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From Wikipedia, the free encyclopedia

(Redirected from Potential barrier)

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a

standard one-dimensional problem that demonstrates the phenomena of wave-
mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection.
The problem consists of solving the one-dimensional time-independent Schrödinger
equation for a particle encountering a rectangular potential energy barrier. It is usually
assumed, as here, that a free particle impinges on the barrier from the left.

Although classically a particle behaving as a point mass would be reflected if its energy
is less than

V0

, a particle actually behaving as a matter wave has a non-zero probability of

penetrating the barrier and continuing its travel as a wave on the other side. In classical
wave-physics, this effect is known as evanescent wave coupling. The likelihood that the
particle will pass through the barrier is given by the transmission coefficient, whereas
the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's
wave-equation allows these coefficients to be calculated.

Calculation[edit]
 Scattering at a finite potential barrier of height

V0

. The amplitudes and direction of left and right moving waves are indicated. In red, those
waves used for the derivation of the reflection and transmission amplitude.

E>V0

for this illustration.

The time-independent Schrödinger equation for the wave function

ψ(x)

reads

H^ψ(x)=[−ℏ22md2dx2+V(x)]ψ(x)=Eψ(x)

where

H^

is the Hamiltonian,
ℏ

is the (reduced) Planck constant,

is the mass,

the energy of the particle and

V(x)=V0[Θ(x)−Θ(x−a)]

is the barrier potential with height

V0>0

and width

Θ(x)=0,x<0;Θ(x)=1,x>0

is the Heaviside step function, i.e.,

V(x)={0if x<0V0if 0<x<a0if a<x

The barrier is positioned between

x=0
 and

x=a

. The barrier can be shifted to any

position without changing the results. The first term in the Hamiltonian,

−ℏ22md2dx2ψ

is the kinetic energy.

The barrier divides the space in three parts (

x<0,0<x<a,x>a

). In any of these parts, the potential is constant, meaning that

the particle is quasi-free, and the solution of the Schrödinger equation can be written as
a superposition of left and right moving waves (see free particle). If

E>V0

{ψL(x)=Areik0x+Ale−ik0xx<0ψC(x)=Breik1x+Ble−ik1x0<x<aψR(x)=Crei
k0x+Cle−ik0xx>a

where the wave numbers are related to

the energy via

{k0=2mE/ℏ2x<0orx>ak1=2m(E−V0)/ℏ20<x<a.
The index

r/l

on the coefficients

and

denotes the direction of the velocity vector. Note that, if the energy of the particle is
below the barrier height,

k1

becomes imaginary and the wave function is exponentially decaying within the
barrier. Nevertheless, we keep the notation

r/l

even though the waves are not propagating anymore in this case. Here we assumed

E≠V0

. The case

E=V0

is treated below.

The coefficients
A,B,C

have to be found from the boundary conditions of the wave function at

x=0

and

x=a

. The wave function and its derivative have to be continuous everywhere, so

ψL(0)=ψC(0)dψLdx|x=0=dψCdx|x=0ψC(a)=ψR(a)dψCdx|x=a=dψRdx|x=a.

Inserting the wave functions, the boundary conditions give the following restrictions on
the coefficients

Ar+Al=Br+Bl

ik0(Ar−Al)=ik1(Br−Bl)

Breiak1+Ble−iak1=Creiak0+Cle−iak0
 ik1(Breiak1−Ble−iak1)=ik0(Creiak0−Cle−iak0).

Transmission and reflection[edit]

At this point, it is instructive to compare the situation to the classical case. In both
cases, the particle behaves as a free particle outside of the barrier region. A classical
particle with energy

larger than the barrier height

V0

would always pass the barrier, and a classical particle with

E<V0

incident on the barrier would always get reflected.

