Semiconductor Junctions


 Bruce Phillips
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1 8 Semiconductor Junctions Almost all solar cells contain junctions between different materials of different doping. Since these junctions are crucial to the operation of the solar cell, we will discuss their physics in this chapter. A pn junction fabricated in the same semiconductor material, such as csi, is an eample of an pn homojunction. There are also other types of junctions: A pn junction that is formed by two chemically different semiconductors is called a pn heterojunction. Inapin junctions, the region of the internal electric field is etended by inserting an intrinsic,i, layer between the ptype and the ntype layers. The ilayer behaves like a capacitor; it stretches the electric field formed by the pn junction across itself. Another type of the junction is between a metal and a semiconductor; this is called a MS junction. The Schottky barrier formed at the metalsemiconductor interface is a typical eample of the MS junction. 8.1 pn homojunctions Formation of a spacecharge region in the pn junction Figure 8.1 shows schematically isolated pieces of a ptype and an ntype semiconductor and their corresponding band diagrams. In both isolated pieces the charge neutrality is maintained. In the ntype semiconductor the large concentration of negativelycharged free electrons is compensated by positivelycharged ionised donor atoms. In the ptype semiconductor holes are the majority carriers and the positive charge of holes is compensated by negativelycharged ionised acceptor atoms. For the isolated ntype semicon 85
2 86 Solar Energy ntype region ptype region Band diagrams: electrons holes Figure 8.1: Schematic representation of an isolated ntype and ptype semiconductor and corresponding band diagrams. ductor we can write n = n n N D, / p = p n n 2 i N D. 8.1a 8.1b For the isolated ptype semiconductor we have p = p p N A, / n = n p n 2 i N A. 8.2a 8.2b When a ptype and an ntype semiconductor are brought together, a very large difference in electron concentration between n and ptype regions causes a diffusion current of electrons from the ntype material across the metallurgical junction into the ptype material. The term metallurgical junction denotes the interface between the n and ptype regions. Similarly, the difference in hole concentration causes a diffusion current of holes from the p to the ntype material. Due to this diffusion process the region close to the metallurgical junction becomes almost completely depleted of mobile charge carriers. The gradual depletion of the charge carriers gives rise to a space charge created by the charge of the ionised donor and acceptor atoms that is not compensated by the mobile charges any more. This region of the space charge is called the spacecharge region or depleted region and is schematically illustrated in Fig Regions outside the depletion region, in which the charge neutrality is conserved, are denoted as the quasineutral regions.
3 + 8. Semiconductor Junctions 87 ntype region space charge region ptype region electrons Efield holes Figure 8.2: Formation of a spacecharge region, when ntype and ptype semiconductors are brought together to form a junction. The coloured part represents the spacecharge region. The space charge around the metallurgical junction results in the formation of an internal electric field which forces the charge carriers to move in the opposite direction than the concentration gradient. The diffusion currents continue to flow until the forces acting on the charge carriers, namely the concentration gradient and the internal electrical field, compensate each other. The driving force for the charge transport does not eist any more and no net current flows through the pn junction The pn junction under equilibrium The pn junction represents a system of charged particles in diffusive equilibrium in which the electrochemical potential is constant and independent of position. The electrochemical potential describes an average energy of electrons and is represented by the Fermi energy. Figure 8.3 a shows the band diagrams of isolated n and a ptype semiconductors. The band diagrams are drawn such that the vacuum energy level E vac is aligned. This energy level represents the energy just outside the atom, if an electron is elevated to E vac it leaves the sphere of influence of the atom. Also the electron affinity χ e is shown, which is defined as the potential that an electron present in the conduction band requires to be elevated to an energy level just outside the atom, i.e. E vac. The band diagram of the pn junction in equilibrium is shown in Figure 8.3 b. Note, that in the band diagram the Fermi energy is constant across the junction, not the vacuum energy. As the Fermi energy denotes the filling level of electrons, this is the level that is constant throughout the junction. To visualise this, we take a look at Fig. 8.4 a, which shows two tubes of different lengths that are partially filled with water. The filling level is equivalent to the Fermi energy in a solid state material. The vacuum energy would be the upper boundary of the tube; if the water was elevated above this level, it could leave the tube. If the two tubes are connected as illustrated in Fig. 8.4 b, the water level in both tubes will be the same. However, the length of the tubes might be different; for leaving the first tube, a different energy can be required than for leaving the second tube. In addition to the Fermi energy being constant across the junction, the the bandedge
4 88 Solar Energy a ntype Si ptype Si b ntype Si ψ ptype Si E vac qvbi V bi l n l p qχ e qχ e qχ e E vac E C E C E 1 E F E G E G E C E F E 1 E V E 2 E V E F E G E 2 qv bi E V E Figure 8.3: a The energy band diagrams of an n and a ptype material that are separated from each other. b The energyband diagram of the pn junction under equilibrium. The electrostatic potential profile green curve is also presented in the figure. a b Figure 8.4: a Two tubes of different length upper boundary represents vacuum level and filling level representing Fermi energy. b If the tubes are connected, the filling level will equalise but the heights of the tube boundaries can be different.
