THE GUNN EFFECT

Kiros Hagos Abay

University of Mysore,

M.G.M.

14 May 2011

Contents• Introduction

• Hot Electrons and the Expression for the Electron Temperature

• The Gunn Effect (NDC)

• Expression for the drift velocity

• Formation and Drift of Space Charge Domains

• Superlattice phenomenon

• Reference

Introduction

We have different effects in a semiconductor.

When a high electric field is applied to a semiconductor, the carriers

(electrons and holes) absorb appreciable energy from the field, and

their temperature rises above that of the lattice temp. ; i.e., they

become “hot”.

The effect of this is a decrease in their mobility.

In certain semiconductors of appropriate band structure, such as

GaAs, the heating of carries result in a transfer of electrons to high-

energy valleys of very low mobility.

In such case;

The application of the electric field may produce a region of Negative Differential Conductance (NDC).

• Because of such a situation is inherently unstable, the sample “break up” into coherent electrical oscillations, which is the Gunn effect discovered by J.B. Gunn in 1963.

• The Gunn-effect diodes are basically made from an n-type GaAs, as this effect is found in n-type materials, so it associate with electrons but holes.

Hot Electrons and the Expression for the Electron Temperature

• If the velocities of a group of electrons , e.g., in a plasma, follow a Maxwell-Boltzmann distribution , then the electron temperature is well-defined as the temperature of that distribution.

• Semiconductor exhibits linear Ohmic behavior that is , J in the region of low electric fields.

• In the high field present in some devices, however, considerable deviation from Ohm’s law is observed, see Fig.1. for n-type germanium

• The deviation becomes significant at some field ɛ1, and for evenɛ1< ɛ the current lies below its expected Ohmic value.

• Above a certain higher field ɛ2, the current actually saturated at a constant value until, at an extremely high field, usually in the 100kV/cm range, the sample undergoes an electrical breakdown.

Dri

ft v

elo

city

, cm

/s

,V/cm

106

107

102 103 104

Fig.1. drift velocity vs electric field in n-type Ge.

• Consider the average electron energy =3kBT/2 (Maxwell-Boltzmann distribution with three degree of freedom).

• At high fields, the electron receives considerable energy from the field because of the acceleration of the electron between collision, and also loses energy to the lattice.

• In the steady state the rates of gain and loss of energy must be equal.

• That is:

=0

•Where

is the electron drift velocity,

is the energy relaxation time and Te is the electron temperature .

(1)

• We have allowed for the possibility that Te may higher than

the lattice temperature T=TL, leading to the concept of hot

electrons.

• By substituting (Te)=3kBT/2, (TL)=3kBT/2 and

in equation (1), we have;

- - 3kB(Te-T)/2 =0

Te=T+ (2)

This is the expression for the electron temperature. The heating would be much greater at higher fields or mobility.

We know that

• = e /me* = ele/me

* , where is the random velocity of the electron, and le is the mean free-path.

• Since , it follows that

• We may thus write

(3)

,

• Where is the familiar low-field mobility. Equation (2) and (3) are the two equations in Te and and can be employed in employed in solving for these unknown.

= (4)

• In the range in which field is not too high, one field, which explain the initial decrement in mobility, just above the field in figure, 1.

• One can explain the current saturation at high field by assuming that the electrons dissipate their energy by emitting optical phonons in the lattice.

• Since these phonons have much greater energy than their acoustic counterparts, they represent the most efficient means for the electron to rid them of the energy gained from the field, thus achieving steady-state conditions.

The Gunn Effect

• J.B. Gunn (May 13, 1928 – December 2, 2008), was an Egyptian born, British physicist, who spent most of his career in the United States.

• The Gunn effect is named after J.B. Gunn, who made the discovery in 1963, while measuring the current of the hot electrons in GaAs and other III-V compounds.

• When he was measuring the current J versus the field in n-type GaAs, he observed an unexpected phenomenon.

,

• Gunn effect is a phenomenon observed in some semiconductors in which a steady electric field of magnitude greater than a threshold value generates electrical oscillations with microwave frequencies.

• The Gunn-effect diodes are basically made from an n-type GaAs, with the concentrations of free electrons ranging from 1014 to 1017 per cubic centimetre at room temperature. Its typical dimensions are 150 x 150 μm in cross section and 30 μm long.

+ Anode

Cathod-

→ v

Fig. 2. Schematic diagram for n-type GaAs diode

• As is increased from zero, the current increases gradually and essentially linear (fig.3a.) until a field is reached. As the field is increased beyond , the current suddenly becomes oscillatory ( vs t, not vs ).

• These oscillations are essentially coherent (waves having a constant phase relation), provided the sample is sufficiently thin.

• The field is necessary for the onset of the Gunn oscillation is called the threshold field.

