carriers concentration and current in...

P.Ravindran, PHY02E Semiconductor Physics, 21 February 2013: Carriers and Current in Semiconductors

Carriers Concentration and Current in Semiconductors

P.Ravindran, PHY02E Semiconductor Physics, 21 February 2013: Carriers and Current in Semiconductors2

Carrier Transport

Two driving forces for carrier transport: electric field and spatial variation of the carrier concentration.

Both driving forces lead to a directional motion of carriers superimposed on the random thermal motion.

To calculate the directional carrier motion and the currents in a semiconductor, classical & nonclassical models can be used.

The classical models assume that variation of E-field in time is sufficiently slow so that the transport properties of carriers (mobility or diffusivity) can follow the changes of the field immediately.

If carriers are exposed to a fast-varying field, they may not be able to adjust their transport properties instantaneously to variations of the field, and carrier mobility and diffusivity may be different from their steady-state values nonstationary

Nonstationary carrier transport can occur in electron devices under both dc and ac bias conditions.


Classical Description of Carrier Transport

Assume thermal equilibrium for a semiconductor having a spatially homogeneous carrier concentration with no applied E-field. No driving force for directional carrier motion. The carriers not in standstill condition but in continuous motion due to kinetic energy. For electron in the conduction band,

where vth is the thermal velocity, mn* is the conductivity effective electron mass.

• The average time between two scattering events is the mean free time and the average distance a carrier travels between collisions is the mean free path. Fig. 2.5 (a)

• Applying V, the E-fields adds a directional component to the random motion of the electron. Fig. 2.5 (b)

*23

2 2

nkin B th

mE k T v


• The mean electron velocity: vn= -μnE

• The directed unilateral motion of carriers caused by E-field is drift velocity.

• Similarly, vp = μpE

• A change in E-field instantaneously results in a change of the drift velocity.


Fick’s First Law: relating diffusion current to carrier concentration gradient.

e = electron flux, De = diffusion coefficient of electrons, dn/dx = electron concentration

gradient

e De

dn

dx

JD, e = electric current density due to electron diffusion,

e = electron flux, e = electronic charge,

De = diffusion coefficient of electrons,

dn/dx = electron concentration gradient

Electron Diffusion Current Density

JD,e ee eDe

dn

dx

Where:


Hole Diffusion Current Density

JD, h = electric current density due to hole diffusion, e = electronic charge, h = hole flux, Dh

= diffusion coefficient of holes, dp/dx = hole concentration gradient

JD,h eh eDh

dp

dx

Total Electron Current Due to Drift and Diffusion

Je = electron current due to drift and diffusion, n = electron concentration, e = electron drift

mobility, Ex = electric field in the x direction, De = diffusion coefficient of electrons, dn/dx =

electron concentration gradient

Je eneEx eDe

dn

dx


Total Currents Due to Drift and Diffusion

Jh = hole current due to drift and diffusion, p = hole concentration,

h = hole drift mobility, Ex = electric field in the x direction,

Dh = diffusion coefficient of holes,

dp/dx = hole concentration gradient

Jh ephEx eDh

dp

dx

Je eneEx eDe

dn

dxJe = electron current due to drift and diffusion, n = electron concentration

e = electron drift mobility, Ex = electric field in the x direction,


dn/dx = electron concentration gradient

Jtotal = Jh+Je



e = electron drift mobility,

Dh = diffusion coefficient of the holes,

h = hole drift mobility

De

e

kT

eand

Dh

h

kT

e

Einstein Relation: diffusion coefficient and mobility are related!


Exposed

As+ Donor

n2

n1

Diffusion Flux

Drift

Net current = 0

Ex

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca

Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion of

electrons towards the right is balanced by their drift towards the left.

Vo

Carrier diffusion due to doping level gradient.This is a common device fabrication step.

Diffusion occurs until an electric field builds up!

We call this the built-in potential.

Note: the As+

are fixed,non-mobile charges!