5 8. Semiconductor Junctions 89 ntype silicon ptype silicon n=n n =N D p=p p =N A ln n ln p p=p n =n i2 /N D n=n p =p i2 /N A Position Figure 8.5: Concentrations profile of mobile charge carriers in a pn junction under equilibrium. energies E C and E V as well as the vacuum energy E vac must be continuous. Hence, the bands get bended, which indicates the presence of an electric field in this region. Due to the electric field a difference in the electrostatic potential is created between the boundaries of the spacecharge region. Across the depletion region the changes in the carrier concentration are compensated by changes in the electrostatic potential. The electrostaticpotential profile ψ is also drawn in Fig. 8.3 b. The concentration profile of charge carriers in a pn junction is schematically presented in Fig In the quasineutral regions the concentration of electrons and holes is the same as in the isolated doped semiconductors. In the spacecharge region the concentrations of majority charge carriers decrease very rapidly. This fact allows us to use the assumption that the spacecharge region is depleted of mobile charge carriers. This assumption means that the charge of the mobile carriers represents a negligible contribution to the total space charge in the depletion region. The space charge in this region is fully determined by the ionised dopant atoms fied in the lattice. The presence of the internal electric field inside the pn junction means that there is an electrostatic potential difference, V bi, across the spacecharge region. We shall determine a profile of the internal electric field and electrostatic potential in the pn junction. First we introduce an approimation, which simplifies the calculation of the electric field and electrostaticpotential. This approimation the depletion approimation assumes that the spacecharge density, ρ, is zero in the quasineutral regions and it is fully determined by the concentration of ionised dopants in the depletion region. In the depletion region of the ntype semiconductor it is the concentration of positively charged donor atoms, N D, which determines the space charge in this region. In the ptype semiconductor, the concentration of negatively charged acceptor atoms, N A, determines the space charge in the depletion region. This is illustrated in Fig Further, we assume that the pn junction is a step junction; it means that there is an abrupt change in doping at the metallurgical junction and the doping concentration is uniform both in the ptype and the ntype semiconductors. In Fig. 8.6, the position of the metallurgical junction is placed at zero, the width of
6 9 Solar Energy a ntype ρ ptype qn D qn A b ξ ma ξ c ψ V bi l n l p Figure 8.6: a The spacecharge density ρ, b the electric field ξ, and c the electrostatic potential ψ across the depletion region of a np junction under equilibrium.