Fig.3. (a) a graphic summary of the Gunn effect. (b) the current J vs in GaAs showing the NDC region ( dashed curve)

(a) (b)

JJ

NDC rigion

0 0

Time

An interesting fact:

• There is a certain field range in which J decreases as

increases (the curve corresponding to this range is shown by

the dashed line in the fig.3b).

• This behavior (contrary to the usual one, in which an increase

in causes an increase in J) is described by saying that the

sample has a Negative Differential Conductance (NDC).

The Gunn Effect (Negative Differential Conductivity)

• To understand the Gunn Effect, consider E-k diagram of the GaAs ( a direct-band semiconductor) as shown below.

C.B

V.B

C

Ex EL

(100)

(111)

E

S

k

Fig.4. E-k diagram for GaAs.

C= central valley

S= satellite valley

C.B= conduction band

V.B= valence band

,

• For GaAs, the band gap Eg is ( =1.4eV). However, in GaAs, there is another minimum S in the conduction band only 0.36eV above the other minima as shown in the above fig 4.

• The effective mass me* of electrons is positive and inversely

proportional to the curvature of the band. As shown in the fig.4, the curvature is much larger in the central valley than in the satellite valley (From Two-Valley Model Theory).

• So for GaAs, (me*)c =0.07me

And (me*)s =1.2me

• Where, me is the free electron mass.

• Effective mass of electron is given by:

Rate of change of the valley curves slope’

(5)

• Since the lower valley slope is shaper then the one in upper valley, thus electron effective mass in lower valley is lower than that in upper valley

• So that, the mobility of electron in upper valley is less due to the higher effective mass

(6)

The conductivity of electrons in a band is given by;

(7) And Ohm’s law

(8)

Let’s consider J vs E curve for a sample of GaAs. For low fields, the free electrons nc present would occupy states near C in the central valley and for them

(9)

As the is increased, some of the free electrons may acquire sufficient energy to be able to transfer themselves to the satellite valley.

If ns electrons were present in the satellite valley, then conductivity due to these would be

(10)

Consider the situation where, for a given there is a population density nc in the central valley and ns in the satellite valley. Then current density

(11)

If the field now increase by , some additional electrons will acquire sufficient energy to transfer from central valley to the satellite valley then,

⇒

.

Where,

(12)

The ratio of is known as the differential conductivity

and represents the slope of the characteristic of the sample.

⇒

In this equation;

the terms T1 and T2 are always positive while T3 is always negative ( is always negative).

Hence for certain fields, could be negative depending on the values of material parameters and the magnitude of .

The fig.5, below shows for GaAs. Thus this shows the Negative Differential Conductivity (NDC).

When the field is low, conductivity is .

When the applied field exceeds Ec, the transfer of electrons to the satellite valley is becomes significant and the conductivity reduces from its low field values.

• As increases further, the term T3 predominant and a negative resistance region observed ( part BC of the curve

ABCD). • When the field is sufficiently large, virtually all electrons get

transferred to the satellite valley and the conductivity becomes positive (determined solely by the mobility of electrons in the satellite valley). Thus the curve goes through a minimum.

,

J

Jpeak­

Jval

Em E

D

C

B

A

Fig. 5. J vs E curve of a Gunn diode

• The Gunn effect has also been observed in InP, GaAsxP1-x, CdTe, ZnSe InAs and other semiconducting compounds.

• All have conduction-band structures similar to that of GaAs, and the inter-valley transfer is responsible for Gunn oscillations in every case.

Expression for the drift velocity:

• The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field.

• In general, an electron will 'rattle around' in a conductor at the Fermi velocity randomly. An applied electric field will give this random motion a small net velocity in one direction.

• If (n1,v1) and ( n2,v2) are the electron densities and drift velocities respectively in the central and secondary valleys of fig.4, the average drift velocity of electrons may be expresses as

Vd = (n1v1+n2v2)/n1+n2 (13)

• Expressing v1= and v2= , we get

(14)

• Here , and are the mobilities of the electrons in the central and secondary (satellite) valleys.

• The expression for electron density in an intrinsic material is given by

Vd =

.

• .

and

From the above two equations, we can get the expression for drift velocity as

(15)⇒

(16) .

• Where Te denotes the electron temperature

• This equation explain the variation of the drift velocity with respect to , where both are the functions of the applied electric field.

Formation and Drift of Space Charge Domains

• Space charge:• Space charge is a concept in which excess electric charge is

treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges.

• when charge carriers have been emitted from some region of a solid—the cloud of emitted carriers can form a space charge region if they are sufficiently spread out, - the solid can form a space charge region.

• Space charge usually only occurs in dielectric media (including vacuum) because in a conductive medium the charge tends to be rapidly screened .

• The sign of the space charge can be either negative or positive.

• This situation is perhaps most familiar in the area near a metal object when it is heated to incandescence in a vacuum.

• Thermionic emission.

