Chapter 5 UEEP613 Microelectronic Fabrication Diffusion
Prepared by Dr. Lim Soo King 4 Jun 01
Chapter 5 Diffusion...131 5.0 Introduction... 131 5.1 Model of Diffusion in Solid... 133 5. Fick s Diffusion Equation... 134 5..1 Constant Diffusivity... 135 5..1.1 Constant Surface Concentration... 135 5..1. Constant Total Dopant... 136 5..1.3 Sheet Resistance of a Diffused Layer... 138 5..1.4 Effect of Successive Diffusion Steps... 139 5.. Concentration Dependent Diffusivity... 140 5..3 Temperature Dependent Diffusivity... 14 Exercises... 144 Bibliography... 146 - i -
Figure 5.1: Schematic of a diffusion system using liquid source... 13 Figure 5.: Mechanism of diffusion in solid... 133 Figure 5.3: Plot of complementary error function... 136 Figure 5.4: A surface Gaussian diffusion with total dopant Q T at the center of silicon... 138 Figure 5.5: A typical Irvin curve for p-type Gaussian profile in an n-type background concentration... 139 Figure 5.6: Diffusivity dependent on doping concentration... 141 Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal silicon... 14 Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon... 143 Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in silicon... 143 Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants... 144 - ii -
Chapter 5 Diffusion 5.0 Introduction Diffusion of impurity atom or dapant in silicon is an important process in silicon integrated circuit. Using diffusion techniques, altering the conductivity in silicon or germanium was disclosed in a patent by William Gardner Pfann in 195. Since then, various ways of introducing dopant into silicon by diffusion have been studied with the goal of controlling the distribution of dopant, the concentration of total dopant, its uniformity, and reproducibility, and for processing large number of device wafer in a batch to reduce the manufacturing cost. Diffusion is used to form emitter, base, and resistor for the bipolar device technology. It is also used to form drain and source regions and to dope polysilicon in MOS device technology. Dopant that spans a wide range of concentration can be introduced by a number of ways. The most common way of diffusion is from chemical source in vapor form at high temperature. The other ways are diffusion from a doped oxide source and diffusion and annealing from ion implanted layer. Ion implantation can provide 10 11 cm - to greater than 10 16 cm -. It is used to replace the chemical or doped oxide source wherever possible and is extensively used in VLSI/ULSI device fabrication. Diffusion of impurities is typically done by placing semiconductor wafers in a carefully controlled, high temperature quartz-tube furnace and passing a gas mixture that contains the desired dopant through it. Its purpose is to introduce dopant into silicon crystal. Mixture of oxygen and dopants such as diborane and phosphine are introduced in the furnace with the exposed wafer surface at temperature ranges between 800 0 C and 1,00 0 C for silicon and 600 0 C and 1,000 0 C for gallium arsenide. The number of dopant atoms that diffused into the semiconductor is related to the partial pressure of the dopant impurity in the gas mixture. The schematic of a diffusion system using liquid source is shown in Fig. 5.1. - 131 -
Figure 5.1: Schematic of a diffusion system using liquid source Dopant can be introduced by solid source such as BN for boron, As O 3 for arsenic, and P O 5 for phosphorus, gases source such as B H 6, AsH 3, and PH 3, and liguid source such as BBr 3, AsCl 3, and POCl 3. However, liquid source is the commonly used method. The chemical reaction for phosphorus diffusion using liquid source POCl 3 is shown as follow. 4POCl 3 + 3O P O 5 + 6Cl (5.1) The P O 5 forms a glass-on-silicon wafer and then reduces to phosphorus by silicon following the equation. P O 5 + 5Si 4P + 5SiO (5.) The phosphorus is released and diffused into silicon Si and chlorine Cl gas is vented. For diffusion in gallium arsenide, the high vapor pressure of arsenic requires special method to prevent loss of arsenic by decomposition or evaporation. These methods include diffusion in sealed ampules with over pressure of arsenic and diffusion in an open-tube furnace with doped oxide capping layer such as silicon nitride. Most of the studies on p-type diffusion have been confined to the use of zinc in the forms of Zn-Ga-As alloys and ZnAs for the sealed-ampule approach or ZnO-SiO for the open-tube approach. The n-type dopants in gallium arsenide include selenium and tellurium. To complete the process, 'drive in' or re-distribution of dopant is done in nitrogen or wet oxygen where silicon dioxide SiO is grown at the same time. - 13 -