7 8. Semiconductor Junctions 91 the spacecharge region in the ntype material is denoted as l n and the width of the spacecharge region in the ptype material is denoted as l p. The spacecharge density is described by ρ = qn D for l n, 8.3a ρ = qn A for l p, 8.3b where N D and N A is the concentration of donor and acceptor atoms, respectively. Outside the spacecharge region the spacecharge density is zero. The electric field which is calculated from the Poisson s equation, in one dimension can be written as d 2 ψ d 2 = d ξ d = ρ, 8.4 ɛ r ɛ where ψ is the electrostatic potential, ξ is the electric field, ρ is the spacecharge density, ɛ r is the semiconductor dielectric constant and ɛ is the vacuum permittivity. The vacuum permittivity is = F/cm and for crystalline silicon ɛ r = The electric field profile can be found by integrating the spacecharge density across the spacecharge region, ξ = 1 ɛ r ɛ ρ d. 8.5 Substituting the spacecharge density with Eqs. 8.3 and using the boundary conditions ξ l n = ξ l p =, 8.6 we obtain as solution for the electric field ξ = q N D l n + ɛ r ɛ for l n, 8.7a ξ = q N ɛ r ɛ A lp for l p. 8.7b At the metallurgical junction, =, the electric field is continuous, which requires that the following condition has to be fulfilled N A l p = N D l n. 8.8 Outside the spacecharge region the material is electrically neutral and therefore the electric field is zero there. The profile of the electrostatic potential is calculated by integrating the electric field throughout the spacecharge region and applying the boundary conditions, ψ = ξd. 8.9 We define the zero electrostatic potential level at the outside edge of the ptype semiconductor. Since we assume no potential drop across the quasineutral region the electrostatic potential at the boundary of the spacecharge region in the ptype material is also zero, ψl p =. 8.1
8 92 Solar Energy Using Eqs. 8.7 for describing the electric field in the n and pdoped regions of the spacecharge region, and taking into account that at the metallurgical junction the electrostatic potential is continuous, we can write the solution for the electrostatic potential as ψ = ψ = q 2ɛ r ɛ N D + l n 2 + q for l n, 8.11a N D l 2 n + N 2ɛ r ɛ A l 2 p q 2 N 2ɛ r ɛ A lp for l p. 8.11b Under equilibrium a difference in electrostatic potential, V bi, develops across the spacecharge region. The electrostatic potential difference across the pn junction is an important characteristic of the junction and is denoted as the builtin voltage or diffusion potential of the pn junction. We can calculate V bi as the difference between the electrostatic potential at the edges of the spacecharge region, V bi = ψ l n ψ l p = ψ ln Using Eq. 8.11a we obtain for the builtin voltage V bi = q N D l 2 n + N 2ɛ r ɛ A l 2 p 8.13 The builtin potential V bi can also be determined with using the energyband diagram presented in Fig. 8.3 b. qv bi = E G E 1 E Using Eqs. 6.1 and 6.22, which determine the band gap, and the positions of the Fermi energy in the n and ptype semiconductor, respectively, E G = E C E V, E 1 = E C E F = k B T ln N C /N D, E 2 = E F E V = k B T ln N V /N A. We can write NV NC qv bi = E G k B T ln k B T ln N A N D NV N = E G k B T ln C N A N D Using the relationship between the intrinsic concentration, n i and the band gap, E G [see Eq. 6.9], [ n 2 i = N C N V ep E ] G, k B T we can rewrite Eq and obtain V bi = k BT N ln A N D q n i
9 8. Semiconductor Junctions 93 This equation allows us to determine the builtin potential of a pn junction from the standard semiconductor parameters, such as doping concentrations and the intrinsic carrier concentration. We can calculate the width of the space charge region of the pn junction in the thermal equilibrium starting from Eq Using Eq. 8.8, either l n or l p can be eliminated from Eq. 8.13, resulting in epressions for l n and l p respectively, l n = 2ɛ r ɛ V bi q N A N D 1 N A + N D, 8.17a 2ɛ r ɛ l p = V bi N D b q N A N A + N D The total spacecharge width, W, is the sum of the partial spacecharge widths in the n and ptype semiconductors. Using Eq wefind W = l n + l p = 2ɛ r ɛ 1 V q bi N A N D The spacecharge region is not uniformly distributed in the n and p regions. The widths of the spacecharge region in the n and ptype semiconductor are determined by the doping concentrations as illustrated by Eqs Knowing the epressions for l n and l p we can determine the maimum value of the internal electric field, which is at the metallurgical junction. By substituting l p from epressed by Eq. 8.17b into Eq. 8.7b we obtain the epression for the maimum value of the internal electric field, ξ ma = 2q ɛ r ɛ V bi NA N D N A + N D Eample A crystalline silicon wafer is doped with 1 16 acceptor atoms per cubic centimetre. A 1 micrometer thick emitter layer is formed at the surface of the wafer with a uniform concentration of 1 18 donors per cubic centimetre. Assume a step pn junction and that all doping atoms are ionised. The intrinsic carrier concentration in silicon at 3 K is cm 3. Let us calculate the electron and hole concentrations in the p and ntype quasineutral regions at thermal equilibrium. We shall use Eqs. 8.1 and 8.2 to calculate the charge carrier concentrations. Ptype region: p = p p N A = 1 16 cm 3. / n = n p = n 2 i p p = / = cm 3 Ntype region: n = n n N A = 1 18 cm 3. / p = p n = n 2 i n n = / = cm 3 We can calculate the position of the Fermi energy in the quasineutral ntype and ptype regions,