To study the quantum case, consider the following situation: a particle incident on the
barrier from the left side (

Ar

). It may be reflected (

Al

) or transmitted (

Cr

).
To find the amplitudes for reflection and transmission for incidence from the left, we put
in the above equations

Ar=1

(incoming particle),

Al=r

(reflection),

Cl=0

(no incoming particle from the right), and

Cr=t

(transmission). We then eliminate the coefficients

Bl,Br

from the equation and solve for

and

The result is:

t=4k0k1e−ia(k0−k1)(k0+k1)2−e2iak1(k0−k1)2
 r=(k02−k12)sin⁡(ak1)2ik0k1cos⁡(ak1)+(k02+k12)sin⁡(ak1).

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are
the same as those from the left. Note that these expressions hold for any energy

E>0

E≠V0

. If

E=V0

, then

k1=0

, so there is a singularity in both of these expressions.

Analysis of the obtained

expressions[edit]

E < V0[edit]
 Transmission probability through a finite potential barrier for

2mV0a/ℏ

= 1, 3, and 7. Dashed: classical result. Solid line: quantum mechanical result.

The surprising result is that for energies less than the barrier height,

E<V0

there is a non-zero probability

T=|t|2=11+V02sinh2⁡(k1a)4E(V0−E)

for the particle to be transmitted through the barrier, with

k1=2m(V0−E)/ℏ2

. This effect, which differs from the classical case, is called

quantum tunneling. The transmission is exponentially suppressed with the barrier width,
which can be understood from the functional form of the wave function: Outside of the
barrier it oscillates with wave vector

k0

, whereas within the barrier it is exponentially damped over a distance

1/k1

. If the barrier is much wider than this decay length, the left and right part are
virtually independent and tunneling as a consequence is suppressed.

E > V0[edit]

In this case

T=|t|2=11+V02sin2⁡(k1a)4E(E−V0),

where

k1=2m(E−V0)/ℏ2

Equally surprising is that for energies larger than the barrier height,

E>V0

, the particle may be reflected from the barrier with a non-zero probability

R=|r|2=1−T.
The transmission and reflection probabilities are in fact oscillating with

k1a

. The classical result of perfect transmission without any reflection (

T=1

R=0

) is reproduced not only in the limit of high energy

E≫V0

but also when the energy and barrier width satisfy

k1a=nπ

, where

n=1,2,…

(see peaks near

E/V0=1.2

and 1.8 in the above figure). Note that the probabilities and amplitudes as
written are for any energy (above/below) the barrier height.

E = V0[edit]

The transmission probability at

E=V0
 [1]
is

T=11+ma2V0/2ℏ2.

This expression can be obtained by calculating the transmission coefficient from the
constants stated above as for the other cases or by taking the limit of

as

approaches

V0

. For this purpose the ratio

x=EV0

is defined, which is used in the function

f(x)

f(x)=sinh⁡(v01−x)1−x
In the last equation

v0

is defined as follows:

v0=2mV0a2ℏ2

These definitions can be inserted in the expression for

which was obtained for the case

E<V0

T(x)=11+f(x)24x

Now, when calculating the limit of

f(x)

as x approaches 1 (using L'Hôpital's rule),

limx→1f(x)=limx→1sinh⁡(v01−x)
(1−x)=limx→1ddxsinh⁡(v01−x)ddx1−x=v0cosh⁡(0)=v0

also the limit of

T(x)

as

approaches 1 can be obtained:

limx→1T(x)=limx→111+f(x)24x=11+v024

By plugging in the above expression for

v0

in the evaluated value for the limit, the above expression for T is successfully
reproduced.

Remarks and applications[edit]

The calculation presented above may at first seem unrealistic and hardly useful.
However it has proved to be a suitable model for a variety of real-life systems. One such
example are interfaces between two conducting materials. In the bulk of the materials,
the motion of the electrons is quasi-free and can be described by the kinetic term in the
above Hamiltonian with an effective mass
m

. Often the surfaces of such materials are covered with oxide layers or are not ideal
for other reasons. This thin, non-conducting layer may then be modeled by a barrier
potential as above. Electrons may then tunnel from one material to the other giving rise
to a current.

The operation of a scanning tunneling microscope (STM) relies on this tunneling effect.
In that case, the barrier is due to the gap between the tip of the STM and the underlying
object. Since the tunnel current depends exponentially on the barrier width, this device
is extremely sensitive to height variations on the examined sample.

