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2/6/2019

INE: Revision

INTRODUCTION TO NANOELECTRONICS Week-6, Lecture-2

Sneh Saurabh 6th February, 2019

Introduction to Nanoelectronics: S. Saurabh Overview 2

A free : Significance in Solid State Physics

Seminal concept in solid state physics Impact on Analysis

. In crystalline solids behave as if . Reduces the complicated problem of they are in vacuum, but with an effective waves in a crystalline solid to that different from their natural mass of waves in vacuum (Dramatically eases (Surprising !!!) the analysis of electronic flow)

. Energy-momentum relation 푝 . Valid for several materials such as Copper (metal) and Silicon () in wide 퐸 푝 = 퐸 + ∗ 2푚 range of energies . Energy- (퐸 − 푘) relation (using . Parabolic 퐸 − 푘 푟푒푙푎푡푖표푛푠ℎ푖푝 푝 = ℏ푘) A ℏퟐ풌ퟐ (퐷푖푠푝푒푟푠푖표푛 푟푒푙푎푡푖표푛푠ℎ푖푝) is not valid for 퐸 푘 = 퐸 + some materials such as graphene 2푚∗

Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 3 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 4

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A free particle + particle in a box !

Wave function of an electron in a confined nanoelectronic device can be treated as combined effect of (a free particle + a particle in a box)

Quantum Mechanics

Bulk Quantum Quantum Quantum (3D) Well (2D) Wire (1D) Dots (0D)

3-D Schrödinger Equation

Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 5 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 6

Schrödinger Equation: 3-D Schrödinger Equation: 3-D Separable solution (1)

Three-dimensional time-independent Schrödinger equation: . Applying quantum mechanics to nanostructures (which are naturally 3-D objects) requires 퐻휓(푥, 푦, 푧)= 퐸휓(푥, 푦, 푧) extending Schrödinger Equation to 3-D where: ℏ 휕 휕 휕 . Useful in finding solutions for problems such as a particle in 3-D box, hydrogen atom etc. 퐻 =− + + + 푉(푥, 푦, 푧) 2푚 휕푥 휕푦 휕푧 . Extending Schrödinger Equation to 3-D is straightforward If the function is separable: 푉 푥, 푦, 푧 = 푉 푥 + 푉 푦 +푉 푧 One-dimensional time-independent Three-dimensional time-independent Schrödinger equation: Schrödinger equation: Then the 3-D Schrödinger Equation can be separated 퐻휓(푥, 푦, 푧)= 퐸휓(푥, 푦, 푧) ℏ 휕 ℏ 휕 ℏ 휕 퐻휓(푥)= 퐸휓(푥) − + 푉 푥 휓 푥, 푦, 푧 + − + 푉 푥 휓 푥, 푦, 푧 + − + 푉 푥 휓 푥, 푦, 푧 2푚 휕푥 2푚 휕푦 2푚 휕푧 where: where: = 퐸 + 퐸 + 퐸 휓(푥, 푦, 푧) ℏ 휕 ℏ 휕 휕 휕 퐻 =− + 푉(푥) 퐻 =− + + + 푉(푥, 푦, 푧) 2푚 휕푥 2푚 휕푥 휕푦 휕푧 The solution can also be separated as: 휓 푥, 푦, 푧 = 휓 푥 휓 푦 휓 푧

Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 7 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 8

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Schrödinger Equation: 3-D Separable solution (2) Schrödinger Equation: 3-D Separable solution (3)

Substituting 흍 풙, 풚, 풛 = 흍풙 풙 흍풚 풚 흍풛 풛 When potential functions are separable, the ℏ 휕 휓 푥 ℏ 휕 휓 푦 These 1-D equations can be solved − + 푉 푥 휓 푥 휓 푦 휓 푧 + − + 푉 푦 휓 푦 휓 푥 휓 푧 + 3D Schrödinger equation can be changed to 2푚 휕푥 2푚 휕푦 three independent 1-D equations: independently and the solution is finally combined as: ℏ 휕 휓 푧 + − + 푉 푧 휓 푧 휓 푥 휓 푦 2푚 휕푧 ℏ 휕 휓 − + 푉휓 = 퐸휓 흍 풙, 풚, 풛 = 흍 흍 흍 = 퐸 + 퐸 + 퐸 휓 푥 휓 푦 휓 푧 2푚 휕푥 풙 풚 풛

Dividing both sides by 흍풙 풙 흍풚 풚 흍풛 풛 휕휓 1 ℏ 휕 휓 푥 1 ℏ 휕 휓 푦 ℏ − + 푉휓 = 퐸휓 − + 푉 푥 휓 푥 + − + 푉 푦 휓 푦 휓 푥 2푚 휕푥 휓 푦 2푚 휕푦 2푚 휕푦 1 ℏ 휕휓 푧 + − + 푉 푧 휓 푧 = 퐸 + 퐸 + 퐸 휓 푧 2푚 휕푧 ℏ 휕휓 − + 푉 휓 = 퐸 휓 2푚 휕푧 Three terms on LHS within square brackets are dependent on one variable only and are independent of other terms. The equation can be satisfied for any given (푥, 푦, 푧) if these terms are equal to constant 푬풙, 푬풚 퐚퐧퐝 푬풛 respectively.

