Semiconductor device structures are traditionally divided into homojunction devices

0. Introduction: Semiconductor device structures are traditionally divided into homojunction devices (devices consisting of only one type of semiconductor material) and heterojunction devices (consisting of two or more different semiconductor materials). Heterojunction devices are generally more difficult and more expensive to fabricate, but provide the undisputed advantage that in addition to the doping type and concentration one can tailor the energy bandgap as well other material properties such as mobility and effective mass, to achieve superior performance. The use of varying bandgap material is sometimes referred to as "bandgap engineering". In some devices such as laser diodes one finds that the use of heterointerfaces is crucial to obtain room temperature operation. The purpose of this text is to provide the background needed to understand heterojunction devices and their design trade-offs. Focus is on heterojunction devices, the fundamentals of hetero-interfaces and the properties of quantum confined structures. In chapter one we briefly discuss the treatment of multilayer structures, the temperature and doping dependence of the energy bandgap, the calculation of the Fermi energy in bulk semiconductors and review the recombination-generation mechanisms in semiconductors, which are also extended to quantum wells. Chapter two contains data related to the most commonly used heterojunction material systems. Chapter three covers two types of unipolar junctions: the metal-semiconductor junction and the n + -n junctions. Chapter four deals with different types of p-n junction. These first four chapters describe the fundamentals of heterojunction interfaces. The emphasis has been on calculating the energy band diagram and the current-voltage characteristics. Chapters five, six and seven expand on three different types of devices: Photodetectors, Light emitting diodes and heterojunction bipolar transistors. The appendices deal with some properties of quantum confined structures, a list of semiconductor equations, physical and material constants, a Principles of Electronic Devices 1.1 Bart J. Van Zeghbroeck 1996

list of symbols, the exact treatment of the MOS capacitor and some useful optics related issues. Principles of Electronic Devices 1.2 Bart J. Van Zeghbroeck 1996

1. General solution to semiconductor devices 1.1 Multilayer structures 1.1.1 Electrostatics The first step when studying multilayer structures is to solve Poisson's equation. Under thermal equilibrium the Fermi energy level is constant, whereas with an external bias one ends up with two quasi-fermi levels, one for electrons and one for holes. 1.1.2 Simplifying multilayer structures Multilayer structures can be simplified by using some of the following rules: a) contact is made to the quasi-fermi level of the majority carriers carriers are injected into the semiconductor with an energy which is directly related to the position of the quasi-fermi level of the majority carriers. Applied voltages equal the difference between the quasi-fermi levels of the majority carriers b) quasi-fermi levels in a depletion region take on the value of the adjacent quasi-neutral region where the carriers are majority carriers This statement assumes that no current flows through the depletion region. Even though this assumption is obviously incorrect it enables to find reasonably accurate carrier densities throughout the depletion region from which the currents can be calculated. c) a multilayer structure can be partitioned if in the separating quasi-neutral region Fn = Fp Principles of Electronic Devices 1.3 Bart J. Van Zeghbroeck 1996

If the two quasi-fermi levels are equal in a quasi-neutral region between two devices, this region contains no excess carriers and the current is mainly carried by the majority carriers. The structure can therefore be split into two devices and can be combined again by using circuit equations d) multilayer structures can only be solved if the carrier transport is also considered Voltages at not-contacted internal layers of multilayer structures can only be obtained if the current through the devices is known. If partitioning (as described under c) is possible these voltages can easily be determined using circuit equations. If not a more detailed analysis is required. Examples of structures where the current transport affects the potential are the Kirk effect in a bipolar junction transistor and a phototransistor with a floating base. e) Built-in voltage φi = Φ1 - ΦN The built-in voltage of a device depends only on the work function of the first and last layer of the device. The work functions of the intermediate layers do not affect the built-in voltage. The built-in voltage is normally chosen to be a positive quantity. 1.2 Parameters of bulk semiconductors A detailed understanding of semiconductor devices requires knowledge of the semiconductor material. The following section describes the temperature and doping dependence of the energy bandgap as well as the complete calculation of the Fermi energy in bulk semiconductors as a function of temperature and the impurity concentrations. 1.2.1 Temperature dependence of the energy bandgap The temperature dependence of the energy bandgap is described by: Principles of Electronic Devices 1.4 Bart J. Van Zeghbroeck 1996

