Sub-bandgap Voltage Electroluminescence and Magneto-oscillations in a WSe2 Light-Emitting van der Waals Heterostructure.

TLDR: A threshold voltage for electroluminescence significantly lower than the corresponding single particle band gap of monolayer WSe2 is observed, interpreted by considering the Coulomb interaction and a tunneling process involving excitons, well beyond the picture of independent charge carriers.

Abstract: We report on experimental investigations of an electrically driven WSe2 based light-emitting van der Waals heterostructure. We observe a threshold voltage for electroluminescence significantly lower than the corresponding single particle band gap of monolayer WSe2. This observation can be interpreted by considering the Coulomb interaction and a tunneling process involving excitons, well beyond the picture of independent charge carriers. An applied magnetic field reveals pronounced magneto-oscillations in the electroluminescence of the free exciton emission intensity with a 1/B periodicity. This effect is ascribed to a modulation of the tunneling probability resulting from the Landau quantization in the graphene electrodes. A sharp feature in the differential conductance indicates that the Fermi level is pinned and allows for an estimation of the acceptor binding energy.

Full text

The University of Manchester Research
Sub-bandgap voltage electroluminescence and magneto-
oscillations in a WSe2 light-emitting van der Waals
heterostructure
DOI:
10.1021/acs.nanolett.6b04374
Document Version
Accepted author manuscript
Link to publication record in Manchester Research Explorer
Citation for published version (APA):
Binder, J., Withers, F., Molas, M. R., Faugeras, C., Nogajewski, K., Watanabe, K., Taniguchi, T., Kozikov, A.,
Geim, A., Novoselov, K., & Potemski, M. (2017). Sub-bandgap voltage electroluminescence and magneto-
oscillations in a WSe2 light-emitting van der Waals heterostructure. Nano Letters.
https://doi.org/10.1021/acs.nanolett.6b04374
Published in:
Nano Letters
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Download date:26. Aug. 2022