10 94 Solar Energy respectively, using Eq. 6.22a. We assume that the reference energy level is the bottom of the conduction band, E C = ev. Ntype region: E F E C = k B T ln N C /n =.258 ln / 1 18 =.9 ev. Ptype region: E F E C = k B T ln N C /n =.258 ln / =.9 ev. The minus sign tells us that the Fermi energy is positioned below the conduction band. The builtin voltage across the pn junction is calculated using Eq. 8.16, [ ] V bi = k BT N ln A N D 1 q n 2 = V i =.81 V. 2 The width of the depletion region is calculated from Eq. 8.18, 2ɛ r ɛ W = 1 V q bi = 14 1 N A N D = cm =.325 μm. A typical thickness of csi wafers is 3 μm. The depletion region is.3 μm which represents.1%of the wafer thickness. It is important to realise that almost the whole bulk of the wafer is a quasineutral region without an internal electrical field. The maimum electric field is at the metallurgical junction and is calculated from Eq q NA N ξ ma = V D ɛ r ɛ bi = N A + N D = Vcm The pn junction under applied voltage When an eternal voltage, V a, is applied to a pn junction the potential difference between the n and ptype regions will change and the electrostatic potential across the spacecharge region will become V bi V a. Remember that under equilibrium the builtin potential is negative in the ptype region with respect to the ntype region. When the applied eternal voltage is negative with respect to the potential of the ptype region, the applied voltage will increase the potential difference across the pn junction. We refer to this situation as pn junction under reversebias voltage. The potential barrier across the junction is increased under reversebias voltage, which results in a wider spacecharge region. Figure 8.7 a shows the band diagram of the pn junction under reversebiased voltage. Under eternal voltage the pn junction is not under equilibrium any more and the concentrations of electrons and holes are described by the quasifermi energy for electrons, E Fn, and the quasifermi energy for holes, E Fp, respectively. When the applied eternal voltage is positive with respect to the potential of the ptype region, the applied voltage will decrease the potential difference across the pn junction. We refer to this situation as pn junction under forwardbias voltage. The band diagram of the pn junction under forwardbiased voltage is presented in Fig. 8.7 b. The potential barrier across the junction is decreased under forwardbias voltage and the space charge region becomes narrower.
11 8. Semiconductor Junctions 95 a ntype Si ψ ptype Si b ntype Si ψ ptype Si +V a V a qψ qψ E vac qχ e qχ e E vac E C E Fn E C E Fn EFp E G qv bi +V a E G E Fp qv bi V a E V E V E E Figure 8.7: Energy band diagram and electrostaticpotential in green of a pn junction under a reverse bias and b forward bias conditions. The balance between the forces responsible for diffusion concentration gradient and drift electric field is disturbed. Lowering the electrostatic potential barrier leads to a higher concentration of minority carriers at the edges of the spacecharge region compared to the situation in equilibrium. This process is referred to as minoritycarrier injection. This gradient in concentration causes the diffusion of the minority carriers from the edge into the bulk of the quasineutral region. The diffusion of minority carriers into the quasineutral region causes a socalled recombination current density, J rec, since the diffusing minority carriers recombine with the majority carriers in the bulk. The recombination current is compensated by the socalled thermal generation current, J gen, which is caused by the drift of minority carriers, which are present in the corresponding doped regions electrons in the ptype region and holes in the ntype region, across the junction. Both, the recombination and generation current densities have contributions from electrons and holes. When no voltage is applied to the pn junction, the situation inside the junction can be viewed as the balance between the recombination and generation current densities, J = J rec J gen = for V a = V. 8.2 It is assumed that when a moderate forwardbias voltage is applied to the junction the recombination current density increases with the Boltzmann factor epev a /k B T, qva J rec V a = J rec V a = ep k B T This assumption is called the Boltzmann approimation