E g (T) = E g (0) - αt2 T + β [1.2.1] The numerical values for germanium, silicon and gallium arsenide are given in the following table: Germanium Silicon GaAs Bandgap at T = 0K E g (0) [ev] 0.7437 1.166 1.519 Coefficient α [10-4 /K] 4.774 4.73 5.405 Coefficient β [K] 235 636 204 The figure below provides a comparison of the energy bandgap versus temperature for germanium, silicon and gallium arsenide. 1.6 Energy Bandgap [ev] 1.4 1.2 1 0.8 0.6 GaAs Si Ge 0.4 0 200 400 600 800 1000 Temperature [K] Fig.1 Energy bandgap versus temperature for GaAs, silicon and germanium Principles of Electronic Devices 1.5 Bart J. Van Zeghbroeck 1996

1.2.2 Doping dependence of the energy bandgap High doping concentrations within a semiconductor causes the energy bandgap to decrease because of the impurity band, rather than the impurity levels associated with the donors or acceptors. As the doping density increases the individual wavefunctions of the electrons/holes occupying the impurity levels overlaps with others thereby causing a splitting of the individual levels and forming an impurity band. Conduction within this band is possible through hopping of the carriers from one impurity to the other. At high impurity densities, typically 10 18 cm -3, this impurity band overlaps with the conduction/valence band and effectively lowers the energy bandgap. The resulting reduction of the energy bandgap is characterized by: E g = 3q2 16πε s q 2 N ε s kt [1.2.1] and the corresponding intrinsic carrier concentration, n' i, is given by: n' i 2 = n i 2 exp( E g /kt) [1.2.2] For silicon this yields: E g = 22.5 N 10 18 cm-3 mev [1.2.3] A plot of E g versus doping density is shown in the figure below. Experimental data for GaAs at 4.2 K can be found in reference 1 and for silicon at room temperature in reference 2. 1 J. Pankove, "Optical Processes in Semiconductors", Dover, 1971, p 42. Principles of Electronic Devices 1.6 Bart J. Van Zeghbroeck 1996

250 200 Delta Eg [mev] 150 100 50 0 0.0E+00 2.0E+19 4.0E+19 6.0E+19 8.0E+19 1.0E+20 1.2E+20 Doping density [cm-3] 2 R.S. Muller and T.I. Kamins, "Device electronics for integrated circuits", second edition, Wiley and Sons, 1986, p 27. Principles of Electronic Devices 1.7 Bart J. Van Zeghbroeck 1996

1.2.3 Calculation of carrier densities at thermal equilibrium This section discusses the calculation of the free carrier densities and the Fermi level for bulk material in thermal equilibrium. The derivation includes the treatment of multiple conduction/valence band minima/maxima. A solution for non-degenerately doped material is provided as a series expansion known as the Joyce-Dixon approximation. An example of a graphical solution is also presented. The electron and hole densities by integrating the density of state functions multiplied with the Fermi-Dirac occupancy probability as expressed by: n = f(e) c (E) de and E v p = [1 - f(e)] c (E) de [1.2.4] E c - where c (E) is the density of states in the conduction band and v (E) is the density of states in the valence band and f(e) is the Fermi-Dirac distribution function: f(e) = 1 1 + exp[(e-e f )/kt] [1.2.5] The density of states in the conduction band is given by 3 : c (E) = M c 2 π 2 E - E c /h 3 (m* n ) 3/2 E E c [1.2.6] = 0 E E c [1.2.7] where M c is the number of equivalent minima in the conduction band and m* n is the density-of-state effective mass for electrons as defined by: 3 see also Appendix 1, section A.1.1. Principles of Electronic Devices 1.8 Bart J. Van Zeghbroeck 1996

m* n = (m* n,1 m* n,2 m* n,3 ) 1/3 [1.2.8] where m* n,1, m* n,2, and m* n,3 are the effective masses along the principal axis of the ellipsoidal energy surface around the conduction band minimum. The density of states can also be written as a function of the effective density of states, N c and N v : c (E) = 2 π N c kt E - E c kt E E c [1.2.9] = 0 E E c [1.2.10] v (E) = 2 π N v kt E v - E kt E E v [1.2.11] = 0 E E v [1.2.12] where the effective density of states in the conduction and valence band are given by: N c = M c 2 2π n kt h 2 [1.2.13] N v = M v 2 2π p kt h 2 [1.2.14] The integrals for the electron and hole densities can be written as a series approximation 4 : E f - E c kt - ln n 1 n - ( 3 N c 8 N c 16-3 9 ) ( n ) 2 +... [1.2.15] N c E v - E f kt - ln p 1 p - ( 3 N v 8 N v 16-3 9 ) ( p ) 2 +... [1.2.16] N v For non-degenerate semiconductors, as defined by: 4 W.B. Joyce and R.W. Dixon, Appl. Phys. Lett. 31, 354 (1977) Principles of Electronic Devices 1.9 Bart J. Van Zeghbroeck 1996