The above model is one-dimensional, while space is three-dimensional. One should

solve the Schrödinger equation in three dimensions. On the other hand, many systems
only change along one coordinate direction and are translationally invariant along the
others; they are separable. The Schrödinger equation may then be reduced to the case
considered here by an ansatz for the wave function of the type:

Ψ(x,y,z)=ψ(x)ϕ(y,z)

For another, related model of a barrier, see Delta potential barrier (QM), which can be
regarded as a special case of the finite potential barrier. All results from this article
immediately apply to the delta potential barrier by taking the limits

V0→∞,a→0

while keeping

V0a=λ

constant.

See also[edit]
● Morse/Long-range potential
● Step potential
● Finite potential well
References[edit]
● ^ McQuarrie DA, Simon JD (1997). Physical Chemistry - A molecular Approach
(1st ed.). University Science Books. ISBN 978-0935702996.
● Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.).
Prentice Hall. ISBN 0-13-111892-7.
● Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck; et al. (1996).
Quantum mechanics. transl. from the French by Susan Reid Hemley. Wiley-
Interscience: Wiley. pp. 231–233. ISBN 978-0-471-56952-7.

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Categories:

Quantum models
Scattering theory
Schrödinger equation
Quantum mechanical potentials
This page was last edited on 22 April 2024, at 15:36 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply.
By using this site, you agre
Rectangular
potential barrier
20 languages

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From Wikipedia, the free encyclopedia

(Redirected from Potential barrier)

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a
standard one-dimensional problem that demonstrates the phenomena of wave-
mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection.
The problem consists of solving the one-dimensional time-independent Schrödinger
equation for a particle encountering a rectangular potential energy barrier. It is usually
assumed, as here, that a free particle impinges on the barrier from the left.

Although classically a particle behaving as a point mass would be reflected if its energy
is less than

V0

, a particle actually behaving as a matter wave has a non-zero probability of

penetrating the barrier and continuing its travel as a wave on the other side. In classical
wave-physics, this effect is known as evanescent wave coupling. The likelihood that the
particle will pass through the barrier is given by the transmission coefficient, whereas
the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's
wave-equation allows these coefficients to be calculated.

Calculation[edit]
 Scattering at a finite potential barrier of height

V0

. The amplitudes and direction of left and right moving waves are indicated. In red, those
waves used for the derivation of the reflection and transmission amplitude.

E>V0

for this illustration.

The time-independent Schrödinger equation for the wave function

ψ(x)

reads

H^ψ(x)=[−ℏ22md2dx2+V(x)]ψ(x)=Eψ(x)

where

H^

is the Hamiltonian,
ℏ

is the (reduced) Planck constant,

is the mass,

the energy of the particle and

V(x)=V0[Θ(x)−Θ(x−a)]

is the barrier potential with height

V0>0

and width

Θ(x)=0,x<0;Θ(x)=1,x>0

is the Heaviside step function, i.e.,

V(x)={0if x<0V0if 0<x<a0if a<x

The barrier is positioned between

x=0
 and

x=a

. The barrier can be shifted to any

position without changing the results. The first term in the Hamiltonian,

−ℏ22md2dx2ψ

is the kinetic energy.

The barrier divides the space in three parts (

x<0,0<x<a,x>a

). In any of these parts, the potential is constant, meaning that

the particle is quasi-free, and the solution of the Schrödinger equation can be written as
a superposition of left and right moving waves (see free particle). If

E>V0

{ψL(x)=Areik0x+Ale−ik0xx<0ψC(x)=Breik1x+Ble−ik1x0<x<aψR(x)=Crei
k0x+Cle−ik0xx>a

where the wave numbers are related to

the energy via

{k0=2mE/ℏ2x<0orx>ak1=2m(E−V0)/ℏ20<x<a.
The index

r/l

on the coefficients

and

denotes the direction of the velocity vector. Note that, if the energy of the particle is
below the barrier height,

k1

becomes imaginary and the wave function is exponentially decaying within the
barrier. Nevertheless, we keep the notation

r/l

even though the waves are not propagating anymore in this case. Here we assumed

E≠V0

. The case

E=V0

is treated below.

