Fachgebiet Angewandte Nanophysik, Institut für Physik

Contact: yong.lei@tu-ilmenau.de; huaping.zhao@tu-ilmenau.deOffice: Unterpoerlitzer Straße 38 (Heisenbergbau) (tel: 3748)

www.tu-ilmenau.de/nanostruk

Vorlesung: Thursday 13:00 – 14:30, F 3001Übung: Friday (G), 11:00 – 12:30, C 110

Prof. Yong Lei & Dr. Huaping Zhao & Dr. Rui Xu

(a) (b2)(b1)

UTAM-prepared free-standing one-dimensional surface nanostructures on Sisubstrates: Ni nanowire arrays (a) and carbon nanotube arrays (b).

Nanostrukturphysik (Nanostructure Physics)

Übung for:

Class 3: Quantum effects and nanostructures

Class 11: Graphene and 2D atomic-thin nanosheets

Übung for:

Class 3: Quantum effects and nanostructures

Q1: Predictions of light-metal interaction regarding

wave theory of light？

1. The higher the intensity of incident light, the greater the energy of

electrons that are emitted from surface.

2. A light wave of any frequency should be able to knock off

electrons, provided a reasonable intensity is maintained.

3. If the incident light is of low intensity, then the metal surface

must be continuously exposed for some time until enough waves

strike the surface to knock off electrons.

3 predictions from wave theory of light：

Q2: What‟s the photoelectric effect and its

experimental observation？

Experimental schematic of photoelectric effect

Frequency (Hz x1014)

The maximum kinetic energy of the photoelectron varies linearly with frequency with a limiting

frequency, below which no photoelectron is produced. The limiting frequency is known as the

threshold frequency.

Q3: Physical and mathematical explanation of the

photoelectric effect?

In 1905, physicist Albert Einstein published a paper to explain

light-metal interaction. To quote him:

“In accordance with the assumption to be considered here, the energy

of a light ray spreading out from a point source is not continuously

distributed over an increasing space, but consists of a finite number of

energy quanta which are localized at points in space, which move

without dividing, and which can only be produced and absorbed as

complete units.”

In simple words, he proposed that in photoelectric effect, light did not

behave like a wave, but rather like a particle, which we refer to as a

„photon‟.

• Physical explanation

The kinetic energy of photoelectron can be expressed by:

• Mathematical expression

2

max 0

1

2K m h h

hv：the incident photon energy；

hv0 ：the minimum energy required to remove an electron from the surface

Light moves like individual particles. The amount of energy carried by a

single photon (light particle) is

E h

h is Planck‟s constant (6.626×10-34J·s) and v is the frequency of light (s-1).

If the speed of light is c (299,792,458 m/s),

hcE

c

Einstein won a Nobel Prize for this!

Einstein won Nobel Prize in Physics in 1921 not for his theory of relativity, but

for successfully explaining photoelectric effect using particle nature of light.

Q4: Application of the photoelectric effect in solar

cells?

Cross section of

solar cell

Energy band diagram

of solar cell

N-Type P-Type P-Type Depletion

zone

Q5: What is De Broglie hypothesis?

Louis de-Broglie suggested that similar to light dual nature "every

moving matter has an associated wave". The wave associated with

the moving particle is known as matter wave or de-Broglie wave.

According to de Broglie‟s hypothesis, massless photons as well as

massive particles must satisfy one common set of relations, that

connect the energy E with the frequency f, and the momentum p with

the wavelength λ

Q6: De Broglie wavelength in terms of momentum?

h h

p m

Supposing a particle of mass ′m′ moving with a velocity v carries a

momentum p , it must be associated with the wave of wavelength

The above relation is known as de-Broglie equation and the

wavelength λ is known as de-Broglie wavelength.

Q7: De Broglie wavelength in terms of kinetic energy?

If a particle has kinetic energy E, then

2 2 221

2 2 2

m pE m

m m

2p mE

2

h h

p mE

Q8: De Broglie wavelength of an particle

accelerated by potential difference?

If an electron is accelerated by a potential difference of V volts, then its

kinetic energy E is given by

21

2E eV m

2eV

m

2

h h m

m m eV

Q9: What is Heisenberg‟s uncertainty principle?

The position and momentum of a particle cannot be simultaneously measured

with high precision. There is a minimum (uncertainty) for the product of

position and momentum - likewise a minimum (uncertainty) for product of

energy and time.

