Doping Silicon with Arsenic

solved DopedSemiconductor

Q. Suppose a pure Si crystal has 5 × 10²⁸ m⁻³ $n_{a} = 5 \times 10^{28}$ m⁻³ atoms. It is doped by 1 × 10⁻⁶ $x_{d} = 1 \times 10^{−6}$ concentration of pentavalent As. Given that the intrinsic carrier concentration is 1.5 × 10¹⁶ m⁻³ $n_{i} = 1.5 \times 10^{16}$ m⁻³, how many holes are there, per cubic metre?

Inputs

atoms

n_{a}

5e+28 m⁻³

doping

x_{d}

1e-06

intrinsic

n_{i}

1.5e+16 m⁻³

Outputs

holes

n_{h}

How it's solved

Solved with DopedSemiconductor

Given

$n_{a} = 5 \times 10^{28}$ m⁻³
$x_{d} = 1 \times 10^{−6}$
$n_{i} = 1.5 \times 10^{16}$ m⁻³

Find $n_{h}$ .

Working

donors from the doping(crystal) $N_{D} = x_{d} \cdot n_{a}$
$N_{D} = 1 \times 10^{−6} \cdot 5 \times 10^{28}$
$N_{D} = 5 \times 10^{22}$ m⁻³
each donor gives an electron(crystal) $n_{e} = N_{D}$
$n_{e} = 5 \times 10^{22}$
$n_{e} = 5 \times 10^{22}$ m⁻³
mass action(crystal) $n_{e} \cdot n_{h} = {n_{i}}^{2}$
$5 \times 10^{22} \cdot n_{h} = {(1.5 \times 10^{16})}^{2}$
$n_{h} = 4.5 \times 10^{9}$ m⁻³
$n_{h} \approx 4.5 \times 10^{9}$ m⁻³to 2 significant figures

Conclusion

Each arsenic atom gives an electron, swamping the crystal's own; electrons and holes keep their product fixed, so the holes almost vanish.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :doping_silicon_with_arsenic do4  kind :numerical5  describes "Suppose a pure Si crystal has %{atoms} atoms. It is doped by %{doping} concentration of pentavalent As. " \6            "Given that the intrinsic carrier concentration is %{intrinsic}, %{holes}, per cubic metre?"7  given atoms: "5e28 m-3"8  given doping: "1e-6"9  given intrinsic: "1.5e16 m-3"10  asks holes: "how many holes are there"11end

solution.rb

1require_relative "../questions/doping_silicon_with_arsenic"2require_relative "../science/semiconductor_electronics_materials_devices_and_simple_circuits/doped_semiconductor"34solve :doping_silicon_with_arsenic, using: DopedSemiconductor do5  objects :crystal67  take atoms: crystal.atoms8  take doping: crystal.doping9  take intrinsic: n_i1011  holes = crystal.holes12  answer holes:, in: "m-3", figures: 21314  conclusion "Each arsenic atom gives an electron, swamping the crystal's own; electrons and holes keep their " \15             "product fixed, so the holes almost vanish."16end

playground.rb

1require_relative "../solutions/doping_silicon_with_arsenic"23play_with :doping_silicon_with_arsenic do4  title "Doping Silicon with Arsenic"5end

The laws it drew on

quantity	written	units	within	from
intrinsic carriers	$n_{i}$	m⁻³	0 … ∞	DopedSemiconductor
atoms	$n_{a}$	m⁻³	—	DopedSemiconductor
doping	$x_{d}$	—	—	DopedSemiconductor
donors	$N_{D}$	m⁻³	—	DopedSemiconductor
electrons	$n_{e}$	m⁻³	—	DopedSemiconductor
holes	$n_{h}$	m⁻³	—	DopedSemiconductor
donors of ground	$N_{D,g}$	m⁻³	—
doping of ground	$x_{d,g}$	—	—
atoms of ground	$n_{a,g}$	m⁻³	—
donors of crystal	$N_{D}$	m⁻³	—
doping of crystal	$x_{d}$	—	—
atoms of crystal	$n_{a}$	m⁻³	—
electrons of ground	$n_{e,g}$	m⁻³	—
electrons of crystal	$n_{e}$	m⁻³	—
holes of ground	$n_{h,g}$	m⁻³	—
holes of crystal	$n_{h}$	m⁻³	—

N_{D,c} = x_{d,c} \cdot n_{a,c}

donors from the doping

n_{e,c} = N_{D,c}

each donor gives an electron

n_{e,c} \cdot n_{h,c} = {n_{i}}^{2}

mass action

N_{D,g} = x_{d,g} \cdot n_{a,g}

donors from the doping(ground)

N_{D} = x_{d} \cdot n_{a}

donors from the doping(crystal)

n_{e,g} = N_{D,g}

each donor gives an electron(ground)

n_{e} = N_{D}

each donor gives an electron(crystal)

n_{e,g} \cdot n_{h,g} = {n_{i}}^{2}

mass action(ground)

n_{e} \cdot n_{h} = {n_{i}}^{2}

mass action(crystal)