EE 4611

PROBLEM SET 3

S. G. Burns

Due: Wednesday, 2 March 2016

Some mathematical warmup. 

1.      Show analytically that  is a solution for   and also show that    is also a solution    Use the closed form integrals in the class WEB notes.  Do not use MATLAB or MATHEMATICA.  You will have to use the chain rule for the first equation. 

The following problems require using the data and curves (not in the text) but embedded in the class notes PowerPoints related to semiconductor device design diffusion and oxidation.  Also recall the erfc and Gaussian graphs. The semiconductor device processis in the overview is in the Prologue and elaborated in a PowerPoint.

2.  Assume that an infinite source profile of boron doping can be approximated by an exponential of the form.  Assume the n-type substrate is doped at ND = 1017 cm-3. 

(a)  Sketch the diffusion profile

(b)  Compute the junction depth in μm.

(c ) Compute a value for the dose, Qo, with correct units.  Although you can use MATLAB or MATHEMATICA for the integration, the exponential function closed form by-hand method is probably much quicker.  Whichever you use, show the problem graphical set up.

3.  Similar to Problem 2 and extracted partially from an old quiz.  Assume a linear donor doping profile defined by                         as sketched.  The substrate is doped at NA = 1 x 1017cm-3.

 

 

 

 

 

 

 


(a)                        Compute a value for the junction depth, xj.

(b)                       What is the resultant dose, Qo?  [You can make your algebraic life easier by utilizing the linear

                   nature of the doping profile!]

(c)           This predeposition step is now capped and we proceed with a limited source diffusion for time   t1 and temperature T1.                     During the time, t1, used for this finite (limited source) drive diffusion, the surface concentration will (INCREASE,                               REMAIN THE SAME, DECREASE), the junction depth, xj, will (INCREASE, REMAIN THE SAME, DECREASE), and              the dose, Qo will (INCREASE, REMAIN THE SAME, DECREASE).  Circle your choices.

(d)                       Suppose the finite (limited) source diffusion in Part (c) was done at temperature T2, where T2> T1 and the time remains the                    same.  Comparing to the results from Part (c), the surface concentration would be (LOWER, ABOUT THE SAME), the                        junction depth, xj, would be (DEEPER, ABOUT THE SAME, NOT AS DEEP), and the dose, Qo would be (LARGER,                        ABOUT THE SAME, SMALLER)  Circle your choices.

4.      Using graphs from class notes, answer the following for oxidations:

(a)     How long to grow a 200Å  SiO2 MOSFET gate dielectric at 1000°C.  For this, use assume a dry oxidation.  Discuss several  reasons why this dry oxidation is preferable to a wet oxidation.

(b)    How long to grow a 0.3 μm   SiO2 masking layer at 1000°C.  For this, assume using a wet  oxidation.  Discuss several reasons why        this wet oxidation is preferable to a dry oxidation.

5.      Assume a p-doped substrate where NA = 5 x 1015 cm-3 .  From an infinite As source ND = 1 x 1019 cm-3, compute the junction depth, xj, for one hour and a two hour diffusion at 1100°C. 

6.      Compare the impurity concentration at 1μm with that of the surface concentration for a 1 hour and 2 hour diffusion of boron into an n-type substrate at 1100°C  To have some consistency in your solutions, use D=3.5 x 10-13 cm2/sec

7.      Refer to the information about ion implantation

 How many ions Dose, Q) are required to obtain a uniform doping profile of ND = 2 x 1017 cm-3 to a depth of 1μm?  How long will it take to ion implant a 12 inch (300 mm) diameter wafer assuming a beam current of 200 μA? Assume singly ionized ions.