Space charge domain:

• If a sample of GaAs is biased such that the field falls in the negative conductivity region, space charge instabilities result, and

the device cannot be maintained in a d-c stable condition.• To understand the formation of these instabilities, consider first

the dissipation of space charge in the usual semiconductor.

(17)

Because of this process,

• Random fluctuations in carrier concentration are quickly neutralized and space charge neutrality is a good approximation for most semiconductors in the usual range of conductivities.

• The above Equation gives a rather remarkable result for cases in which the conductivity is negative. For these cases, is negative also and space charge fluctuations build up exponentially in time rather than dying out.

• Let us see how this occurs in a GaAs sample biased in the negative conductivity regime. The velocity-field diagram for n-type GaAs is illustrated in Fig. 6a .

• .

Fig. 6. Buildup and drift of a space charge domain in GaAs:

(a) Velocity-field characteristic for n-type GaAs;

(b) Formation of a dipole;

(c) Growth and drift of a dipole for conductions of negative conductivity.

•Under normal conditions, this dipole would die out quickly.

• However, under conditions of negative conductivity, the charge within the dipole, and therefore the local electric field, builds up as shown in Fig.6 .

• this buildup takes place in a stream of electrons drifting from the cathode to the anode, and the dipole (now called a domain) drifts along with the stream as it grows.

• Eventually, the drifting domain will reach the anode, where it gives up its energy as a pulse of current in the external circuit.

Superlattice phenomenon

• Superlattice is a periodic structure of layers of two (or more) materials. Typically, the thickness of one layer is several nanometers.

• The lattice spacing 0f normal semiconductor sample is of the order of 5.

• This may be realized if by some means the size of the Brillouin zone is reduced so that the electron velocity at the zone edge falls below the hot electron limit as mentioned above.

.• We know that the electron effective mass me

* is positive near the bottom 0 of the conduction band i.e., at k=0 in figure 7, while it is negative near its top.

• Hence, if it were possible to accelerate electron from 0 to A by applying an appropriate d.c. field over the superlattice sample, the electron effective mass me

*, the mobility and the sample conductivity would all be negative.

A B

E

k

Fig. 7. First Brillouin zone

,• As the electric field over the superlattice sample is slowly

raised, the electron energy E and its wave function k would first trace the path 0A of figure 7, if there is no electron-phonon interaction hampering their motion.

• At A the carriers suffer a Bragg reflection and switch back to B (since the wave vector changes to

• The electrons then begin to trace the path BO of the curve under the influence of the applied electric field .

.

.

,• the sample conductance once again changes from negative to

positive as the wave vector crosses the point 0 of the E-k curve. Thus in a superlattice a high frequency oscillation can be realized by applying d.c. field of appropriate magnitude.

Fig .8, Gunn oscillation

• Owning to increased lattice spacing the electron-phonon interaction impending the carrier motion would be smaller.

k

E

,

• The original Brillouin zone, as a result, will be subdivided into a number of minizones or minibands separated by mingaps as shown in the figure. 9 , where the super lattice period d is fixed as thrice the lattice period. The oscillation frequency of electrons may be computed as detailed below.

• The equation of motion of electrons on the basis of Newton’s 2nd law when electron-phonon collisions are absent is given by

(18)

from

Fig. 9. Energy band diagram of a superlattice

,• Where is the applied electric field intensity,

is the external force,

ħ is the Planck’s constant ;

• on integrating the above equation over a minizone, we get

(19)

(20)

Or

Where denotes the Bloch frequency of oscillation.

• For such a superlattice if a suitable magnetic field H is applied perpendicular to the plane of figure.9, the electrons would stat cycling about the magnetic field with a frequency (qH/2me

*) in the plane of the diagram.

• Such a circular motion is known to be resolved into two mutually perpendicular simple harmonic motions (SHMs) in the plane of the circle, one along the k-axis and the other perpendicular to it.

• When SHMs produced by the electric field is absorbed.

Such a phenomenon is known as Stark resonance.

At each well, a series of energies are available, much like that of a harmonic resonator.

These states form what is known as a Wannier-Stark energy ladder

• Given , we may easily derive the magnitude of the electron effective mass in a superlattice.

• Bloch oscillation is a phenomenon from solid state physics. It describes the oscillation of a particle (e.g. an electron) if a constant force is acting on it.

• Bloch oscillations can control Josephson Junctions

,

• Act much like bipolar transistors

• Bloch oscillations are just another strange quantum phenomenon

• They can be used for frequencies in the terahertz range

• Bloch oscillator transistors are an interesting way of amplifying signals.

References:

Ben. Streetman ,1995, Solid State Electronics Devices, 4th

edition, Englewood cliffs, New Jersey 07632.

S.M. Sze and Kwok K.Ng, 2006, Physics of Semiconductor Devices , 2nd edition, California , Interscience

A.J Saxana, 2010, An Introduction to Solid Electronics Devices. , 2nd edition

M.A. Omar, 2007, Elementary Solid State Physics, (revised), Delhi, India.

,