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Schottky diode operation1 Ideal junction characteristics The shottky-barrier diode is formed by a metal contact (anode) to a semiconductor(the cathode), instead of the more common junction between P- and N-typesemiconductors. Shottky diodes differ from PN-junction devices in that rectificationoccurs because of differ in work function between the metal contact and thesemiconductor, rather a nonuniform doping profile. Conduction is not controlled byminority carrier recombination in the semiconductor, but by thermionic emission ofmajority carriers over the barrier created by the unequal work functions. The Schottkydiode is, therefore, a majority carrier device whose switching speed is not limited byminority carrier effects. Many metals can create a Shottky barrier on either silicon or GaAssemiconductors. For GaAs the most common are platinum, titanium and gold.Fig.1Band structure of the metal and semiconductor before contact. E0 is the free-space energy level, EC isthe bottom of the conduction band, and Ev is on the top of the valence band. Efm and Efs are the Fermilevels in the metal and semiconductor, respectively. Figure 1 shows the energy band diagrams of a metal and an N-typesemiconductors. The difference between the Fermi level Ef for each material andthe free-space energy level E0 is the work function , qφm or qφs ,where q is theelectron charge . The work function is, therefore, the average energy required toremove an electron from the material .The electron affinity, qX , is the energyrequired to free space. X is a constant for each material, and must remain constant
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throughout it. However, the Fermi level of the semiconductor, and hence its work function, can be expected to vary with doping density. When the metal and semiconductor are in equilibrium and are not in contact, the energy levels are constant throughout the materials .The Fermi levels are generally unequal, indicating that the electrons in one material (in this case the metal) have less energy, on the average, than these in the semiconductor. Therefore, when the materials are joined, some of the electrons in the semiconductor move spontaneously into the metal and collect on the surface. These leave behind ionized donor locations, which are positively charged, and create a negative surface charge where they collect on the surface of the metal. An electric field is set up between these positive charges and the electrons that eventually inhibits further electron flow into the metal. The positively charged region is called a depletion region, since it is almost completely depleted of mobile electrons. The shape of the energy diagram of the metal-semiconductor junction is governed by three rules: 1. In equilibrium, the Fermi levels for the semiconductor and metal must be constant throughout the system; 2. The electron affinity must be constant; 3. The free-space energy level must be continuous. Fig.2(a)Band structure of the Shottky junction; (b) charge densities at the junction (the negative componentis the surface electron concentration on the metal); (c) electric field in the depletion region. Figure 2a shows the resulting band structure when the metal and semiconductor are joined. In order to satisfy all three rules simultaneously, the valence and
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conduction bands of the semiconductor are forced to bend at the junction; the upwardbend of the conduction band of the N-type semiconductor indicates depletion region.The resulting potential difference across this region, as shown in the figure, is simplythe difference between the work function, φbi = φm - φs. This is called the built-inpotential of the junction. The positively charge depletion region in the semiconductor, can be considered tobe an area of stored charge. Indeed charge has been moved onto the metal contact,one "plate " of the capacitor, and off the semiconductor, the other "plate ", by theapplication of the built-in potential difference. Before it is possible to determine thecapacitance, it is necessary to find the quality of charge that has been moved, which isequal to the depletion zone charge. The depletion zone charge density to known:because the depletion zone charge is due to donor atoms, all of which are ionized, it isequal to the doping density. The junction area is, of course, known, but the width ofdepletion zone still must be found be found in order to determine the total charge. The electric field in the depletion zone is found by applying the Gauss law to theregion. It