ \frac{2q N_{a} \Delta \left(V_{\text{Si}}\right)}{\varepsilon_{\text{Si}}} } \end{array}\]

    by multiplying the slope of the \(E(x)\) line by the width of the depletion region, \(x_{p}\).

    We can now use Equation \(\PageIndex{4}\) to find the electric field in the oxide: \[\begin{array}{l} E_{\text{ox}} &= \frac{\varepsilon_{\text{Si}}}{\varepsilon_{\text{ox}}} E_{\text{Si}} \\ &= \frac{1}{\varepsilon_{\text{ox}}} \sqrt{ 2q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right) } \end{array}\]

    Finally, \(\Delta \left(V_{\text{ox}}\right)\) is simply the product of \(E_{\text{ox}}\) and the oxide thickness \(x_{\text{ox}}\): \[\begin{array}{l} \Delta \left(V_{\text{ox}}\right) &= x_{\text{ox}} E_{\text{ox}} \\ &= \frac{x_{\text{ox}}}{\varepsilon_{\text{ox}}} \sqrt{ 2q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right) } \end{array}\]

    Note that \(\varepsilon_{\text{ox}}\) is simply one over \(c_{\text{ox}}\) the oxide capacitance, which we described earlier. Thus \[\Delta \left(V_{\text{ox}}\right) = \frac{1}{c_{\text{ox}}} \sqrt{2q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right) }\]

    And the threshold voltage \(V_{t}\) is then given as \[\begin{array}{l} V_{t} &= \Delta \left(V_{\text{Si}}\right) + \Delta \left(V_{\text{ox}}\right) \\ &= \Delta \left(V_{\text{Si}}\right) + \frac{1}{c_{\text{ox}}} \sqrt{ 2q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right) } \end{array}\]

    which is not that hard to calculate! Equation \(\PageIndex{10}\) is one of the most important equations in this discussion of field effect transistors, as it tells us when the MOS device is turned on.

    Equation \(\PageIndex{10}\) has several "handles" available to the device engineer to build a device with a given threshold voltage. We know that as we increase the acceptor density \(N_{a}\) the Fermi level gets closer to the valance band, and hence \(\Delta \left(V_{\text{Si}}\right)\) will change somewhat. But as we said, it will always be around \(0.8\) to \(1 \mathrm{~V}\), so it will not be the driving term which dominates \(V_{T}\). Let's see what we get with an acceptor concentration of \(10^{17}\). Just for completeness, let's calculate \(E_{f} - E_{v}\). \[\begin{array}{l} p &= N_{a} \\ &= N_{v} e^{\frac{E_{f}-E_{v}}{kT}} \end{array}\]

    Thus, \(E_{f} - E_{v} = kT \ \ln \left(\frac{N_{v}}{N_{a}}\right)\).

    In silicon, \(N_{v}\) is \(1.08 \times 10^{19}\) and this makes \(E_{f} - E_{v} = 0.117 \mathrm{~eV}\), which we will call \(\Delta (E)\). It is conventional to say that a surface is inverted if, at the silicon surface, \(E_{c} - E_{f}\), the distance between the conduction band and the Fermi level is the same as the distance between the Fermi level and the valance band in the bulk. With a little time spent looking at Equation \(\PageIndex{4}\), you should be able to convince yourself that the total energy change in going from the bulk to the surface in this case would be \[\begin{array}{l} q \Delta \left(V_{\text{Si}}\right) &= E_{g} - 2 \Delta (E) \\ &= 1.1 \mathrm{~eV} - 2 \times \left(0.117 \mathrm{~eV}\right) \\ &= 0.866 \mathrm{~eV} \end{array}\]

    Band diagram where conductance and valance bands curve down at the left ends, so E_c is only a small distance Delta E above the Fermi level. Delta E is also the distance between the Fermi level and the maximum, stabilized value of the valance band, and is equal to 0.117 eV. There is a difference E_g = 1.1 eV between the stabilized horizontal portions of E_c and E_v.
    Figure \(\PageIndex{5}\): Example of finding \(\Delta \left(V_{\text{Si}}\right)\)

    Using \(N_{A}=10^{17}\), \(\varepsilon_{\text{Si}} = 1.1 \times 10^{-12} \ \frac{\mathrm{F}}{\mathrm{cm}}\) and \(q = 1.6 \times 10^{-19} \mathrm{~C}\), we find that \[\sqrt{2 q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right)} = 1.74 \times 10^{-7}\]

    We saw earlier that if we have an oxide thickness of \(250 \ \AA\), we get a value for \(c_{\text{ox}}\) of \(1.3 \times 10^{-7} \ \frac{\mathrm{F}}{\mathrm{cm}^2}\), or \(\frac{\mathrm{Coulombs}}{\mathrm{V} \cdot \mathrm{cm}^2}\), and so \[\begin{array}{l} \Delta \left(V_{\text{ox}}\right) &= \frac{1}{c_{\text{ox}}} \sqrt{ 2q \varepsilon_{\text{Si}} N_{a} \Delta \left(V_{\text{Si}}\right) } \\ &= \frac{1}{1.3 \times 10^{-7}} 1.74 \times 10^{-7} \\ &= 1.32 \mathrm{~V} \end{array}\]

    and \[\begin{array}{l} V_{t} &= \Delta \left(V_{\text{Si}}\right) + \Delta \left(V_{\text{ox}}\right) \\ &= 0.866 + 1.32 \\ &= 2.18 \mathrm{~V} \end{array}\]

     

    This page titled 3.4.3: Threshold Voltage is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Bill Wilson via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.