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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.
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.