n N c /e 3 and p N v /e 3, or E v + 3kT E f E c - 3kT this reduces to: n = N c exp ( E f - E c kt ) and p = N v exp ( E v - E f kt ) [1.2.17] In the special case of intrinsic material where n = p = n i, this becomes: n i = N c exp ( E i - E c kt ) = N v exp ( E v - E i kt ) [1.2.18] from which one finds the mass action law: n p = N c N v exp -E g /kt = n i 2 [1.2.19] and the position of the intrinsic Fermi energy, E i : E i = E c + E v 2 + kt 2 ln N v = E c + E v + 3 N c 2 4 kt ln m* p + kt m* n 2 ln M v [1.2.20] M c Combining the above equations enables to write the carrier densities as a function of the intrinsic density, n i and the intrinsic Fermi energy, E i : n = n i exp ( E f - E i kt ) and p = n i exp ( E i - E f kt ) [1.2.21] The equations can also be solved for the Fermi energy: E f = E i + kt ln (n/n i ) and E f = E i - kt ln (p/n i ) [1.2.22] Charge neutrality must be fulfilled in a homogenous piece of semiconductor under thermal equilibrium, or the sum of all the positive charges equals the sum of all the negative charges: N d + + p = N a - + n [1.2.23] Principles of Electronic Devices 1.10 Bart J. Van Zeghbroeck 1996

where N d + and N a - are the densities of the ionized donor and acceptor atoms: N d + = N a - = N d 1 + 2 exp(e f - E d )/kt N a 1 + 4 exp(e a - E f )/kt [1.2.24] [1.2.25] These expressions resemble that for the Fermi-Dirac distribution function, except for the ground-state degeneracy factors 2 and 4. The factor 2 accounts for the fact that an electron occupying the donor level can have spin up or spin down which represents a twofold degeneracy. The same is true for electrons occupying the acceptor level, but in addition there are two degenerate valence bands which bring the total ground-state degeneracy factor to 4. Combination of the above equations allows to solve for the Fermi energy in a material with arbitrary donor and acceptor doping: N d 1 + 2 exp(e f - E d )/kt + n i exp ( E i - E f kt ) = N a 1 + 4 exp(e a - E f )/kt + n i exp ( E f - E i kt )[1.2.26] This equation can be solved graphically by plotting both sides of the equation as a function of the Fermi energy. The intersection of the two curves occurs at the Fermi energy from which the electron and hole density can be found. This procedure is illustrated with the figure below: Principles of Electronic Devices 1.11 Bart J. Van Zeghbroeck 1996

Carrier density [cm-3] 1.0E+22 p 1.0E+20 n 1.0E+18 1.0E+16 1.0E+14 p + Nd 1.0E+12 n + Na 1.0E+10-0.5 0 0.5 1 1.5 Fermi Energy [ev] Fig.1 Electron and hole density as well as the density of positive and negative particles versus Fermi energy in silicon with E g = 1.12 ev, E d = E c - 50 mev, E a = E v + 50 mev, N a = 10 16 cm -3 and N d = 10 14 cm -3. Principles of Electronic Devices 1.12 Bart J. Van Zeghbroeck 1996

1.3 Recombination and generation mechanisms 1.3.1 Light absorption This section first outlines the general procedure to obtain the absorption coefficient in a direct bandgap material. The result for an indirect bandgap is described next. Conservation of energy and momentum imposes the following relation on the photon energy, E ph, and the electron and hole momentum k n and k p : /h k n + /h k p = E ph /c e ph = h/λ e ph 0 [1.3.1] E ph = E g + / h 2 k n 2 2m n * + / h 2 k p 2 2m p * [1.3.2] where e ph is the unit vector along the propagation direction of the photon and E g is the energy bandgap of the semiconductor. These equations can be solved for the electron and hole momentum assuming the photon momentum to be negligible compared to both the electron and hole momentum k = 2m r * /h E ph - E g, with 1 m * = 1 r m * + 1 n m * p [1.3.3] where m r * is the reduced mass. Applying Fermi's golden rule one can then calculate 5 the absorption coefficient α: α = K E ph - E g [1.3.4] where the absorption coefficient is defined as the proportionality constant between the optical power, P opt, and the optical power absorbed per unit length: 5 Yariv, A. Quantum Electronics, Wiley and sons, third edition, p 238. Principles of Electronic Devices 1.13 Bart J. Van Zeghbroeck 1996