should be obvious that the electric field is in the negative x direction (fig2),and that it is maximum at the junction. It must also be zero at the edge of thedepletion region, because E=-dφ/dx=0, as evidenced by the flat band at this point.Secondly, the voltage across the junction, found by integrating the electric field, mustequal φbi. Applying the Gauss law in one dimension,(1) dE(x) / dx = ρ(x) / εs = qNd / εs(2) E(x) = Emax(1-x/d) where(3) F=-qNdd/εsEmax - maximum electric fieldd- depletion widthNd- doping density (assumed uniform)εs -dielectric permittivity of the semiconductorεs -=13.1∗ ε0 for GaAsεs -=11.9∗ ε0 for silicon ε0 =8.854∗ 10 F cm-1 -14An assumption used in deriving (1) to (3) is that the edge of the depletion region isabrupt; i.e., there is no gradual variation in charge density between the depletionregion and undepleted semiconductor. This assumption is called the depletionapproximation. In fact, a narrow transition region does exist, but it effect is negligiblefor most purposes. Since E (x) is a simple triangle function, it is easily integrated to give(4) φbi = Emax d/2 = qd2 Nd / 2εsThe resulting depletion width d, is(5) d= 2 φbi εs / q Nd
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The charge contained in the depletion region is found from the donor density and thedimensions of the region, which are known. The depletion charge, QJ , is(6) QJ = qWd Nd = W 2q φbi es Ndwhere W is the area of junction. This relates directly to the junction capacitance.2 Ideal I/V characteristic and junction capacitanceFig.3Fig.3 biased Schottky junction: (a) forward bias; (b) reverse bias. The Fermi levels are offset by anamount to the applied voltage. Figure 3 shows a biased Schottky junction. Since biased is applied, the junctionis no longer in equilibrium, and the requirement that the Fermi levels be constantthroughout the diode no longer applies. Instead, the Fermi levels (which shouldrightly be called quasi-Fermi levels for the nonequilibrium case), move with appliedvoltage. The offset from their equilibrium position is simply equal to qV, where V isthe applied voltage. The voltage across the junction then is φbi -V, where V is definedas positive with polarity that forward-biased the junction. The expression for electricfield E (x), maximum electric field Emax , depletion region d, and charge QJ are stillvalid for the biased diode as long as the potential φbi -V. the resulting expressions forcharge and depletion width are as follows:(7) QJ = W 2q es Nd(φbi -V)(8) d= 2 es (φbi -V) / q Nd
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The capacitance of charge with junction voltage. Taking this derivative, the junctioncapacitance is found:(9) d QJ /dV = C(V) = W q es Nd /2(φbi -V) = Wes/dThis can be put into the form(10) C(V) = CJ0/(1-Vφbi)1/2which is most useful for circuit analysis. CJ0 is the junction capacitance at zero biasvoltage. The exponent 1/2 in the denominator of (10) comes from the assumption that thedoping density Nd is constant throughout the semiconductor. In practice, Nd may notbe uniform, thus changing the exponent .One of the most dramatic examples of this isthe Mott diode, the capacitance of which has relatively weak dependence on voltage. The junction can be found by several methods, which give the same generalvoltage dependence. The following derivation is simple and intuitively satisfying. Electron conduction occurs primarily by thermionic emission over the barrier.This emission occurs equally in the both directions in equilibrium, at zero bias, givingno net current. When forward bias is applied, electron energy is increased relative tothe barrier height, allowing increased electron emission from the semiconductor intothe metal. The current component in the opposite direction stays constant.The electron density at junction, n1 , can be found from the Maxwell-Boltzmanndistribution. It is given by(11) n1 = Nd exp(-qφbi /KT)under zero bias conditions. The current in each direction is equal, and must beproportional to this electron density. Under bias, the potential barrier becomes φbi-V,and therefore the density of forward-conducted electrons is(12) n2 = Nd exp[-q(φbi-V )/KT]where K is Boltzmanns constant (1.37∗ 10-23 J/K) and T is absolute temperature. Thecurrent is proportional to the difference between these densities,(13) I(V)= I0 [exp(qV/KT) -1]Equation (13) is called the ideal diode equation. In order to compensate for nonidealbehavior, it is usually modified to form(14) I(V)= I0 [exp(qV/nKT) -1]Where n is a number close to 1.0, usually between 1.05 and 1.25, called the slopeparameter or ideality factor. Calculation of the current parameter I0 is much more complicated task, and probablyfutile , since I0 can be dominated by second-order effects such as leakage chargegeneration , and tunneling. Nevertheless, an ideal expression for I0 can be found byassuming that all current conduction is by thermionic emission. It is given by