5.1 Model of Diffusion in Solid At high temperature, point defects such as vacancy and self interstitial atom are generated in a single crystalline solid. When concentration gradient of the host or impurity atom exists, such point defect affects the atom movement namely as diffusion. Diffusion in solid can be treated as the atomic movement of diffusant either impurity atom or host atom in crystal lattice by vacancy and self intersitial. There are several types of diffusion mechanisms. They are vacancy diffusion, intersitial diffusion, divacancy or diffusion assisted by a double vacancies, and interstitialcy diffusion. Figure 5. illustrates the diffusion mechanisms by vacancy, interstitial, and interstitialcy. At elevated temperature, the atom in crystal lattice vibrates in its equilibrium site. Occasionally, the atom acquires sufficient energy to leave its equilibrium site and becomes a self interstitial atom. If there is an impurity atom (red color) around or a neigboring host atom, it can occupy this vacant site and this type of diffusion is termed as diffusion by a vacancy. This type of diffusion is illustrated by diffusion mechanism 1 shown in Fig. 5.(a). If the the migrating atom is a host atom, it is called self diffusion. If it is a impurity atom, then it is called impurity diffusion. (a) Vacancy diffusion and interstitial diffusion (b) Interstitialcy diffusion Figure 5.: Mechanism of diffusion in solid If the movement of impurity atom is in between equilibrium site of the crystal lattice that does not involve occupying lattice site, it is called interstitial diffusion as illustrated by diffusion mechanism shown in Fig. 5.(a). Diffusion assisted by a double vacancy or divacancy is a diffusion mechanism involving impurity atom has to move to a second vacancy that is at the nearest neighbor of the original vacancy site. - 133 -
Interstitialcy diffusion is shown in Fig. 5.(b). The mechanisms are shown by four steps. In step 1, a self interstitial host atom displaces an impurity atom (read color) from the lattice site and makes this impurity atom as interstitial atom (step ). This interstitial impurity atom then displaces a host atom (step 3) from its equilibrium site and makes this host atom to become interstitial atom (step 4). Vacancy and interstitialcy diffusions are commonly happened for phosphorus P, boron B, arsenic As, antimony Sb impurity diffusion in silicon. However, for phosphorus P and boron B diffusion, interstitialcy diffusion is more dominant than vacancy diffusion. Vacancy diffusion is more dominant than interstitialcy diffusion for arsenic As and antimony Sb diffusion. Group 1 and VIII elements have small ionic radii and are fast diffuser in silicon. The diffusion is normally involved interstitial diffusion. 5. Fick s Diffusion Equation In 1855, Adolf Fick published the theory of diffusion. His theory was based on the analogy between material transfer in a solution and heat transfer by conduction. Fick s assumed that in dilute liquid or gaseous solution, in the absence of convection, the transfer solute atom per unit area in one direction flow can be described by Fick s first law of diffusion shown in equation (5.3). J C(x, t) D x (5.3) where J is the local rate of transfer of solute per unit area or the diffusion flux, C is the concentration of solute is a function of x and t, x is the coordinate axis in the direction of solute flow, t is the diffusion time, and D is the diffusivity or at time it is called diffusion coefficient or diffusion constant. The negative sign of the equation denotes that the solute flows to the direction of lower concentration. From the law of conservation of matter, the change of solute concentration with time must be the same as the local decrease of the diffusion flux in the absence of a source or sink. Thus, C(x, t) J(x, t) t x (5.4) Substitute equation (5.3) into equation (5.4) yields equation of Fick s second law in one dimensional form, which is - 134 -