· represents electrons (majority carriers in this

case)


Built-In Potential and Concentration

V2 = potential at point 2, V1 = potential at point 1,

n2 = electron concentration at point 2,

n1 = electron concentration at point 1

V2 V1 kT

eln

n2

n1

Built-In Field in Nonuniform Doping

Ex = electric field in the x direction,

b = characteristic of the exponential doping profile,

e = electronic charge .

Ex

kT

be

Exposed

As+ Donor

n2

n1

Diffusion Flux

Drift

Net current = 0

Ex

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca

Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion of

electrons towards the right is balanced by their drift towards the left.

Vo


Carrier creation: Photoinjected charge carriers

If we shine light on a semiconductor, we will generate new charge carriers (in addition to those thermally generated) if Ephoton>Egap.

If the light is always on and of constant intensity, some steady state concentration of electrons and holes will result.


Carrier creation: Photoinjected charge carriers

Consider an n-type semiconductor with a doping concentration of 5 x 1016 cm-3.

What are the carrier concentrations?Let’s define some terms;nno majority carrier concentration in the n-type semiconductor in the dark(only thermally ionized carriers) (i.e. the electron concentration in n-type)

pno minority carrier concentration in the n-type semiconductor in the dark(only thermally ionized carriers) (i.e. the hole concentration in n-type)

Note: the no subscript implies that mass action law is valid!

Let’s consider the case of n-type material


When we have light:

With light of Ephoton>Egap hitting the semiconductor, we get photogenerationof excess charge carriers.

nn excess electron concentration such that::nn = nn-nn0

&

pn excess hole concentration such that::pn = pn-pn0

Note that photogenerated carriers excited across the gap can only be created in pairs i.e. pn = nn and now (in light) nnpn≠ni

2 i.e. mass action not valid!


If the temperature is constant, nn0 and pn0 are not time dependent, so

and

Consider the case of ‘weak’ illumination, which creates a 10% change in nn0

i.e. nn = 0.1nn0

Or if the doping level is nno=5 x 1016cm-3, then

nn = 0.1nn0= 0.5 x 1016cm-3

And pn =nn = 0.5 x 1016cm-3

Which change is more important? Majority or minority?

dnn

dt

dnn

dt

dpn

dt

dpn

dt

Carrier density change under illumination


Recall the intrinsic carrier concentration

For Sini is roughly 1.5x1010cm-3

At room temperature

Since pno=ni2/nn0

= (1.5x1010)2/5x1016

pno =4500 cm-3


An extremely important concept!

Minority carrier concentration can be controlled over many orders of magnitude with only a small change in majority concentration.

pn =nn = 0.5 x 1016cm-3


Carrier creation followed by recombination


Carrier creation followed by recombination

Mostly majority carriers in the dark

Almost equalCarrier concentrationIn light

The extra minority carriers recombine once the generation source is removed.

How quickly do the carriers recombine?


Minority carrier lifetime h for n-type

h = average time a hole exists in the valance band from its generation

until its recombination

And so 1/ h is the average probability (per unit time) that a hole will

recombine with an electron.

h depends on impurities, defects and temperature.

The recombination process in a real semiconductor usually involves a

carrier being localized at a recombination center.

can be short (nanoseconds) allowing fast response (e.g. switch)

or slow (seconds) for a photoconductor or solar cell


Excess Minority Carrier Concentration

pn = excess hole (minority carrier) concentration,

t = time,

Gph = rate of photogeneration,

h = minority carrier lifetime (mean recombination time)

h = average time a hole exists in the valance band from its

generation until its recombination

dpn

dt Gph

pn

h


Carrier concentration versus time with pulsed illumination

t’ is time after illumination is removed


Continuous illumination provides increased conductivity

Often used as a switchor motion detector


Carrier diffusion away from high concentration

holes in this p-type example


Carrier motion: via diffusion (due to concentration gradient) and drift (due to electric field)

Both diffusion and drift occur in semiconductors.

Note here that holes (minority carriers) drift and diffuse in the same direction; but electrons (majority carriers) do not!

With light we alter minority carrier concentration

carriers concentration and current in...

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