Trouble replicating a calculation in Silicon Nanoelectronics

[Achmed]

Trouble replicating a calculation in "Silicon Nanoelectronics"

I'm reading the textbook "Silicon Nanoelectronics" and I've encountered an equation for the transmission probability, which you can see among the attachments.

In this equation, T is the transmission probability for a rectangular-shaped potential barrier with width d and height ϕ, where m* is the effective mass of silicon and q the elementary charge.

They go on to say that "From Equation (3.1), the barrier width
giving transmission probabilities of 1 × 10^-3 and 1 × 10^-6 at a barrier height of 100
mV can be estimated to be 10 and 5 nm, respectively."

I really, really want to replicate this estimation/calculation but I can't seem to do in. When I plug in the given numbers (ϕ = 100 mV, d = 5 or 10 nm, q = elementary charge,


ħ = reduced Planck constant, and m* = effective mass silicon), I can't seem to get even remotely close to the listed probabilities. Perhaps the problem is the effective mass of silicon? I am not certain what value I should take, but I went with 0,2 times the mass of an electron (see: http://ecee.colorado.edu/~bart/book/effmass.htm)

Can anyone please show me how the writer approximately got 10nm and 5nm using the equation and the given probabilities? I know it's not a precise calculation, nor a precise equation, but I'd still like to see how he got this estimation.

 

[phyzguy]

I think you have the transmission probabilities swapped. The wider barrier will give a lower probability. Using your numbers I got 9.6 nm and 4.8nm for transmission probabilities of 1E-6 and 1E-3, respectively. Why don't you post your calculations and we'll see if we can find what is wrong.

 

[Achmed]

I think you have the transmission probabilities swapped. The wider barrier will give a lower probability. Using your numbers I got 9.6 nm and 4.8nm for transmission probabilities of 1E-6 and 1E-3, respectively. Why don't you post your calculations and we'll see if we can find what is wrong.

That's what I tought, too: It must be a sloppy mistake by the author (it is written this way by the author).

I don't know what I'm doing wrong, but I can't really show you anything insightful. I just plugged in the numbers into the formula, and my calculator keeps giving me 0 as the answer.

 

[phyzguy]

I can't give you any more help unless you show an attempt. Show what numbers you are plugging in and some intermediate steps in the calculation.

 

[Achmed]

I can't give you any more help unless you show an attempt. Show what numbers you are plugging in and some intermediate steps in the calculation.

q = 1.60217657 × 10^-19
ϕ = 100 x 10^6

(qϕ)^1/2 ≈ 4 x 10^-6

2m* = 0.4 * 9.10938291×10^ −31 ≈ 3.64 x 10^-31
ħ^2 = (1.05457173 × 10^-34)^2 ≈ 1,1 x 10^-68

(2m*/ħ^2 )^1/2 ≈ 5.75 x 10^18

-2 * 5.75 x 10^18 * 4 x 10^-6 ≈ -4.6 x 10^(13)

10^-3 = exp[-4.6 x 10^(13)d]

ln(10^-3) = -4.6 x 10^(13)d

d ≈ 1.5 x 10^-13 , which is obviously extremely incorrect.

 

[Cthugha]

ϕ = 100 x 10^6

That is 100 megavolts. That would be a huge barrier. You were talking about 100 millivolts before.

 

[Achmed]

That is 100 megavolts. That would be a huge barrier. You were talking about 100 millivolts before.

Wow... that solves it.