C(x, t) J(x, t) D t x x (5.5) When the concentration of solute is low, the diffusivity at a given temperature can be considered as a constant then equation (5.5) shall become C(x, t) D t J(x, t) x (5.6) Equation (5.6) is another form of Fick s second law of diffusion. In equation (5.6), D is given in unit of cm /s or m /h and C(x, t) is in unit of atom/cm 3. The solution for equation (5.6) for various initial condition and boundary condition shall be dealt in next sub-section. 5..1 Constant Diffusivity The solution of diffusion equation shown in equation (5.6) has constant diffusivity or diffusion coefficient for constant surface concentration and constant total dopant will be discussed in this sub-section. The sheet resistance of a diffused layer of constant diffusivity will be discussed too. 5..1.1 Constant Surface Concentration For the case of constant surface concentration, the initial condition at time t = 0 is C(x, 0) = 0 and the boundary conditions are C(0, t) = C s and C(, t) = 0. The solution of equation (5.6) is equal to C erfc x Dt C(x, t) S (5.7) where C s is the surface concentration, D is the constant diffusivity, x is the distance, t is the diffusion time, and erfc is the complementary error function. The plot of complementary error function erfc of equation (5.7) is shown in Fig. 5.3. - 135 -
Figure 5.3: Plot of complementary error function Since erfc(x) = 1 erf(x), equation (5.7) is also equal to x C(x, t) CS 1 erf (5.8) Dt From the result shown in Fig. 5., the error function solution is approximately a triangular function, so that the total amount of dopant per unit area introduced can be approximated by QT CS Dt. A more accurate answer for the total amount of dopant introduced per unit area is Q T 0 CS1 erf x CS dx Dt Dt (5.9) 5..1. Constant Total Dopant If a thin layer of dopant is deposited onto the silicon surface with a fixed or constant total amount of dopant Q T per unit area. This dopant diffuses only into the silicon and all the dopants remain in the silicon. The initial and boundary conditions are initial condition C(x, 0) = 0 and boundary condition - 136 -
C (x, t)dx Q 0 equation (5.6) shall be T and C(, t) = 0. The solution of the diffusion equation shown in C(x, t) Q T x exp (5.10) Dt 4Dt If x = 0, equation (5.10) is equal to surface concentration C S, which is C S QT C(0, t) (5.11) Dt Combining equation (5.10) and (5.11) yields equation (5.1). C(x, t) x CS exp (5.1) 4Dt Equation (5.10) is often called the Gaussian distribution and the diffusion concentration is referred to dopant concentration of the pre-deposited thin layer source or drive-in diffusion from a fixed total dopant concentration. Impurity atom distribution from ion implantation into amorphous material can be approximated by Gaussian function. For the case whereby a thin layer of dopant is deposited in the center the silicon surface, the diffusion profile looks as what is shown in Fig. 5.4. The diffusion flux will be equal to C(x, t) Q T x exp (5.13) Dt 4Dt whereby the assumption is that half of the Q T will diffuse virtually. - 137 -
Figure 5.4: A surface Gaussian diffusion with total dopant Q T at the center of silicon 5..1.3 Sheet Resistance of a Diffused Layer For a diffused layer that form pn junction, an average sheet resistance R S is defined and related to junction depth x j, the carrier mobility, and the impurity distribution C(x i ) by the equation (5.14). 1 1 R (5.14) S q x 0 j x j C(x)dx q eff 0 C(x)dx Empirical expression of mobility versus impurity distribution C has been determined for concentration above 10 16 cm -3 in silicon. The donor dopant mobility n is 1360 90 n 9.0cm / Vs (5.15) 17 0.91 1 (C /1.3x10 ) For acceptor concentration in silicon for acceptor dopant, the mobility p is 468 49.7 p 49.7cm / Vs (5.16) 17 0.7 1 (C /1.6x10 ) - 138 -
