OTR Tutorial Semiconductor Basics€¦ · · 2017-11-24Institute of Photonics and Quantum Electronics (IPQ) Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany OTR

Institute of Photonics and Quantum Electronics (IPQ)

Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

OTR Tutorial –Semiconductor Basics

Heiner Zwickel

Yilin Xu

November 24, 2017

Institute of Photonics and Quantum Electronics (IPQ) www.ipq.kit.edu 24 November 2017

Direct and Indirect Semiconductors

Direct semiconductor (e.g. GaAs, InP) Indirect semiconductor (e.g. Si, Ge)

Direct semiconductor

Maximum of valence band andminimum of conduction band atsame kµ.Indirect semiconductor

Maximum of valence band andminimum of conduction band atdifferent kµ.

CB

CB

WCWC

CB

WC


Intrinsic and Extrinsic SemiconductorIntrinsic semiconductor: Pure semiconductor with negligible amount of

impurities. Electron and hole carrier concentrations in thermal

equilibrium are determined by material properties and temperature:

Extrinsic semiconductor: Doping changes carrier concentrations in

thermal equilibrium. Donors “donate” negatively charged electrons to

the conduction band (n-type). Acceptors “accept” additional electrons,

and positively charged “holes” are created in the valence band (p-type).

T in p n= =

donors

acceptorsTypical doping concentrations: 1015 cm-3 to 1018 cm-3


Intrinsic and Extrinsic SemiconductorIntrinsic semiconductor: Pure semiconductor with negligible amount of

impurities. Electron and hole carrier concentrations in thermal

equilibrium are determined by material properties and temperature:

Extrinsic semiconductor: Doping changes carrier concentrations in

thermal equilibrium. Donors “donate” negatively charged electrons to

the conduction band (n-type). Acceptors “accept” additional electrons,

and positively charged “holes” are created in the valence band (p-type).

2

T in p n=

Mass-action law holds in thermal equilibrium for intrinsic and (non-degenerately doped) extrinsic semiconductor:

Neutrality condition:

T in p n= =

𝑛𝑇 + 𝑛𝐴− = 𝑝 + 𝑛𝐷

+

n-type: p-type:

majorities: majorities:

minorities: minorities:

𝑛𝑇 ≈ 𝑛𝐷 𝑝 ≈ 𝑛𝐴𝑝𝑛0 ≈ Τ𝑛𝑖

2 𝑛𝐷 𝑛𝑝0 ≈ Τ𝑛𝑖2 𝑛𝐴

At room temperature:


Carrier Concentration at Thermal Equilibrium

Density of states in the conduction band (rC , number of electron states

per energy interval), and in the valence band (rV, number of hole states

per energy interval):

Carrier concentration in conduction band (nT) and valence band (p):

f(W) is the Fermi-Dirac distribution function. f (W) is the probability that

a state at energy W is occupied by an electron.

1 f(W) is the probability that a state at energy W is not occupied by

an electron, i. e., that it is occupied by a hole.

𝜌𝐶 𝑊 =1

2𝜋22𝑚𝑛

ħ2

32

𝑊 −𝑊𝐶𝜌𝑉 𝑊 =

1

2𝜋22𝑚𝑝

ħ2

32

𝑊𝑉 −𝑊

𝑛𝑇 = න𝑊𝐶

∞

𝜌𝐶 𝑊 𝑓 𝑊 𝑑𝑊 𝑝 = න−∞

𝑊𝑉

𝜌𝑉 𝑊 [1 − 𝑓 𝑊 ]𝑑𝑊


Fermi-Dirac Distribution Function

k Boltzmann’s constant

kT Thermal energy

kT = 25meV at T = 293K

WF Fermi energy

• At Fermi energy, f(WF) = 0.5

• Position of Fermi level:

• Intrinsic: Between WV and WC

• n-type: WF moves towards WC

• p-type: WF moves towards WV

• Transition region:

(0.88 > f > 0.12) → width 4kTDf = 4kT/h = 24.2 THz

𝑓 𝑊 =1

1 + ⅇ𝑊−𝑊𝐹𝑘𝑇


Boltzmann Approximation

If the Fermi level is far away (> 3kT) from the band edges WC and WV

(as is the case for doping concentration of nD << NC and nA << NV ),

then Boltzmann’s approximation holds:

Solving the integrals for the carrier concentrations with Boltzmann’s

approximation gives:

NC and NV are called effective density of states (per Volume). Within

kT from the band-edge, there are 0.752NC,V states. For intrinsic

semiconductors follows:

1

1 + 𝑒𝑥≈

1

𝑒𝑥𝑥 ≫ 1 𝑓𝑉(𝑊) ≈ 𝑒−

𝑊𝐹−𝑊

𝑘𝑇

𝑓𝐶(𝑊) ≈ 𝑒−𝑊−𝑊𝐹𝑘𝑇

𝑛𝑖 = 𝑛𝑇𝑝 = 𝑁𝐶𝑁𝑉𝑒−𝑊𝐺2𝑘𝑇

𝑁𝐶 = 22𝜋𝑚𝑛𝑘𝑇

ℎ2

3

2with

𝑝 ≈ 𝑁𝑉𝑒−𝑊𝐹−𝑊𝑉

𝑘𝑇 𝑁𝑉 = 22𝜋𝑚𝑝𝑘𝑇

ℎ2

3

2with

𝑁𝐶,𝑉 ≈ 1019cm−3𝑛𝑇 ≈ 𝑁𝐶𝑒−𝑊𝐶−𝑊𝐹

𝑘𝑇

𝑊𝐹 =1

2𝑊𝐶 +𝑊𝑉 +

𝑘𝑇

2ln𝑁𝑉𝑁𝐶

and


Visual Summary of Carrier Concentrations (1)

Intrinsic semiconductor in thermal equilibrium.