2x p

2E t

Complementarity

Energy/Velocity

Position Time

momentum precise, x unknown

A sinewave of wavelength λ implies that the

momentum is precisely known.

Wave spreads over all of space: the probability

to find its „position‟ is completely uncertain

avg

x

Adding several waves of different λ together

will produce an interference pattern – localized

wave (precise position). The process spreads

the momentum values (more uncertain

momentum).

Each wavelength represents

a different value of

momentum according to the

De Broglie relationship

h h

p m

Q10: How to derive 1D time-dependent

Schrödinger‟s wave equation?

For a system, the total energy E can be described by

Where „V‟ is the potential energy and „T‟ is the kinetic energy.

Then, we can rewrite the equation as

E T V

2

2

pE V

m

Considering a complex plane wave (1D):

After multiplying (3) to (2), we get

,

i kx tx t Ae

2

, , , ,2

pE x t x t V x t x t

m

(1)

(2)

(3)

(4)

: wave function,

k: wave number,

i: imaginary unit

： , f is frequency

2 /k

p: momentum,

m: particle mass,

2 f

Schrödinger Equation describes the change of a physical quantity over time in which the

quantum effects like wave-particle duality are significant.

As we know from wave-particle duality

2hp

2k

where „λ‟ is the wavelength and „k‟ is the wavenumber. We have

p k

(5)

(6)

2

, , , ,2

kE x t x t V x t x t

m (7)

2

, , , ,2

pE x t x t V x t x t

m (4)

Taking (6) into (4), we have

Taking the 1st derivative to x of (3) , we have

,i kx t

ikAe ik x tx

(8)

2

2 2

2,

i kx tk Ae k x t

x

(9)

Taking (9) into (7), we have

2 2

2, , , ,

2E x t x t V x t x t

m x

(10)

,

i kx tx t Ae

(3)

Then taking the 2nd derivative to x, we have

2

, , , ,2

kE x t x t V x t x t

m (7)

As we know from wave-particle duality

E (11)

2 2

2, , , ,

2E x t x t V x t x t

m x

(10)

Taking (11) into (10), we have

2 2

2, , , ,

2x t x t V x t x t

m x

(12)

Now we take the 1st derivative to t of (3), we have

,i kx t

i Ae i x tt

(13)

Taking (14) into (12), we have

2 2

2, , , ,

2i x t x t V x t x t

t m x

(15)

So we can say that

(14) 1

,x t ii t t

,

i kx tx t Ae

(3)

2 2

2, , , ,

2x t x t V x t x t

m x

(12)

From 1D Schrödinger Equation to 3D Schrödinger Equation

3D time-dependent Schrödinger equation

2 2

2, , , ,

2i x t x t V x t x t

t m x

1D time-dependent Schrödinger equation

2 2 22

2 2 2x y z

Q11: How to derive 1D time-independent

Schrödinger‟s wave equation?

Assuming that the wave function can be written in the form:

,x t x t

Substitute this form of the solution into Schrödinger‟s wave equation (15)

(2)

(1)

2 2

2, , ,

2i x t x t V x x t

t m x

2 2

22i x t x t V x x t

t m x

(3) 2 2

22i x t t x V x x t

t m x

(15)

Therefore, we have

(4)

2 2

2

1 1

2i t x V x

t t m x x

The time-dependent portion of equation (4) is then written as

1i t

t t

Since the right side of Equation (4) is a function of position x only and the left

side is a function of time only, each side of this equation must be equal to a

constant. We denote this separation of constant by

Where again the parameter is called a separation constant.

(5)

A solution of Equation (5) can be written in the form

/i tt e

(6)

The form of the above solution is a classical exponential form of a sinewave

where is the radian frequency . We have that . Then，

so that the separation constant is equal to the total energy E

of the particle.

2 f /

/ /E

1i t

t t

(5)

E

(7)

Now，the time-independent portion of Schrödinger‟s wave equation can be written

from Equation (4) as

Where the separation constant is the total energy E of the particle.

2 2

2

1

2x V x E

m x x

(4)

2 2

2

1 1

2i t x V x

t t m x x

3D time-independent Schrödinger equation

2

2

2V x r E r

m

2 2

22V x x E x

m x

(8)

(9)

2 2 22

2 2 2x y z

Q12: What is the physical meaning of the wave

function?

we have the wave function

Where is the complex conjugate function. Therefore,

(2)

/,

i E tx t x t x e

(1) 2 *, , ,x t x t x t

* ,x t

/* *,i E t

x t x e

Then the product of the wave function and its complex conjugate is given by

/ /* *, ,i E t i E t

x t x t x e x e

Therefore, we have that

(4) 2 2*,x t x x x

is the probability density function and is independent of time.

