Semiconductor Devices and Electronic Circuits

Logic gate combinations

A combinational circuit is analysed by writing the Boolean expression at each gate output, starting from the inputs, and then simplifying the result. De Morgan's theorems, \overline{A+B}=\bar{A}\bar{B} and \overline{AB}=\bar{A}+\bar{B}, are the main tools for this. They show that an OR gate with inverted inputs is equivalent to a NAND gate, and an AND gate with inverted inputs is equivalent to a NOR gate. A NAND gate with its two inputs tied together behaves as an inverter, which is why NAND gates can be used to build NOT, AND and OR functions.

Simplification with Boolean identities usually reduces the gate count. Absorption, X+XY=X, and the identity \bar{A}+AX=\bar{A}+X remove redundant terms, while C\cdot C=C removes repeated factors. A function that reduces to a single variable or a single two-input operation needs only one gate, so the minimum number of gates is found from the simplified form and not from the original expression. A function of the form \bar{B}C can be realised with three two-input NAND gates, and no two-gate NAND network produces it.

To find the output for a given input, substitute the input values into the expression and evaluate each inverted term separately. The same method identifies which input pair gives a specified output level.

For timing diagrams, an OR of two signals is high whenever either signal is high, so the output consists of the union of the high intervals of the inputs. Each output interval must be read from the input waveforms one time slot at a time.

For fault diagnosis, the output of each gate is compared with its truth table for the measured inputs. A NOT gate is correct when its output is always the complement of its input, and a NOR gate is correct when its output equals the complement of the OR of its inputs in every interval. A mismatch at a gate output, while its inputs are correct, identifies that gate as faulty.

Op-Amp feedback circuits

Negative feedback stabilises an operational amplifier. Feeding a fraction \beta of the output back to the inverting input gives a closed-loop gain A_{cl}=\dfrac{A}{1+A\beta}, which is smaller than the open-loop gain A. For a constant gain-bandwidth product, the closed-loop bandwidth is extended by roughly the factor (1+A\beta). In a series-shunt (non-inverting) configuration, the input impedance is increased by (1+A\beta) and the output impedance is decreased by the same factor.

In the inverting summing amplifier, the inverting input is a virtual ground at 0 V, so each input current flows through its own resistor into the feedback resistor. The output is V_0=-R_f\sum_i V_i/R_i, so the output is the weighted negative sum of the inputs.

The non-inverting amplifier has closed-loop gain 1+R_f/R_1. The linear relation V_{out}=\text{gain}\times V_{in} holds only while the output stays within the supply rails. If the calculated output exceeds the rails, the op-amp saturates at approximately the supply voltage.

With positive feedback, a fraction of the output is fed back to the non-inverting input. A Schmitt trigger has two switching thresholds, \pm\beta V_{sat}, so the circuit shows hysteresis. When the input amplitude exceeds the thresholds, the output switches between +V_{sat} and -V_{sat} and produces a square wave rather than a sine wave.

In an open-loop comparator with no feedback, even a small difference between the two inputs drives the output to the positive or negative saturation level. The output is therefore a square wave whose peak-to-peak value equals twice the saturation voltage. For an input \sin\omega t, the frequency is f=\omega/2\pi and the period is T=1/f.

BJT operating characteristics

A bipolar junction transistor has an emitter that is heavily doped so that it injects a large density of carriers into the base. The base is lightly doped and thin, and the collector is moderately doped. Both electrons and holes take part in current conduction, which gives the device its name.

The common-emitter current gain \beta=I_C/I_B depends on temperature, because recombination in the base and the minority carrier lifetime change with temperature. Kirchhoff's current law gives I_E=I_B+I_C, so the collector current is always smaller than the emitter current when the base current is positive.

To determine the operating region of a biased transistor, first find the Thevenin equivalent seen by the base. If the base-emitter voltage is below about 0.7 V for a silicon device, the base-emitter junction is not forward biased, the transistor is in cutoff, the collector current is zero, and the output equals the supply voltage.

When an emitter resistor is present, the base-emitter loop equation is V_{BB}=I_BR_B+V_{BE}+I_ER_E, with I_E=(\beta_{DC}+1)I_B. Solving for I_B gives I_B=(V_{BB}-V_{BE})/[R_B+(\beta_{DC}+1)R_E]. The collector resistor does not appear in this loop. The result is valid only if the transistor is in the forward-active region, which should be checked by comparing the collector and base voltages.

Hall effect in semiconductors

In a Hall experiment, a current flowing along x in a magnetic field along z deflects the carriers in the y direction. Charge accumulates on one face until the transverse electric field balances the magnetic force, and a Hall voltage appears across the sample. For a single carrier type, the Hall resistivity is \rho_{xy}=B/(nq), which is proportional to the field and independent of the scattering time. Its magnitude therefore increases linearly with B and passes through the origin.

The sign of the Hall voltage shows the sign of the majority carriers. Electrons deflected in the Lorentz force accumulate on one face, which becomes negative, and holes give the opposite sign. The Hall field always opposes the magnetic deflection, so the direction of the field follows from the carrier charge and the directions of I and B.

In an intrinsic semiconductor, both electrons and holes contribute to the current, so the Hall voltage does not vanish in general. The Hall coefficient for mixed carriers is R_H=(p\mu_h^2-n\mu_e^2)/[e(p\mu_h+n\mu_e)^2], which depends on the mobilities as well as the densities. The sign of the Hall voltage is determined by which carrier type dominates this combination, not by the effective masses alone.

Because the Hall coefficient depends on mobility in a mixed system, it does not directly give a single carrier concentration. For a single carrier type, |R_H|=1/(ne), which can be used to estimate n. Using the measured Hall voltage, current and field, the field strength can be obtained from V_H=IB/(net), where t is the sample thickness along the field.

Metal-semiconductor Schottky junction

A metal-semiconductor contact is rectifying (Schottky) when the metal work function is larger than that of the n-type semiconductor. Electrons flow from the semiconductor into the metal until the Fermi levels align, which leaves a depletion layer and bends the bands upward at the interface. The resulting barrier controls conduction, so the junction behaves as a diode.

In forward bias, the current rises exponentially as the barrier is lowered. In reverse bias, the current saturates at a small value set by the barrier height, and at a large reverse voltage the junction breaks down. Equilibrium band diagrams therefore show equal Fermi levels on both sides, and any diagram with unequal Fermi levels is not an equilibrium picture.