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(15) I0 = A ** T2 Wexp(-qφbi /KT)where A ** is the modified Richardson constant, W is the junction area, and φbi is thebarrier height (difference between the Fermi level and the peak of the conductionband). A ** is approximately 96 A cm-2 K -2 for silicon 4.4 A cm-2 K -2 for GaAs. Thelow value of thee Richardson constant for GaAs implies that the knee of the I/Vcharacteristic occurs at higher applied voltages for GaAs diodes.3 Deviations from the Ideal Case Real Schottky diodes do not always follow the expressions derived before.Deviations from ideal behavior arise from imperfections in fabrication or factors,which are not included in this relatively simple theory. A few of the major limitationsare given below.1.Schottky Barrier Lowering It was assumed that the Schottky barrier height remained constant under allconditions of applied voltage. In fact, the barrier height varies with applied voltagebecause conduction electrons experience a force from their image charges in themetal. This force attracts the electrons toward the metal surface, effectively loweringthe barrier, and allowing voltage-dependent deviations from ideal behavior. In theory,this "image force" should give the reverse current a fourth-power dependence of biasupon voltage, rather than the constant value implied by (14). This effect is usually not observed, because carrier generation in the depletionregion at high reverse bias and tunneling effects dominate reverse leakage. At forward biases above approximately 0.1 V, the effect is to cause theideality factor n to deviate slightly from unity. For a diode that is ideal expect forbarrier lowering, the ideality factor is(16) n = 1/(1-dφbi /dV)where dφbi /dV is the variation in barrier height with applied voltage. The relation forthis quantity is.(17) dφb 1 q 3 Nd φbi - V- φfc - KT -3/4 dV 4 8π2 εs3 q where φfc is the potential difference between the Fermi level and bottom of theconduction band. As for the reverse case, this quantity rarely dominates the idealityfactor; for Nd = 1017 cm-3 , n is only 1.02.2. Surface Imperfections
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The semiconductor surface must be extremely clean in order to realize I/Vcharacteristics approximately the ideal. However, in spite of scrupulous care infabrication, the junction experiences at least a small amount of contamination due toimpurities. The deposition of the junction metal may also damage the crystal structureof the surface, especially if sputtering techniques are used. Formation of undesiredchemical compounds between the junction metal and the semiconductors may alsooccur, especially if the diode is subjected to high temperatures. Although diodes arerarely exposed to high temperatures in use, they are frequently exposed to hightemperatures as part of the fabrication process, such as annealing to repair sputteringdamage, attaching to a circuit, or as innocent bystanders when other componentssoldered into the mixer. The effect is to increase both the ideality factor and, in somecases, reverse conduction. Surface imperfections are probably the major cause ofnonideal behavior in Schottky diodes.3.Tunneling Thermal emission is not only mechanism by which electrons can cross thepotential barrier at the junction. Quantum mechanical tunneling through the barrier isalso possible, and may have a significant effect on I/V characteristic at lowtemperatures and high doping densities. Tunneling is often responsible for "soft" I/Vcharacteristics (i.e., high n) at low currents. It is of particular significance in devicesdesigned for cryogenic operation because, as temperatures are lowered, the currentcomponent due to tunneling does not decrease as rapidly as the thermioniccomponent. Tunneling also increases the noise temperature of the diode.4.Series Resistance Schottky junctions generally require lightly doped semiconductors withrelatively high bulk resistivities. A lightly doped substrate would not be practical fordiode fabrication because it would result in high series resistance and poor ohmiccathode contacts. Practical diodes are, therefore fabricated on a lightly doped, thinepitaxial layer that is grown on a heavily doped, low resistance substrate. Thisstructure allows the lightly doped region