One major difference between classical and quantum mechanics is that in

classical mechanics, the position of a particle or body can be determined

precisely, whereas in quantum mechanics, the position of a particle found in

terms of a probability.

(3)

Q13: What‟s the boundary conditions of the wave

function？

Since the function represents the probability density function, then

for a single particle, we must have that

Equation (5) allows us to normalize the wave function and is a boundary

condition to determine some wave function coefficients.

2

, 1x t dx

2

,x t

(5)

Q14: Schrödinger‟s wave equation of an electron

in free space?

For simplicity, we assume that potential function V(x)=0 for all x. Then, the

time-independent wave equation can be written as

2

2 2

20

mx E V x x

x

2

2 2

20

mEx x

x

A general solution to this differential equation can be written in the form

2 2

exp expix mE ix mE

x A B

(1)

(2)

0V x

Recall that the time-dependent portion of the solution is

(4)

/i E tt e

, exp 2 exp 2i i

x t A x mE Et B x mE Et

Then the total solution for the wave function is given by

,x t x t

(3)

(2) 2 2

exp expix mE ix mE

x A B

Assume that we have a particle traveling in free space with the +x direction, which will

be described by the +x traveling wave. We can write the traveling-wave solution in

the form

(5) , expx t A i kx t

k is wave number and is 2

k

(4) , exp 2 exp 2i i

x t A x mE Et B x mE Et

0B

, exp 2i

x t A x mE Et

(4)

Comparing Equation (5)

with Equation (4)

Therefore, the wavelength is given by

2

h

mE

From de Broglie‟s wave-particle duality principle, the wavelength is also given by

h

p

An electron in free particle with a well-defined energy will also have a well-defined

wavelength and momentum.

(6)

(7)

2

, expx t A i x t

, exp 2i

x t A x mE Et

1 22mE

2

hmE

The probability density function is

* *, ,x t x t AA

which is a constant independent of position. This means that, a free

particle with a well-defined momentum can be found anywhere along

X with equal probability. This result is in agreement with the

Heisenberg uncertainty principle in that a precise momentum implies

an undefined position.

(5) , expx t A i kx t

Q15: Schrödinger‟s wave equation of an electron

in Infinite potential well?

The time-independent Schrödinger‟s wave equation is given by

2

2 2

20

mx E V x x

x

If E is finite, the particle cannot penetrate these infinite potential barriers, so the

probability of finding the particle in regions I and III is zero - the wave function must

be zero in both regions I and III (and also at the position 0 and a).

V x V x

Region I Region II Region III

0 a

(1)

The time-independent Schrödinger‟s wave equation in region II, where V=0, becomes

A particular form of solution to this equation is given by

2

2 2

20

mEx x

x

1 2cos sinx A kx A kx

2

2mEk

where

(2)

(3)

(4)

2

2 2

20

mx E V x x

x

0V x

Applying the boundary condition at x=0, we must have

At x=a, we have

nk

a

0 0x x a

ka nThis equation is valid if , where the parameter n is a positive integer (n=1, 2,

3…). The parameter n is referred to as a quantum number:

One boundary condition is that the wave function must satisfy x

(5)

(6)

(7)

V x V x

Region I Region II Region III

0 a

1 2 1 2 1cos sin cos0 sin0 0x A kx A kx A A A

1 2 2 2cos sin 0cos sin sin 0x A kx A kx ka A ka A ka

The coefficient A2 can be found from the normalization boundary condition that was

given by

Evaluating this integral gives

* 1x x dx

*x x

If we assume that the wave function solution is a real function, then x

Substituting the wave function into equation, we have

(8)

(10) 2 2

20

sin 1a

A kxdx

2

2A

a

(11)

(9)

Finally, the time-independent wave solution is given by

2

sinn x

xa a

The solution represents the electron in the infinite potential well.

The free electron was represented by a traveling wave, and now the bound particle is

represented by a standing wave.

(12) Where n=1, 2, 3….

20 sinx a A ka (6)

nk

a

(7)

2

2A

a

(11)

2( ) sinx kx

a

nk

a

For the particle in an infinite potential

well, the wave function is given by

The parameter k in the wave solution was defined by Equation (4) and (7), we obtain

(13) 2 2

2 2

2mE n

a

nk

a

(7)

2

2mEk (4)

2 2 2

22n

nE E

ma

(14)

Where n=1, 2, 3….

