The whole topic is one sentence: we plant dopant atoms into silicon at a chosen depth and a chosen amount, then let heat spread them by a rule that says atoms flow from crowded places to empty places. Everything else — the error function, the Gaussian, the Arrhenius factor — is just the precise bookkeeping for how many atoms are where, and how that changes with time.
This page builds every symbol the parent Ion Implantation and Diffusion note uses, starting from a smart 12-year-old who has never seen any of them. Read top to bottom; each block earns the next.
Before any symbol, fix the picture in your head. A wafer is a thin flat disc of silicon. We only ever care about how things change going straight down into it . So we draw one axis:
x x = 0 x
That single axis is the backbone of every formula on the parent page.
x x how far below the surface we are looking. It is a length (cm). The surface is x = 0 x
Now we need two different ways to count dopant atoms. They are not the same and confusing them is the #1 beginner error.
N ( x ) N ( x ) how many dopant atoms sit inside one cubic centimetre of silicon at depth x . Units: atoms/cm³ (often written cm⁻³).
Picture: at each depth, sample a tiny 1 cm³ box and count what's inside. N
Q Q how many dopant atoms sit under one square centimetre of surface, added up over ALL depths. Units: atoms/cm² (cm⁻²).
Picture: drill one thin vertical column straight down and count every dopant in that column. That total, per cm² of opening, is the dose.
The bridge between them is the key relationship you will meet again and again:
Q = ∫ 0 ∞ N ( x ) d x
Intuition Why the integral?N ( x ) d x d x Q Q area under the N ( x ) curve. Concentration is a density ; dose is its total .
Common mistake "Dose and concentration are basically the same number."Why it feels right: both count dopant atoms.
Fix: Q area (a total down the column), N volume (a local density). They even have different units. The whole area under the N Q
Link forward: doses and profiles are what Doping and PN Junctions turns into working device regions.
To describe movement we need one more counter.
J J how many atoms cross an imaginary plane, per second, per cm² of that plane. Units: atoms/(cm²·s).
Picture: hold a hoop flat at depth x J deeper (increasing x
Why we need it: diffusion is a story about atoms crossing planes , so we need a word for that crossing rate. J
The parent note says flux is driven by how steep the concentration is. "Steepness" is a derivative — so let us earn that symbol from zero.
Intuition What a derivativeis
Walk along the depth axis a tiny step d x d N derivative ∂ x ∂ N = d x d N change in N divided by the tiny step that caused it. It is the slope of the N x
Why the curly ∂ d N two things at once — depth x and time t ∂ N / ∂ x x partial derivative . (More in Diffusion Equation .)
If N ∂ N / ∂ x negative .
A steep fall = large-magnitude slope = strong driving force for diffusion.
Definition Second derivative∂ 2 N / ∂ x 2 of the derivative : how the slope itself changes with depth. Picture: it measures the curvature — is the concentration curve bulging up (a hill, negative curvature) or scooped up (a valley, positive curvature)? This is exactly what governs whether atoms are piling up or draining away at a point (Fick's 2nd law).
Definition Diffusion coefficientD D a single number saying how easily atoms wander through this material at this temperature. Units: cm²/s. Big D D
Why cm²/s and not cm/s? Because diffusion spreads like D t Fick's Laws .
D why , meet three symbols.
Definition Absolute temperatureT T how much random jiggling energy the atoms have. Hotter = more violent jiggling.
Definition Activation energyE a E a the energy "wall" an atom must clear to hop from one lattice site to the next. Units: electron-volts (eV) or joules. Picture: an atom sitting in a valley; to move to the next valley it must first climb over a hill of height E a
Definition Boltzmann's constantk k 8.617 × 1 0 − 5 k T the typical jiggling energy one atom has at temperature T
Now the Arrhenius factor e − E a / k T the fraction of atoms lucky enough to have more than the barrier energy E a
Intuition Why an exponential?Random thermal energies follow a decaying curve: very few atoms have huge energy at any moment. The chance of clearing a wall of height E a e − E a / k T Arrhenius Equation , and it makes D = D 0 e − E a / k T
Here D 0
Both ion-implant profiles and drive-in diffusion end up shaped like a bell curve (Gaussian). Two symbols describe it.
Definition Projected rangeR p Δ R p
R p average depth where implanted ions come to rest — the centre of the bell, buried inside the wafer.Δ R p straggle , the standard deviation: how far, typically, an ion stops from R p half-width of the bell. Small Δ R p
A Gaussian N p e − ( x − R p ) 2 /2Δ R p 2 N p R p Δ R p
Definition Error functionerf erfc erf ( z ) the running area under a bell curve from the centre out to z , scaled so that going all the way out gives 1. So erf erfc ( z ) = 1 − erf ( z ) the area still remaining beyond z — it falls from 1 (at the surface) to 0 (deep down). That falling shape is exactly a diffusion profile fed by an endless surface source.
Why these and not something simpler? Because they are the exact answers to the diffusion equation for the two source conditions — nothing simpler solves it. Details in Diffusion Equation and Fick's Laws .
Ion implantation counts atoms electrically, so it borrows three circuit symbols.
I t q
I beam current (amperes) — charge arriving per second.t q one ion, 1.6 × 1 0 − 19 I t I t / q A Q = I t / ( q A ) exact dose: you are literally counting charge.
concentration N per volume
dose Q per area = area under N
slope dN dx - the driving force
flux J - atoms crossing a plane
Fick second law - diffusion equation
second derivative - curvature
erfc and Gaussian profiles
beam current I and time t
Ion implantation and diffusion
Related builds: Thermal Oxidation , Photolithography , and MOSFET Structure all reuse these same counting and depth ideas within Semiconductor Fabrication .
Cover the right side and answer each — if any stumps you, re-read that section.
What does x x = 0 Depth below the wafer surface; x = 0
Difference between N Q N Q Q = ∫ 0 ∞ N d x
What is flux J J Atoms crossing a plane per cm² per second; positive = net flow deeper (increasing x
What does ∂ N / ∂ x The slope (steepness) of the concentration-vs-depth curve at fixed time.
Why the curly ∂ d N x t ∂ / ∂ x x
What does the second derivative ∂ 2 N / ∂ x 2 Curvature — how the slope changes with depth; drives pile-up/drain in Fick's 2nd law.
What is D Diffusion coefficient — how easily atoms wander; cm²/s because spreading scales as
D t (area-like).
In words, what is e − E a / k T The fraction of atoms with enough thermal energy to clear the hopping barrier E a
What do R p Δ R p R p Δ R p
What does erfc ( z ) z Falls smoothly from 1 (at surface) to 0 (deep) — the constant-source diffusion shape.
How does implantation get exact dose from a beam? Q = I t / ( q A ) I t q A
