• Random Motion is due to the interaction between charged carriers and ionized impurities atoms and vibrating lattice atoms.

• Scattering process: Collision of carriers with ionized impurity atoms (called ionized impurity scattering), and thermally vibrating lattice atoms (called phonon or lattice scattering).

Carrier Transport Phenomena

Carrier Transport Phenomena

Random motion of carriers without applied field

Random motion as well as a net movement along the direction of field

No net carrier displacement

JkTvm th212* 102245.62

321 −×==

Random thermal velocity, vth

Net carrier displacement and thus a net velocity along the field.

This velocity is called the drift velocity, which gives drift current.

Charged Carriers Collisions

• Mean scattering time τ: is the average time between the two collisions.

• During this time the carriers moves an average distance l , called mean free path.

• Random thermal velocity, vth, is the average velocity between the collisions:

JKTvm th212* 102245.62

321 −×==

Typical value of vth ∼ 107 cm/s

*meτμ =

Drift mobility:

When an electric field is applied to the crystal, electrons and holes experience a net acceleration in a direction in addition to its random motion. As the charged particle accelerates, the velocity increases and then it suddenly collide with a vibrating atom and loses the gained velocity. The particle will again begin to accelerate and gain energy until it is again involved in a scattering process. Throughout this process the particle will gain an average drift velocity.

Average drift velocity (for low fields), <vd>= (eτ / m* )E = µEµ is called mobility (cm2/ V-s):

Mobility:

22 // cmAmporscmCvJ ddrift −= ρ

Drift Current Density:

If we have a positive volume charge density ρ moving with an average drift velocity vd, the drift current density,

For example if the volume charge density is due to holes then: dpp epvJ =

For low electric fields, the drift velocity is proportional to the electric field.

Ev pdp μ=

Where μ is the proportionality factor and is called the hole mobility. Its unit is cm2/ V-s

Therefore:

pEeJ pp μ=

densityCurrentAIj =

eporenDensityeCh =−=ρarg

Drift Current density

EEpneJ pndrf σμμ =+= )(

)( pn pne μμσ +=Conductivity:

nEeJ nn μ=

Since both electrons and holes contribute to the drift current, the total drift current density is,

Similarly the drift current density due to electrons is,

Where σ is the conductivity of the semiconductor material. Its unit is Ω-1 cm-1

The reciprocal of conductivity is resistivity ρ,

.)(

11

pn pne μμσρ

+==

For intrinsic material,

ipnpn nepne )()( μμμμσ +=+=

For n type material, n >> p. therefore,

nepne npn μμμσ ≅+= )( (Assuming that μn and μp are comparable)

For p type material, p >> n. therefore,

pe pμσ ≅

Temperature dependence of mobility

There are two collisions or scattering mechanisms that dominate in a semiconductor and affect the carrier mobility:

(1) Phonon or lattice scattering and (2) Ionized impurity scattering.

Phonon or lattice scattering:The thermal energy at temperature above absolute zero causes the atoms to randomly vibrate about their lattice position within the crystal. Charged carriers collide with vibrating atoms and are scattered. The lattice scattering increases with temperature and thus mobility decreases.

23−∝ TLμ

Ionized impurity scattering:

A coulomb interactions between the electrons or holes and the ionized impurities cause collisions and the scattering of the carriers. As the temperature increases this type of scatterings decreases, increases the mobility:

II N

T 23∝μ −+ += adI NNN NI is concentration of

the ionized carries

Therefore:

LI μμμ111

+=

LI

dtdtdtτττ

+=

Scattering time:

Temperature dependence of mobility:

23−∝ TLμ II N

T 23∝μ

dt/τ is the probability of a scattering event occurring in differential time dt. Now recall:

*meτμ =

Doping dependence of mobility

( ) ( )αμμμ

refNNN

/10

min ++=

As the doping levels in semiconductors change, the ionization scattering changes. The mobility follows an empirical relationship as:

Parameters for silicon at 300K:

Velocity Saturation:

vd,sat ≈ vth ∼ 107 cm/s

KTvm th 23

21 2* =

Ref: D. A. Naeman, Semiconductor Physics and Devices

31710 −= cmNdExample: Given: Si sample,

I) Calculate σ at 27ºC and 127ºC.II) The above n-type is further doped with Na = 9 × 1016 cm-3, Calculate the conductivity of the sample at 27ºC

Example: Consider the electron mobility in Si, μn = 1350 cm2/V-s and

. Calculate: (1) The mean scattering time and (2) The mean free path.

0* 26.0 mmn =

Examples

An electron has a mobility of 1000 cm2/V-s at 300 K. if effective mass is 0.2 mo,Calculate mean free time and mean free path of the electron.

If a silicon sample is doped with Nd = 2 x 1017 calculate the conductivity and theresistivity of the sample at 300K and assuming full ionization. Repeat your calculations for the case if the sample is doped with Nd = 2 x 1014.