The total energy of the particle in infinite potential can only have discrete

values. This result means that the energy of the particle is quantized：quantum effect

Q16: Quantum effect in Quantum dots.

Quantum Dots (QDs)

When semiconductor particles are small enough, quantum effect comes into

play, which limit the energies at which electrons and holes can exist in particles.

https://www.sigmaaldrich.com/technical-documents/articles/materials-science/nanomaterials/quantum-dots.html

Übung for:

Class 11: Graphene and 2D atomic-thin nanosheets

1. How to synthesize graphene?

2. How to characterize graphene?

3. The Raman features of graphene. (D, G, and 2D bands.)

4. How to determine the layer numbers of graphene.

5. The basic properties of graphene.

6. The applications of graphene.

7. The features of 2D materials and the family of 2D materials.

8. The approaches to produce 2D materials.

9. 2D transition metal dichalogenides and their crystal structure.

10.The phase engineering of 2D TMDs.

11.The properties and applications of 2D materials.

12.Van de Waals heterostructures based on 2D materials.

1. How to synthesize graphene?

2. How to characterize graphene?

• Morphology

• Thickness

• Quality (defect)

• Atomic structure

3. The Raman features of graphene. (D, G, and 2D bands.)G band (∼1580 cm−1): associated with highly ordered graphite;D peak (∼1350 cm−1): associated with edge defects;2D band (∼2700 cm−1): characteristic of few layer graphene

4. How to determine the layer numbers of graphene.

AFM

5. The basic properties of graphene.

• Extremely low density.

• Mechanically strong.

• Highly flexible.

• Optically transparent.

• Excellent conductor of electron and heat.

• Semi‐metal: zero‐bandgap semiconductor.

• Tunable bandgap.

6. The applications of graphene.

7. The features of 2D materials and the family of 2D materials.

Layer Materials

Xenes which are atomically thin materials of a single element, with atoms arranged ina honeycomb lattice, e.g. graphene (carbon atoms), silicene (silicon atoms),germanene (germanium atoms), phosphorene (phosphorus atoms)

TMDs transition metal dichalcogenides of the form MX2, where M stands for atransition metal (from the 4th, 5th, or 6th group of the periodic table ofelements) and X is a chalcogen, such as S, Te, or Se. The TMDs based on M = Moand W and others based on Hf, Pd, Pt and Zr are semiconductors. The moststudied TMD semiconductors are MoS2 and WS2, which were the first atomicallythin semiconductors

SMCs semimetal chalcogenides of the form M2X2, where M is a semimetal (Ga or In)and X a chalcogen (S or Se); they are semiconducting materials

MXenes which have a hexagonal lattice and are of the form MAX, where M is atransition metal, A is an element from group 13 or 14, and X carbon or nitrogen

LDHs Layered double hydroxides

MOFs Metal‐organic frameworks

COFs Covalent‐organic frameworks

Families of 2D materials

8. The approaches to produce 2D materials.

• Top‐down

• Bottom‐up

Top‐Down: Mechanical Cleavage

Ion intercalation Ion exchange

Sonication‐assisted exfoliation

Electrochemical lithiation & exfoliation

Top‐Down

Nano Lett. 2012, 12, 1538‐1544

Adv.Mater. 2012, 24, 2320–2325

ACS Nano 2013, 7, 2768‐2772

Bottom‐Up

Chemical Vapor Deposition

Chemical Synthesis

9. 2D transition metal dichalogenides and their crystal structure.

Crystal structure of 2D TMDs

Crystal structure of 2D TMDs monolayer

10. The phase engineering of 2D TMDs.

Phase engineering of 2D TMDs monolayer

1T’Intermediate

1TOctahedral

2HTrigonal Prismatic

Nature Chemistry 2013, 5, 263

Phase engineering of 2D TMDs monolayer

11. The properties and applications of 2D materials.

2D materials covering a broad spectral range

12. Van de Waals heterostructures based on 2D materials.

Van der Waals Heterostructures based on 2D materials Van der Waals bonding enables stacking of different materials without need to form chemical bonds

Lego on atomic scale

Science  2016, 353, aac9439

Adv. Mater. 2019, 1903800

Physics Today 69, 9, 38 (2016)

Thank you!