dp opt dx = - α P opt [1.3.5] For GaAs, K equals 11700 cm -1 ev -1/2. The generation rate, g op of electron-hole pairs due to light absorption can then be calculated from: g op = n t light = α q P opt (x) A E ph [1.3.6] This expression assumes that every photon absorbed creates one electron-hole pair. It also implies that other photon absorption processes such as free carrier absorption 6 have been ignored. The above analysis is only valid for direct bandgap semiconductors. In indirect bandgap semiconductor one finds that momentum can not be conserved unless another particle is involved. The dominant process is that where a phonon is either absorbed or emitted, yielding: 7 α = A (E ph - E g - E p )2 1 - exp(- E p /kt) + A (E ph - E g + E p )2 exp(e p /kt) - 1 [1.3.7] for each indirect bandgap which exists in the material of interest. Absorption in silicon at room temperature is primarily due to the absorption/emission of acoustic phonons and 6 Pankove, J. Optical processes in semiconductors, Dover publications, p 74. 7 Pankove, J. Optical processes in semiconductors, Dover publications, p 37. This derivation assumes the photon energy to be constant, which can be justified on the basis that the momentum supplied by the phonon does not vary rapidly with changing photon energy. This approximation is more correct for optical phonons than for acoustic phonons. Principles of Electronic Devices 1.14 Bart J. Van Zeghbroeck 1996

emission of optical phonons with an approximate phonon energy of 18 mev and 65 mev. The absorption coefficient in silicon can be approximated as: α SI = 6x10 3 (E ph - 1.1 ev) 2 + 8x10 4 (E ph - 2.5 ev) 2 + 1.29x10 6 Eph - 3.2eV [1.3.8] where the last term was added to account for the first direct bandgap in silicon. This expression is plotted in the figure below and compared to experimental data: 8 1.0E+07 Absorption coefficient [cm-1] 1.0E+06 1.0E+05 1.0E+04 1.0E+03 1.0E+02 1.0E+01 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Eph [ev] Fig.1.1 Absorption in silicon versus photon energy. Dots represent experimental values. 1.3.2 Band-to-Band recombination Band-to-band recombination is found to be a dominant recombination mechanism in direct bandgap semiconductors. This type of recombination is the inverse process of electron- 8 Data were taken from R.S. Muller and T.I. Kamins, "Device Electronics for Integrated Circuits", Wiley and sons, second edition, p. 24. Principles of Electronic Devices 1.15 Bart J. Van Zeghbroeck 1996

hole pair generation due to light absorption. It requires a filled state in the conduction band as well as an empty state with the same crystal momentum in the valence band, assuming the photon has negligible momentum. The recombination rate is therefore proportional to the product of the electron and the hole density. The generation however is independent of the carrier densities. It only depends on the material properties of the material. The recombination can be written as: R = v th σ n p = b n p [1.3.9] Where the constant b is also called the bimolecular recombination rate. The generation must equal the recombination under thermal equilibrium conditions, or for np = n i 2, yielding for the net recombination: U b-b = R - G = (np - n i 2 ) v th σ = b (np - n i 2 ) [1.3.10] with v th = 3kT m* 107 cm/s [1.3.11] For GaAs the bimolecular recombination rate b equals 1.5 x 10-10 cm 3 /s. For carrier densities which are close to the thermal equilibrium values, this equation is frequently linearized yielding for n-type material: U b-b = p n ' 1 with τ τ p = and p p b n n ' = p n - p n0 [1.3.12] n0 where τ p is the radiative carrier lifetime of the holes and p n0 n n0 are the thermal equilibrium carrier densities. 1.3.3 Trap assisted recombination a) Bulk recombination Principles of Electronic Devices 1.16 Bart J. Van Zeghbroeck 1996

The Shockley-Hall-Read recombination theory predicts that the net recombination rate under conditions which are close to thermal equilibrium is given by: 9 U SHR = N t v th n v th p σ n σ p (pn - n i 2 ) v th p σ p [p + n i exp((e i -E t )/kt)] + v th n σ n [n + n i exp((e t -E i )/kt)] [1.3.13] The largest recombination rate is obtained for E i = E t. This confirms the experimental fact that recombination centers in the middle of the bandgap are the most efficient. Equation [1.3.13] can be simplified further if σ n = σ p = σ, yielding: U SHR = (pn - n i 2 ) [n + p + 2 n i cosh((e i -E t )/kt)] τ 0, with 1 τ 0 = N t v th σ [1.3.14] The above expression can be further reduced for holes in n-type material (n>>p) to: U SHR = p n ' τ p with 1 τ p = N t v th σ [1.3.15] where τ p is the recombination time constant, also called the excess carrier lifetime, and p n ' p-p n0 is the excess carrier concentration where p n0 is the thermal equilibrium hole concentration. b) Surface recombination Surface recombination through localized states at the surface, called surface states takes the same form as that of bulk recombination except that the carrier concentrations per unit volume and the trap density per unit area at the surface determine the recombination rate (also per unit area) as indicated with the subscripts s. 9 A derivation can be found in R.S. Muller and T.I. Kamins, "Device Electronics for Integrated Circuits", Wiley and sons, second edition, p. 224 Principles of Electronic Devices 1.17 Bart J. Van Zeghbroeck 1996