The coefficients
A,B,C

have to be found from the boundary conditions of the wave function at

x=0

and

x=a

. The wave function and its derivative have to be continuous everywhere, so

ψL(0)=ψC(0)dψLdx|x=0=dψCdx|x=0ψC(a)=ψR(a)dψCdx|x=a=dψRdx|x=a.

Inserting the wave functions, the boundary conditions give the following restrictions on
the coefficients

Ar+Al=Br+Bl

ik0(Ar−Al)=ik1(Br−Bl)

Breiak1+Ble−iak1=Creiak0+Cle−iak0
 ik1(Breiak1−Ble−iak1)=ik0(Creiak0−Cle−iak0).

Transmission and reflection[edit]

At this point, it is instructive to compare the situation to the classical case. In both
cases, the particle behaves as a free particle outside of the barrier region. A classical
particle with energy

larger than the barrier height

V0

would always pass the barrier, and a classical particle with

E<V0

incident on the barrier would always get reflected.

To study the quantum case, consider the following situation: a particle incident on the
barrier from the left side (

Ar

). It may be reflected (

Al

) or transmitted (

Cr

).
To find the amplitudes for reflection and transmission for incidence from the left, we put
in the above equations

Ar=1

(incoming particle),

Al=r

(reflection),

Cl=0

(no incoming particle from the right), and

Cr=t

(transmission). We then eliminate the coefficients

Bl,Br

from the equation and solve for

and

The result is:

t=4k0k1e−ia(k0−k1)(k0+k1)2−e2iak1(k0−k1)2
 r=(k02−k12)sin⁡(ak1)2ik0k1cos⁡(ak1)+(k02+k12)sin⁡(ak1).

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are
the same as those from the left. Note that these expressions hold for any energy

E>0

E≠V0

Categories:

Quantum models
Scattering theory
Schrödinger equation
Quantum mechanical potentials
This page was last edited on 22 April 2024, at 15:36 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply.
By using this site, you agre

A breakthrough in FET research came with the work of Egyptian engineer Mohamed
[3]
Atalla in the late 1950s. In 1958 he presented experimental work which showed that
growing thin silicon oxide on clean silicon surface leads to neutralization of surface
states. This is known as surface passivation, a method that became critical to the
semiconductor industry as it made mass-production of silicon integrated circuits
[19][20]
possible.

The metal–oxide–semiconductor field-effect transistor (MOSFET) was then invented by

[21][22]
Mohamed Atalla and Dawon Kahng in 1959. The MOSFET largely superseded
[2]
both the bipolar transistor and the JFET, and had a profound effect on digital
[23][22] [24]
electronic development. With its high scalability, and much lower power
[25]
consumption and higher density than bipolar junction transistors, the MOSFET made
[26]
it possible to build high-density integrated circuits. The MOSFET is also capable of
[27]
handling higher power than the JFET. The MOSFET was the first truly compact
[6]
transistor that could be miniaturised and mass-produced for a wide range of uses. The
[20]
MOSFET thus became the most common type of transistor in computers, electronics,
[28]
and communications technology (such as smartphones). The US Patent and
Trademark Office calls it a "groundbreaking invention that transformed life and culture
[28]
around the world".

CMOS (complementary MOS), a semiconductor device fabrication process for

MOSFETs, was developed by Chih-Tang Sah and Frank Wanlass at Fairchild
[29][30]
Semiconductor in 1963. The first report of a floating-gate MOSFET was made by
[31]
Dawon Kahng and Simon Sze in 1967. A double-gate MOSFET was first
demonstrated in 1984 by Electrotechnical Laboratory researchers Toshihiro Sekigawa
[32][33]
and Yutaka Hayashi. FinFET (fin field-effect transistor), a type of 3D non-planar
multi-gate MOSFET, originated from the research of Digh Hisamoto and his team at
[34][35]
Hitachi Central Research Laboratory in 1989.

Basic information