𝜌𝐶 𝑊

𝜌𝑉 𝑊

𝜌𝐶 𝑊 𝑓(𝑊)



Visual Summary of Carrier Concentrations (2)

Extrinsic semiconductor (n-type) in thermal equilibrium.

𝜌𝐶 𝑊

𝜌𝑉 𝑊




Currents in Semiconductors

Drift current due to an electric field E : mn,p carrier mobility

e elementary charge

s conductivity

Diffusion current due to a gradient of carrier concentration:

Diffusion coefficients Dn and Dp for electrons and holes (Einstein-relation):

Diffusion lengths Ln and Lp for electrons and holes:

tn , tp are the minority carrier lifetimes of electrons and holes.

റ𝐽𝐹 = റ𝐽𝑛,𝐹 + റ𝐽𝑝,𝐹 = 𝑒𝑛𝑇𝜇𝑛 + 𝑒𝑝𝜇𝑝 𝐸 = 𝜎𝐸

റ𝐽𝐷 = റ𝐽𝑛,𝐷 + റ𝐽𝑝,𝐷 = 𝑒𝐷𝑛 grad 𝑛𝑇 − 𝑒𝐷𝑝 grad 𝑝

𝐿𝑛 = 𝐷𝑛𝜏𝑛 𝐿𝑝 = 𝐷𝑝𝜏𝑝and

𝐷𝑛 = 𝜇𝑛𝑈𝑇 = 𝜇𝑛𝑘𝑇

𝑒𝐷𝑝 = 𝜇𝑝𝑈𝑇 = 𝜇𝑝

𝑘𝑇

𝑒

UT = kT/e is called temperature voltage ( 25 mV @ T = 293 K)


pn-junction in Thermal Equilibrium

Electrons diffuse into the p-type semiconductor, and holes into the n-type

semiconductor. The positively and negatively charged donor and

acceptor ions in the space charge region (SCR) build up an electric field

that counteracts diffusion.

In thermal equilibrium, there are zero net electron and hole currents, i.e.

diffusion and drift currents compensate each other:

Before contact: After contact:

The built-in potential UD of the pn-junction is given by:

2 2ln lnD A D A

D T

i i

n n n nkTU U

en n= =

റ𝐽𝑝 = റ𝐽𝑝,𝐹 + റ𝐽𝑝,𝐷 = 0റ𝐽𝑛 = റ𝐽𝑛,𝐹 + റ𝐽𝑛,𝐷 = 0


Current-Voltage Characteristics of pn-DiodeApplying an external voltage to the pn-junction → no equilibrium

2

T in p n>

2

T in p n<

Under reverse bias condition (U < 0, “+” at n-type, “-” at p-type),

charge carriers are removed to increase the SCR width:

Under forward bias condition (U > 0, “+” at p-type, “-” at n-type),

charge carriers are injected to reduce the SCR width:

Assuming only diffusion currents outside the SCR, the current-voltage

characteristics of the pn-diode follows as:

Concentration of minority charge carriers at the edges of the SCRincrease/decrease exponentially with the applied voltage U.

𝑝𝑛 𝑙𝑛 = 𝑝𝑛0 𝑒𝑈𝑈𝑇

𝐼 = 𝑒𝐹𝐷𝑛𝐿𝑛

𝑛𝑝0 +𝐷𝑝

𝐿𝑝𝑝𝑛0 ⅇxp

𝑒𝑈

𝑘𝑇− 1 = 𝐼𝑆 ⅇxp

𝑒𝑈

𝑘𝑇− 1

Saturation current IS

For example, for holes in the n-type region:


Band Diagrams

In thermal equilibrium, the Fermi level is flat,

i.e. no net current flows.

In non-equilibrium, the Fermi level splits upinto the quasi Fermi levels (QFL) WFn and

WFp.

Gradients of the QFL indicate, that there is a

current (due to combined effects of drift and

diffusion)

U > 0 reduces barrier for carriers

U < 0 increases barrier for carriers


Depletion-Layer and Diffusion CapacitanceDepletion-layer capacitance (dominating if reverse biased): The total

width of the SCR varies according to the applied voltage. Thus also the

amount of charges in the SCR changes. Taking the formula for a

parallel plate capacitor yields:

Only half of the charge is modulated effectively:

Diffusion capacitance (dominating if forward biased): When applying a

small AC signal, not all the minority carriers at the edge of the SCR

follow the signal instantaneously. The stored minority charge (here:

holes in an n-type semiconductor) is given by:

G0

𝐶𝑆 = 휀0휀𝑟𝐹

𝑤(𝑈)𝑤 𝑈 = 𝑙𝑛 − 𝑙𝑝 =

2𝜀0𝜀𝑟

𝑒

𝑛𝐷+𝑛𝐴

𝑛𝐷𝑛𝐴(𝑈𝐷 − 𝑈)with

𝑄 = 𝑒𝐹න

0

∞

𝑝𝑛0 ⅇxp𝑈

𝑈𝑇ⅇxp −

𝑥

𝐿𝑝𝑑𝑥 = 𝑒𝐹𝐿𝑝𝑝𝑛0ⅇxp

𝑈

𝑈𝑇

𝑄 = 𝐼𝜏 𝐺0 =𝛿𝐼

𝛿𝑈=

𝐼

𝑈𝑇

𝐶𝐷 =1

2

𝛿𝑄

𝛿𝑈=1

2

𝛿𝐼

𝛿𝑈

𝛿𝑄

𝛿𝐼=1

2𝐺0𝜏 =

1

2

𝐼

𝑈𝑇𝜏

Documents

OTR Tutorial Semiconductor Basics€¦ · · 2017-11-24Institute of Photonics and Quantum Electronics (IPQ) Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany OTR