Compare these values with experimental values.

If the sample in the above example has a 1 mm length and a 100 volts appliedon both sides of the sample calculate the drift current going through thesample. Calculate the average drift velocity of the carries.

If Ge has an intrinsic mobility of 3900 for electrons and 1900 for holes respectively, calculate the intrinsic conductivity and resistively of this material.

Carrier Diffusion:

Diffusion is the process whereby the thermal random motion of particles causes them to flow from a region of high concentration toward a region of low concentration.

Diffusion Current:

In semiconductor, the net flow of charge carriers (electrons or holes) results in a diffusion current. The carrier diffusion current is proportional to the density gradient or spatial derivative, thus the diffusion current density,

dxdp

dxdnJ dif =∝∝

For electrons, dxdneDJ ndifn =,

For holes,dxdpeDJ pdifp −=,

x

n(x)

← Electron diffusion

→ Electron currentp(x)

← hole diffusion

← hole current

x

Where D is the diffusion coefficient, has units of cm2/s and is a positive quantity.

lvD th=Where νth is the thermal velocity and l is the mean free path of the carrier

Total Current Density:

dxdpeD

dxdneDEepEenJ

JJJcurrenttotalThe

currentHoledxdpeDEepJ

currentElectrondxdneDEenJ

pnxpxn

pn

pxpp

nxnn

−++=

+=

→−=

→+=

μμ

μ

μ

,

Example 5.3: Given: n-type GaAs at T = 300ºK. n(x) varies linearly from 1018 to 7×1017 cm-3 over a distance of 0.1cm. Calculate the diffusion current density due to the electron diffusion. Assume Dn = 225 cm2/s.

Graded Impurity Concentration:If the semiconductor is not uniformly doped, still it reaches thermal equilibrium by diffusion and latter diffusion induced drift process.

Induced Electric Field:Consider a semiconductor that is non uniformly doped with donor impurity atoms. The doping concentration decreases as x increases as shown in the following figure.

x

Ec

Ev

EFi

EF

The electric field (induced) is defined as

( )dx

xdVx −=E

V(x) = E(x)/e.

dxdE

edxEEd

edxxdE

eFiFiF

x1)(1)(1E =

−==

dxdE

eFi

x1E =

( )

dxxdn

xnkT

dxdEor

nxnkTEEor

kTenxn

Fi

iFiF

EEi

FiF

)()(

,

)(ln,

/)(

0

0

0

0

−=

=−

= −

dxxdn

xnekT

x)(.

)(1.E 0

0

−=

( ) ( )xNxn d=0

dxxdN

xNekT d

dx

)(.)(

1.E −=

Assuming,

Again at thermal equilibrium the total electron current is zero. Therefore,

relationEinsteine

kTDDThus

ekTD

Similarly

ekTD

De

kTor

dxxdnD

dxxdn

xnekTxnor

dxxdneDxenJ

p

p

n

n

p

p

n

n

nn

nn

nxnn

==

=

=∴

=+−

=+⎭⎬⎫

⎩⎨⎧−

=+=

μμ

μ

μ

μ

μ

μ

,

,,

0.,

0)()()(

1)(,

0)(E)(

00

00

00

Example 5.4:The total current in a semiconductor is constant and is composed of electron drift current and hole diffusion current. The electron concentration is constant and is equal to 1016 cm-3. The hole concentration is given by

cm-3 ( x ≥ 0), where L =12 μm. Dp = 12 cm2/s and μn = 1000 cm2/V-s. The total current density is J = 4.8 A/cm2. Determine (a) the hole diffusion current density versus x, (b) Calculate electron drift current at x = 0 and (c) the electric field at x = L.

( ) Lxexp −= 1510

Hall effect:

deVBIpH

zx=

WdepVLI

x

xp =μ

Carriers experience a force, Fy = evx×Bz

ZxH BeveE =

WEV HH =

Example 5.5:Consider the geometry shown in previous page, determine the majority carrier concentration and mobility based on the following data. L = 0.1 cm, W = 0.01 cm, d = 0.001 cm, Ix = 0.75 mA, Vx = 15 V, Bz = 1000 gauss and VH = 5.8 mV.

The unit of Bz is Tesla (MKS unit, weber/m2) or gauss (CGS unit, Maxwell/cm2). 1 Tesla = 104 gauss. 1 Weber = 108 Maxwell.

In GaAs, Nd (x) = Nd0 exp(-x/L) for 0 ≤ x ≤ L, where L = 0.1 μm. and Nd0 = 5 ×1016 cm-3. Assume μn = 6000 cm2/V-s and T =300 K. Derive the expression for the electron diffusion current density versus distance over the given range of x. (b) Determine the induced electric field that generates a drift current density that compensates the diffusion current density.

Example: