MTLE-6120: Advanced Electronic Properties of Materials - - PowerPoint PPT Presentation

MTLE-6120: Advanced Electronic Properties of Materials Low-dimensional materials

References:

◮ 2D materials: Science 353, aac9439 (2016) ◮ Carrier lifetimes: Adv. Opt. Mat. 5, 1600914 (2017) ◮ Graphene mobility: C. Ullal, J. Shi and R. Sundararaman preprint (2018)

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2D materials: the new frontier

◮ Design materials at the atomic scale by combining different 2D layers ◮ Works with many classes of layered materials ⇒ tremendous flexibility

Figure adapted from Science 353, aac9439 (2016)

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Graphene and Hexagonal Boron Nitride

◮ Planar hexagonal structure (sp2 bonding) ◮ Graphene semi-metallic (more details later) ◮ Boron nitride (hBN): insulator which can withstand 0.5 V per atomic layer! ◮ Graphene/hBN spacing ≈ 3.3 ˚

A: capacitance?

Figure adapted from Science 353, aac9439 (2016)

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Transition metal dichalcogenides (TMDC): trigonal

◮ Non-planar structure: two group VI atoms vertically aligned ◮ Metallic or semiconducting depending on transition metal group ◮ Transition metal ⇒ d bands play important role ◮ Low dielectric screening ⇒ strongly bound excitons

Figure adapted from Science 353, aac9439 (2016)

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Transition metal dichalcogenides (TMDC): octahedral

◮ Non-planar structure: two group VI atoms vertically anti-aligned ◮ Metallic or semiconducting depending on transition metal group ◮ Transition metal ⇒ d bands play important role ◮ Low dielectric screening ⇒ strongly bound excitons

Figure adapted from Science 353, aac9439 (2016)

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Phosphorene

◮ Group V or Group IV-VI ◮ Distorted hexagons due to higher bond order in one direction ◮ Anisotropic electronic and optical properties in plane ◮ IV-VI materials non-centrosymmetric ⇒ 2D piezoelectric

Figure adapted from Science 353, aac9439 (2016)

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Group III chalcogenides

◮ Structure like trigonal TMDC, but with two metal layers ◮ Unusual VBM: away from Γ ⇒ high DOS at VBM ◮ High DOS ⇒ short lifetime ⇒ ultrafast response ◮ Ferromagnetic instability for degenerate p-doping

Figure adapted from Science 353, aac9439 (2016)

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Summary of properties

◮ Capacitance: ultra-thin dielectrics ◮ Tunneling devices: electronic wavefunctions

can couple across a 2D layer

◮ Optical properties:

◮ Strongly-bound excitons: LEDs (potentially) ◮ Indirect (inter-layer) excitions: solar cells ◮ Highly-confined plasmons (nano-photonics)

◮ Electronic properties: high mobility ◮ Heterostructures: mix and match all of the above!

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Band structure of graphene and heterostructures

Γ M K Γ A L H A −4 −2 2 4

E − Ef [e V] (a) Graphene

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0 10 1 10 2 10 3 10 4

−4 −2 2 4 −4 −2 2 4

E − Ef [e V] (b) Graphite

−4 −2 2 4 −4 −2 2 4

E − Ef [e V] (c) Graphene/hBN

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0 10 1 10 2 10 3 10 4

Lifetime [fs] −4 −2 2 4

Graphene Graphite Graphene/hBN

◮ Dirac point: linear E vs k ◮ Graphite: hybridiztaion

across layers breaks this

◮ hBN spacer reduces coupling;

preserves Dirac point

◮ Dirac point ⇒ low DOS ⇒

large lifetime ⇒ high mobility

Figure adapted from Adv. Opt. Mat. 5, 1600914 (2017)

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Carrier lifetimes in graphene

1 10 100 1000

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 τ [fs] E-EF [eV] Ag Graphene Graphite Graphene/hBN

◮ At Fermi level, graphene lifetime 100× that of silver ◮ Reduced somewhat by graphite, enhanced slightly by hBN ◮ Feature of low DOS: drops below Ag once away from Dirac point

Figure adapted from Adv. Opt. Mat. 5, 1600914 (2017)

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Carrier mobility

◮ Drude formula:

µ = eτ m∗

◮ Fermi Golden rule:

τ −1 ∝ g(EF )T (for T ≫ TD)

◮ Noble metals: τ ∼ 30 fs, m∗ ≈ me

⇒ µ ∼ 50 cm2V/s

◮ Semiconductors: τ ∼ 200 fs, m∗ ≈ 0.3me

⇒ µ ∼ 1000 cm2V/s

◮ Graphene: τ ∼ 2000 fs, claimed µ > 105 cm2V/s ⇒ m∗ < 0.02me! ◮ What is the effective mass for the band structure E = vF |

k| (with vF ∼ 8.3 × 105 m/s) near Dirac point?

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Why are carriers massless in graphene?

◮ In general, define ˙

• p = m∗ ˙
• v

◮ Since

v = ∇

pE, this yields (m∗)−1 = ∇ p∇ pE ◮ Near the Dirac point in graphene, E = vF |

k| = vF | p|

◮ If

p along x, v = ∇

pE = vF ˆ

x

◮

v follows direction of p without changing magnitude!

◮ What mass does this correspond to? ◮ Conventional explanation in literature: linear dispersion (Dirac equation)

⇒ massless particles in relativity

◮ Important: this is an analogy: vF ∼ c/400; many aspects of relativistic

particles like photons do not apply

◮ Examine mass tensor at more carefully:

(m∗)−1 = ∇

p∇ pvF |

p| = vF /p

• at

p = pˆ x with eigenvalues 0 and vF /p, i.e. ∞ and p/vF for m∗

◮ At Dirac point, p → 0 ⇒ m∗ T → 0 while m∗ L → ∞

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Mobility of graphene

◮ Derivation outline of usual Drude formula:

◮ Momentum

p = p0 − eEt where t = time since last collision

◮ Average over

p0 and t ⇒ p = −eEτ

◮ Drift velocity

vd = v = p/m∗ = −eEτ/m∗

◮ Mobility µ = vd/E = −eτ/m∗

◮ Issue for graphene: m∗ strongly dependent on

p (singular at Dirac point)

◮

p = p0 − eEt still true, but v = p/m∗

◮ True only for an appropriately averaged m∗:

(m∗)−1 =

• d

pP( p)(m∗)−1( p)

• d

pP( p) = v2

F

2kBT

◮ At room temperature, this yields m∗/me ≈ 0.013 ◮ At liquid He temperature, m∗/me ≈ 3.6 × 10−4 ◮ Conduction dominated by the transverse mass i.e.

v ⊥ p! (What does that look like?)

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Graphene vs graphane

◮ Graphane = hydrogenated graphene (proposed material) ◮ Non-planar: sp3 hybridization (like methane, diamond) ◮ Graphene: lim→∞ conjugated benzene rings ◮ Graphane: lim→∞ cyclohexane rings ◮ Will graphane be a metal, a semi-metal or a semiconductor?

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Band theory of bond conjugation

◮ Alternating single and double bonds ⇒ resonance structures ◮ ⇒ symmetry between bonding and antibonding orbitals / energies ◮ Finite system: discrete levels; always have gap ◮ Benzene has smaller gap than cyclohexane ◮ Infinite 2D limit → Dirac point in graphene ◮ What about 1D limit of a linear conjugated hydrocarbon?

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Conductive polymers

◮ Polyethene (polyethylene): all single bonds, excellent insulator ◮ Polyethyne (polyacetylene): conjugated bonds, highly conductive! ◮ Control band structure with functional groups on monomer ◮ Control Fermi level with occasional functional groups

during polymerization (like dopants)

◮ Conductive polymers and organic semiconductors:

2000 Nobel Prize in chemistry

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