12 96 Solar Energy The generation current density, on the other hand, is almost independent of the potential barrier across the junction and is determined by the availability of the thermallygenerated minority carriers in the doped regions, The eternal netcurrent density can be epressed as J gen V a J gen V a = J V a = J rec V a J gen V a ] qva = J [ep 1, k B T 8.23 where J is the saturation current density of the pn junction, given by J = J gen V a = Equation 8.23 is known as the Shockley equation that describes the currentvoltage behaviour of an ideal pn diode. It is a fundamental equation for microelectronics device physics. The saturation current density is also known as dark current density; its detailed derivation for the pn junction is carried out in Appendi B.1. The saturationcurrent density is given by J = qn 2 i DN L N N A + D P L P N D The saturationcurrent density depends in a comple way on the fundamental semiconductor parameters. Ideally the saturationcurrent density should be as low as possible and this requires an optimal and balanced design of the ptype and ntype semiconductor properties. For eample, an increase in the doping concentration decreases the diffusion length of the minority carriers, which means that the optimal product of these two quantities requires a delicate balance between these two properties. The recombination of the majority carriers due to the diffusion of the injected minority carriers into the bulk of the quasineutral regions results in a lowering of the concentration of the majority carriers compared to the one under equilibrium. The drop in the concentration of the majority carriers is balanced by the flow of the majority carriers from the electrodes into the bulk. In this way the net current flows through the pn junction under forwardbias voltage. For high reversebias voltage, the Boltzmann factor in Eq becomes very small and can be neglected. The net current density is given by JV a = J, 8.26 and represents the flu of thermally generated minority carriers across the junction. The current densityvoltage JV characteristic of an ideal pn junction is schematically shown in Fig The pn junction under illumination When a pn junction is illuminated, additional electronhole pairs are generated in the semiconductor. The concentration of minority carriers electrons in the ptype region and holes in the ntype region strongly increases, leading to the flow of the minority carriers
13 8. Semiconductor Junctions 97 J ln J J V ln J Slope q kt V Reverse bias Forward bias Figure 8.8: JV characteristic of a pn junction; a linear plot and b semilogarithmic plot. across the depletion region into the quasineutral regions. Electrons flow from the ptype into the ntype region and holes from the ntype into the ptype region. The flow of the photogenerated carriers causes the socalled photogeneration current density, J ph, which adds to the thermalgeneration current, J gen. When no eternal electrical contact between the ntype and the ptype regions is established, the junction is in opencircuit condition. Hence, the current resulting from the flu of photogenerated and thermallygenerated carriers has to be balanced by the opposite recombination current. The recombination current will increase through lowering of the electrostatic potential barrier across the depletion region. The band diagram of the illuminated pn junction under opencircuit condition is presented in Fig. 8.9 a. As we have seen in Section 7.6, under nonequilibrium the Fermi level is replaced by quasifermi levels that are different for electrons and holes and denote their electrochemical potential. Under opencircuit condition, the quasifermi level of electrons, denoted by E Fn, is higher than the quasifermi level of holes denoted by E Fp by an amount of qv oc. This means that a voltmeter will measure a voltage difference of V oc between the contacts of the pn junction. We refer to V oc as the opencircuit voltage. The bands in the quasineutral regions i.e. outside the depletion region are flat. Under the assumption that the charge density in each of the regions is homogeneous, the bands follow a parabolic shape in the depletion region. In this case there is generation anywhere in the device. Under open circuit condition, the eternal current density is zero. So we have dn 1 J n = J n, drift + J n, diff = qμ n ne + qd n d = qμ nn q dp 1 J p = J p, drift + J p, diff = qμ p pe qd n d = qμ p p q The carrier densities are given by: de C d de V d + k B Tμ n dn d, k B Tμ p dp d. 8.27a 8.27b n = N C ep E C E Fn k B T and p = N V ep E Fp E V k B T. 8.28
14 98 Solar Energy a ntype Si ψ ptype Si b ntype Si ψ ptype Si V oc E vac qψ qψ qχ e qχ e E vac E C E Fn E Fn E Fp qv oc E V qv bi V oc E C E Fn E Fp qv oc qv bi E Fp E V E E Figure 8.9: Energy band diagram and electrostaticpotential in green of an illuminated pn junction under the a opencircuit and b shortcircuit conditions. So we find for the derivatives dn d = n k B T dec d de Fn d We then find for each current component 1 de J n = qμ n n C q d 1 de J p = qμ p p V q d Hence, the total current is given by and dp d = + k B Tμ n dn J = J n + J p = μ n n de Fn d d = μ nn de Fn d, dp k B Tμ p d = μ pn de Fp d. + μ pn de Fp d p dev k B T d de Fp d 8.3a 8.3b = Note that the last step implies that the current density at V = V oc is zero. The current density can only be zero if de Fn d de Fp, 8.32 d which implies that the quasifermi levels are horizontal in the entire band diagram of the solar cell. Figure 8.9 b shows the band diagram of the shortcircuited pn junction. Under this situation, the photogenerated current will also flow through the eternal circuit. Under
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