Introduction to Nanoelectronics: S. Saurabh Device under Equilibrium 9 Introduction to Nanoelectronics: S. Saurabh Device under Equilibrium 10

3-D Schrödinger Equation: Problem : Quantization of Energy

Problem 15: Find the expression of total energy for a particle Problem 14: of mass 푚 trapped in a quantum well as shown in the Figure alongside. Find the and the possible energies of a particle of mass 푚 trapped in an infinite having a shape of a cube of dimension 퐿. 풏ퟐ흅ퟐℏퟐ Particle trapped in 1-D box: 푬 = 풏 ퟐ풎푳ퟐ

ℏퟐ풌ퟐ Answer: A Free particle: 푬 = ퟐ풎 For a particle trapped in 1-D box: ⁄ 휓 푥, 푦, 푧 = sin sin sin Given: 푳 ≪ 푳 and 푳 ≪ 푳 휓 푥 = sin 풙 풚 풙 풛 ퟐ ퟐ 흅 ℏ ퟐ ퟐ ퟐ 풏ퟐ흅ퟐℏퟐ 푬 = 풏 + 풏 + 풏 푬 = ퟐ풎푳ퟐ ퟏ ퟐ ퟑ 풏 ퟐ풎푳ퟐ Answer: ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ 풏 흅 ℏ ℏ 풌풚 ℏ 풌풛 퐸 = 퐸 + 퐸 + 퐸 = ퟐ + + ퟐ풎푳풙 ퟐ풎 ퟐ풎

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Quantum Wire: Quantization of Energy Things to follow …

Problem 16: Find the expression of total energy for a particle of mass 푚 trapped in a quantum wire as shown in the Figure alongside. . Equilibrium behavior of electrons in nanoelectronic structures: No voltage applied, 풏ퟐ흅ퟐℏퟐ Particle trapped in 1-D box: 푬 = With/without contacts, With/without confinement 풏 ퟐ풎푳ퟐ . Non-equilibrium behavior of electrons in ℏퟐ풌ퟐ A Free particle: 푬 = nanoelectronic structures: Voltage Applied, With ퟐ풎 contacts, With/without confinement Given: 푳풙 ≪ 푳풛 and 푳풚 ≪ 푳풛 . Nanoelectronic Devices: and Novel devices Answer: ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ 풏ퟏ 흅 ℏ 풏ퟐ 흅 ℏ ℏ 풌풛 퐸 = 퐸 + 퐸 + 퐸 = ퟐ + ퟐ + ퟐ풎푳풙 ퟐ풎푳풚 ퟐ풎

Introduction to Nanoelectronics: S. Saurabh Device under Equilibrium 13 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 14

Boundary Conditions

. Practically, all materials have certain boundary (it has to end somewhere) !

. Is treating electrons as free particle without any boundary condition correct?

Device under Equilibrium . In strict sense, applying no boundary condition is not correct.

. When dimensions are large, actual boundary conditions do not matter (for most of the Periodic material). Boundary . Choose a boundary condition that is convenient. . Periodic Boundary Condition: Most widely used boundary condition because it is Conditions mathematically convenient.

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Periodic Boundary Conditions : Meaning Periodic Boundary Conditions : Validity

. Graphene and Carbon Nanotube (CNT) have been studied . Assume that the finite material is repeated in the extensively unconfined direction infinite number of times  Flat structure: Graphene  Ring-shaped structure: CNT

. This introduces periodicity in the expected wave function: . Properties of Graphene and CNT is very much the same when the width of Graphene or the circumference of CNT is in tens of 휓 푧 = 휓(푧 + 퐿) nanometer

. Another way of interpreting is that the material is in form . Properties of Graphene and CNT are noticeably different when the of a ring width of Graphene or circumference of CNT is only a few nanometer

. ⇒ Periodic Boundary Conditions is equivalent to boundary conditions existing for flat material with finite size if dimensions are large enough

Introduction to Nanoelectronics: S. Saurabh Device under Equilibrium 17 Introduction to Nanoelectronics: S. Saurabh Device under Equilibrium 18

Periodic Boundary Conditions : Implications (1) Periodic Boundary Conditions : Implications (2)

. Electrons are modeled as free particle in a crystal . Separation between two allowed 휓 푧 = 퐴푒 Δ푘 = . Applying periodic boundary condition: 휓 푧 = 휓 푧 + 퐿 . When dimension is large (L is large) then the separation between allowed wavenumbers () 퐴푒 = 퐴푒 decrease ⇒ 푒 = 1 . In the limiting case when 퐿 →∞ the separation reduces to zero and the discrete distribution of wavenumbers changes to continuous distribution ⇒ 푘퐿 = 2푛휋 ⇒ 푘 = [Wavenumber is quantized] . Momentum is also quantized: [Using: 푝 =ℏ푘]

ℏ . Only a certain values of wavenumber is allowed 푘 = ⇒ 푝 = = depending on the size of the given specimen . Separation between two allowed momentum states . Separation between two allowed wavenumbers () Δ푝 = Δ푘 = − = . This allows counting the number of electron states in a material

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References

 David J. Griffiths, “Introduction to Quantum Mechanics”

 Arthur Beiser, "Concepts of Modern Physics”

 S. Datta, "Quantum Transport: Atom to "

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