The sheet resistance R S is also equal to R S x (5.17) j where is the resistivity. Thus, the effective conductivity is equal to 1 R S x j (5.18) Once the surface resistance, surface concentration, and junction depth are known, one can design a diffused layer. There is a useful design curve called Irvin curve that can be used to determine the surface concentration C S versus the effective conductivity on background concentration C B, which shown in Fig. 5.5. Figure 5.5: A typical Irvin curve for p-type Gaussian profile in an n-type background concentration 5..1.4 Effect of Successive Diffusion Steps Since there are often multiple diffusion steps in a fully integrated circuit process, they must be added in some ways before the final profile can be predicted. It is clear that if all the diffusion steps occurred at a constant - 139 -
temperature where the diffusivity is the same then the effective Dt product is given by ( Dt) eff D 1(t1 t...) D1t1 D1t... (5.19) In other words doing a single step in a furnace for a total time of t 1 + t is the same as doing two separate steps, one for time t 1 and one for time t. Mathematically, one could increase the time t by a numerical factor D /D 1 and re-write equation (5.19) as D Dt (5.0) ( Dt) eff D1t1 D1 t D1t1 D 1 Thus, the derived formula for the total effective Dt for a dopant that is diffused at a temperature T 1 with diffusivity D 1 for time t 1 and then diffused at temperature T with diffusivity D for time t. The total effective Dt is given by the sum of all the individual Dt products. 5.. Concentration Dependent Diffusivity At high concentration, when the diffusion conditions are closed to the constant surface concentration case or the constant total dopant case, the measured impurity profiles are not the same as the constant diffusivity cases. For high concentration case, it can be represented by concentration dependent diffusivity. Anderson and Lisak obtained the concentration dependent diffusivity equation by changing the diffusivity of equation (5.5) with equation (5.19), which is D C Di n i r (5.1) where D i is the constant diffusivity at low concentration or intrinsic diffusivity; C is doping concentration; and n i is the intrinsic concentration, and r is a constant. We shall further discuss this equation. Based on many experiment results, the diffusivity of common dopants in silicon has been characterized and found to depend linearly or sometimes quadratically on the carrier concentration as shown in Fig. 5.6. - 140 -
Figure 5.6: Diffusivity dependent on doping concentration The effective diffusivity under extrinsic condition based on equation (5.6) can be written as D eff A D 0 D n n i D n n i for n-type dopant (5.) D eff A D 0 D p n i D p n i for p-type dopant (5.3) D 0 and D + etc. are chosen because on the atomic level. These different terms are thought to occur because of the interaction with neutral and charged point defect. The diffusivity under intrinsic condition for an n-type dopant (n = p = n i ) is * 0 D A D D D (5.4) The concentration dependent diffusivity of common dopant in single crystal silicon is shown in Fig. 5.7. - 141 -
By re-writing equation (5.) and (5.4), the diffusivity measured under extrinsic conditions can be elegantly described by equation (5.5). n n 1 eff * ni ni DA DA (5.5) 1 0 where D 0 / D and D / D. Si B In As Sb P Unit D o, 0 560 0.05 0.6 0.011 0.14 3.85 cm s -1 D o, E 4.76 3.5 3.5 3.5 3.65 3.66 ev D +, 0 0.95 0.6 cm s -1 D +, E 3.5 3.5 ev D -, 0 31.0 15.0 4.44 cm s -1 D -, E 4.15 4.08 4.0 ev D -, 0 44. cm s -1 D -, E 43.7 ev Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal silicon 5..3 Temperature Dependent Diffusivity The diffusivity determined experimentally over range of diffusion temperature can be expressed in Arrhenius form as D D E a exp kt o (5.6) where k is the Boltzmann constant and T is the temperature. The activation energy E a has a typical value between 3.5 to 4.5eV for impurity dopant in silicon. Plots of the diffusivity of common dopant in silicon are shown in Fig. 5.8 and 5.9 corresponding to the intrinsic diffusivity and activation energy shown in Fig. 5.10. Figure 5.10 also represents an Arrhenius fit to the diffusivity under intrinsic condition. - 14 -
Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in silicon - 143 -
Si B In As Sb P Unit D o 560 1.0 1. 9.17 4.58 4.70 cm s -1 E a 4.76 3.5 3.5 3.99 3.88 3.68 ev Exercises Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants 5.1. Given the solution of Fick s diffusion equation that satisfies the initial and boundary conditions is - 144 - C erfc x Dt C(x, t) S. Prove that the total number of dopant atoms per unit area of the semiconductor is QT(t) Cs Dt. 5.. Find the diffusivity and total impurity from a known impurity profile. Assume that boron is diffused into an n-type silicon single crystal substrate with doping concentration of 10 15 cm -3 and that the diffusion profile can be described by a Gaussian function. Using diffusion time of one hour, one obtains a measured junction depth of.0m and surface concentration of 1.0x10 18 cm -3. 5.3. A boron diffusion process such that the surface concentration is 4.0x10 17 cm -3, thickness x = 3.0m, substrate concentration C B = 1.0x10 15 cm -3. Calculate the drive in time if the diffusion temperature is 1,100 o C. 5.4. Boron pre-deposition is performed at 950 o C for 30 minutes in a neutral ambient. Given the activity energy of boron E a = 3.46eV, D 0 = 0.76 cm /sec and the boron surface concentration is C s = 1.8x10 0 cm -3. (i). Calculate the diffusion length. (ii). Determine the total amount of dopant introduced. 5.5. Assume the measured phosphorus profile can be represented by a Gaussian function with a diffusivity D =.3x10-13 cm s -1. The measured surface concentration is 1.0x10 18 cm -3, and the measured junction depth is 1.0m at a substrate concentration of 1.0x10 15 atoms/cm 3. (i). Calculate the diffusion time.
(ii). Find the total dopant in the diffused layer. 5.6. For boron diffusion in silicon at 1,000 o C, the surface concentration is maintained at 1.0x10 19 cm -3 and the diffusion time 1.0hr. If the diffusion coefficient of boron at 1,000 o C is.0x10-14 cm /s, calculate (i). The total dopant Q T diffused into silicon. (ii). The location where the dopant concentration reaches 1.0x10 18 cm -3. 5.7. Calculate the effective diffusion coefficient at 1,000 o C for two different box shaped arsenic profile grown by silicon epitaxy, one doped at 1.0x10 18 cm -3 and the other doped at 1.0x10 0 cm -3. - 145 -
Bibliography 1. JD Pummer, MD Del, and Peter Griffin, Silicon VLSI Technology Fundamentals, Practices, and Modeling, Prentice Hall, 000.. Hong Xiao, Introduction to Semiconductor Manufacturing Technology, Pearson Prentice Hall, 001. 3. SM Sze, VLSI Technology, second edition, McGraw-Hill, 1988. 4. CY Chang and SM Sze, ULSI Technology, McGraw-Hill, 1996. - 146 -
Index A Activation energy... 14 Adolf Fick... 134 Antimony... 134 Arrhenius... 14 Arsenic... 13, 134 Arsenic trioxide... 13 As O 3... See Arsenic trioxide B Boltzmann constant... 14 Boron... 134 C Chlorine... 13 Complementary error function... 135 D Diborane... 131 Diffusion... 131 Diffusion coefficient... 134 Diffusion time... 135 Diffusivity... 134 Divacancy diffusion... 133 Dopant... 131 E Effective conductivity... 139 F Fick s first law of diffusion... 134 Fick s second law of diffusion... 135 G Gaussian function... 137 I Impurity diffusion... 133 Interstitial diffusion... 133 Interstitialcy diffusion... 134 Intrinsic diffusivity... 140 Irvin curve... 139 M Mobility... 138 P P O 5... See Phosphorus pentoxide Phosphine... 131 Phosphorus... 13, 134 Phosporus pentoxide... 13 pn junction... 138 Point defect... 133 S Selenium... 13 Self diffusion... 133 Semiconductor Gallium arsenide... 131, 13 Germanium... 131 Sheet resistance... 135, 138, 139 Silicon... 138 Silicon dioxide... 13 T Tellurium... 13 U Ultra large scale integration... 131 V VLSI... 131 W William Gardner Pfann... 131 Z Zinc arsenate... 13 ZnAs... See Zinc arsenate Zn-Ga-As alloy... 13-147 -