Doping Silicon with Arsenic

Q. Suppose a pure Si crystal has 5 × 10²⁸ m⁻³ atoms. It is doped by 1 × 10⁻⁶ concentration of pentavalent As. Given that the intrinsic carrier concentration is 1.5 × 10¹⁶ m⁻³, how many holes are there, per cubic metre? expects 4.5 × 10⁹ m⁻³

What the draft establishes

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The web of concepts

given read from the words equation asked

The equations it used

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Check an answer

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?

The playground

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}

The solution

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 Ruby

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

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

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

The laws

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)