to be used for the junction and the heavilydoped region to minimize series resistance. A high-quality ohmic contact can be madeto this heavily doped substrate. The undepleted epitaxial layer may still contribute to series resistance (Rs)because the epitaxial layer must be made thick enough to contain the depletion region,even at high reverse bias. For example, for a diode with an epitaxial doping density of2X1017 , the depletion depth at zero applied voltage is approximately 750 angstroms.This requires an epitaxial thickness of 1000-2000 angstroms to contain the depletionlayer at 5-6 V reverse bias. At forward bias, however, there may be 500-1500angstroms of undepleted, high-resistance expitaxial material under the junction. The remaining bulk resistance of the substrate and its ohmic contact, as well as theundepleted epitaxial area, may leave several ohms of resistance in series with thejunction. The resistance creates power losses, which are often substantial, especiallyin millimeter-wave mixers, where the junction area is very small. Series resistanceoften creates a lower limit to the diode size, which can be used; sub-micron diameterdiodes can be fabricated with present technology, but series resistance usually limitspractical sizes to 1.5-2.0µm.
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It is difficult to describe a general procedure for estimating series resistance,because it is strongly dependent on diode structure. Estimation is further complicatedby the fact that the skin effect causes the diode current to exist in the surface of thesubstrate, rather than in its bulk, at frequencies, above approximately 50 GHz.Similarly, at high frequencies, the whisker or other connecting wire may have severalohms of series resistance due to the skin effect. For example, the series resistance of a dot-matrix diode at high frequencies isestimated by first determining the current path in the chip. The current path is shownin figure (4). The current flows from the anode through the expitaxial layer, andspreads out because of the skin effect along the top surface of the chip to its edge.Because the substrate resistivity is much lower then that of the epitaxial layer, thecurrent exists primarily in the substrate. It then flows down the sides of the chip to themounting surface.Fig.4Current distribution in the dot-matrix diode. Therefore, the series resistance consists of three components: the undepletedepitaxial layer under the junction, the spreading resistance of the top side of the diodebetween the anode and the sides of the chip, and the resistance of the edges. Becauseit is thin compared to the diode diameter, there is little current spreading in theepitaxial layer. The resistance of the epitaxial layer is that of a cylinder of material:(18) R d1 = (t - d)/qNd µa 2where t is the epitaxial layer thickness, d is the depletion width, and µ is the electronmobility. The spreading resistance component is found by first approximating the chipas a cylinder with the anode in the center, and is approximately(19) R d2 = ln(b/a)/2πδqNd µwhere a is the anode diameter, b is the diameter of the chip ( which can beapproximated as a side length for the more common square chip ), and δ is the skindepth in the substrate material. The sidewall resistance is given by
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(20) R d3 = h/4πbδqNdwhere h is the chip height and w is the side width. This estimate for R d3 may be lowbecause mechanical damage or roughness of the side of the chip increases R d3 . The high-frequency series resistance is the sum of the three components:( 21) R s = R d1 + R d2 + R d3Of course, the series resistance measured at do includes only R d1 ; without skin effect,R d2 and R d3 are the bulk do resistance of the substrate, which is negligible comparedto R d1 . Diode series resistance specified by manufacturers invariably the do value.5.Edge Effects The expressions in section 2 all assume that the electric field is perpendicular tothe junction over its entire area. However, practical diodes are formed with a smallanode on a large semiconductor surface, so the fringing electric field near the edge ofthe metal anode is greater then that in the center. The current density is, therefore,greatest at the edge of the junction, and may be relatively low at the center. As aperiphery than its area. For this reason, small diodes are sometimes fabricated withmetal geometries, such as a cross shape, which increase the periphery to reduce seriesresistance, and to reduce the area in order to minimize capacitance.
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