N U s = st v th σ n σ p (p s n s - n i 2 ) σ p [p s + n i exp((e i -E st )/kt)]+σ n [n s + n i exp((e st -E i )/kt)] [1.3.16] This expression can be further reduced for n-type material (n>>p) to U s = s p' with s = N st v th σ [1.3.17] where s is the surface recombination velocity and p' is the excess carrier concentration. Note that when every carrier which arrives at the surface recombines, the recombination velocity equals the thermal velocity. The continuity equation at the surface takes the following simplified form under steady state conditions: J ps = - J ns = ± q U s = ± q s p' [1.3.18] where J ps and J ns are the hole respectively electron current densities at the surface. The sign is determined by the direction in which the carriers flow relative to the positive x direction. Notice that for every hole which flows towards the surface and recombines an electron also flows towards the surface, so that no net current flows within the sample. 1.3.4 Auger recombination Auger recombination can be looked upon as the inverse process of impact ionization. Three particles are involved: one electron, one hole and an additional hole or electron. The electron and hole recombine and give off the excess energy to the additional particle. Since three particles are involved the recombination rate is dependent on the product of the densities of the three particles and can be written in the following form: U A = Γ n n (np - n i 2 ) + Γ p p (np - n i 2 ) [1.3.19] where Γ n describes the process involving two electrons and Γ p the process involving two holes. For holes in n-type material this can be reduced to Principles of Electronic Devices 1.18 Bart J. Van Zeghbroeck 1996

U A = p n ' τ A with 1 τ A = Γ n N d 2 [1.3.20] The recombination rate is therefore proportional to the square of the donor concentration, so that Auger recombination is expected to dominate only in highly doped semiconductors. For p-type GaAs the Auger coefficient Γ equals 0.625 x 10-29 cm 6 s -1. 10 1.3.5 Recombination in a quantum well a) Band-to-band recombination Band-to-band recombination in quantum wells can be expressed as: U b-b,qw = Β 1 (N 1 P 1 - N i1 2 ) + Β 2 (N 2 P 2 - N i2 2 ) +... [1.3.21] Where the indices 1 and 2 refer to the quantized levels in the quantum well. In addition it was assumed that transitions between energy levels with different quantum number are forbidden. 11 Because of the different wavefunction of each quantized level the bimolecular recombination constant is also different for each possible transition. Frequently one assumes only the first level to be populated so that [1.3.21] reduces to: U b-b,qw = Β (NP - N i1 2 ) [1.3.22] 10 S. Tiwari, S. L. Wright and D. J. Frank, "Compound semiconductor heterostructure bipolar transistors", IBM Journal of Res. and Develop. Vol. 34, July 1990, p 550-567. 11 A first order calculation of the matrix elements suggests that transitions between energy levels of a quantum well with different quantum number are forbidden since the corresponding wavefunctions are orthogonal. A more detailed analysis for a GaAs/AlGaAs quantum well (as well as experimental results) reveals that some of those "forbidden" transitions are allowed. Principles of Electronic Devices 1.19 Bart J. Van Zeghbroeck 1996

For GaAs the recombination constant B is 5 x 10-5 cm 2 s -1. 12 b) Schockley Hall Read recombination Schockley Hall Read (SHR) recombination in quantum wells takes the following form: U SHR,qw = (NP - N i1 2 ) (N + P + 2 N i1 ) τ 0qw [1.3.23] This expression assumes that the recombination rate is independent of the quantized state the carriers are in. Generation is assumed to depend only on the first quantized state which in most cases is dominant. 12 Ch. S. Harder, B. J. Van Zeghbroeck, M. P. Kesler, H. P. Meier, P. Vettiger, D. J. Webb and P. Wolf, "High-speed GaAs/AlGaAs optoelectronic devices for computer applications", IBM Journal of Res. and Develop. Vol. 34, July 1990, p 568-584. Principles of Electronic Devices 1.20 Bart J. Van Zeghbroeck 1996