❙✉❜✲❜❛♥❞❣❛♣ ✈♦❧t❛❣❡ ❡❧❡❝tr♦❧✉♠✐♥❡s❝❡♥❝❡ ❛♥❞
♠❛❣♥❡t♦✲♦s❝✐❧❧❛t✐♦♥s ✐♥ ❛ ❲❙❡
2
❧✐❣❤t✲❡♠✐tt✐♥❣
✈❛♥ ❞❡r ❲❛❛❧s ❤❡t❡r♦str✉❝t✉r❡
❏♦❤❛♥♥❡s ❇✐♥❞❡r✱
∗
✱
†
✱
‡
❋r❡❞❞✐❡ ❲✐t❤❡rs✱
¶
✱
§
▼❛❝✐❡❥ ❘✳ ▼♦❧❛s✱
†
❈❧❡♠❡♥t ❋❛✉❣❡r❛s✱
†
❑❛r♦❧ ◆♦❣❛❥❡✇s❦✐✱
†
❑❡♥❥✐ ❲❛t❛♥❛❜❡✱
k
❚❛❦❛s❤✐ ❚❛♥✐❣✉❝❤✐✱
k
❆❧❡❦s❡② ❑♦③✐❦♦✈✱
¶
✱
§
❆♥❞r❡ ❑✳ ●❡✐♠✱
§
✱
⊥
❑♦st②❛ ❙✳ ◆♦✈♦s❡❧♦✈✱
¶
✱
§
❛♥❞ ▼❛r❡❦ P♦t❡♠s❦✐
∗
✱
†
†
▲❛❜♦r❛t♦✐r❡ ◆❛t✐♦♥❛❧ ❞❡s ❈❤❛♠♣s ▼❛❣♥❡t✐q✉❡s ■♥t❡♥s❡s✱ ❈◆❘❙✲❯●❆✲❯P❙✲■◆❙❆✲❊▼❋▲✱
✷✺ ❘✉❡ ❞❡s ▼❛rt②rs✱ ✸✽✵✹✷ ●r❡♥♦❜❧❡✱ ❋r❛♥❝❡
‡
❋❛❝✉❧t② ♦❢ P❤②s✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❲❛rs❛✇✱ P❛st❡✉r❛ ✺✱ ✵✷✲✵✾✸ ❲❛rs❛✇✱ P♦❧❛♥❞
¶
❙❝❤♦♦❧ ♦❢ P❤②s✐❝s ❛♥❞ ❆str♦♥♦♠②✱ ❯♥✐✈❡rs✐t② ♦❢ ▼❛♥❝❤❡st❡r✱ ❖①❢♦r❞ ❘♦❛❞✱ ▼❛♥❝❤❡st❡r
▼✶✸ ✾P▲✱ ❯❑
§
◆❛t✐♦♥❛❧ ●r❛♣❤❡♥❡ ■♥st✐t✉t❡✱ ❯♥✐✈❡rs✐t② ♦❢ ▼❛♥❝❤❡st❡r✱ ❖①❢♦r❞ ❘♦❛❞✱ ▼❛♥❝❤❡st❡r✱ ▼✶✸
✾P▲✱ ❯❑
k
◆❛t✐♦♥❛❧ ■♥st✐t✉t❡ ❢♦r ▼❛t❡r✐❛❧s ❙❝✐❡♥❝❡✱ ✶✲✶ ◆❛♠✐❦✐✱ ❚s✉❦✉❜❛ ✸✵✺✲✵✵✹✹✱ ❏❛♣❛♥
⊥
▼❛♥❝❤❡st❡r ❈❡♥tr❡ ❢♦r ▼❡s♦s❝✐❡♥❝❡ ❛♥❞ ◆❛♥♦t❡❝❤♥♦❧♦❣②✱ ❯♥✐✈❡rs✐t② ♦❢ ▼❛♥❝❤❡st❡r✱
❖①❢♦r❞ ❘♦❛❞✱ ▼❛♥❝❤❡st❡r✱ ▼✶✸ ✾P▲✱ ❯❑
❊✲♠❛✐❧✿ ❥♦❤❛♥♥❡s✳❜✐♥❞❡r❅❢✉✇✳❡❞✉✳♣❧❀ ♠❛r❡❦✳♣♦t❡♠s❦✐❅❧♥❝♠✐✳❝♥rs✳❢r
P❤♦♥❡✿ ✰✹✽ ✷✷ ✺✺ ✸✷ ✼✾✷❀ ✰✸✸ ✹✼✻ ✽✽ ✼✽ ✼✻
❆❜str❛❝t
❲❡ r❡♣♦rt ♦♥ ❡①♣❡r✐♠❡♥t❛❧ ✐♥✈❡st✐❣❛t✐♦♥s ♦❢ ❛♥ ❡❧❡❝tr✐❝❛❧❧② ❞r✐✈❡♥ ❲❙❡
2
❜❛s❡❞ ❧✐❣❤t✲
❡♠✐tt✐♥❣ ✈❛♥ ❞❡r ❲❛❛❧s ❤❡t❡r♦str✉❝t✉r❡✳ ❲❡ ♦❜s❡r✈❡ ❛ t❤r❡s❤♦❧❞ ✈♦❧t❛❣❡ ❢♦r ❡❧❡❝tr♦❧✉✲
✶

♠✐♥❡s❝❡♥❝❡ s✐❣♥✐✜❝❛♥t❧② ❧♦✇❡r t❤❛♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐♥❣❧❡ ♣❛rt✐❝❧❡ ❜❛♥❞ ❣❛♣ ♦❢ ♠♦♥♦✲
❧❛②❡r ❲❙❡
2
✳ ❚❤✐s ♦❜s❡r✈❛t✐♦♥ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❈♦✉❧♦♠❜ ✐♥t❡r❛❝✲
t✐♦♥ ❛♥❞ ❛ t✉♥♥❡❧✐♥❣ ♣r♦❝❡ss ✐♥✈♦❧✈✐♥❣ ❡①❝✐t♦♥s✱ ✇❡❧❧ ❜❡②♦♥❞ t❤❡ ♣✐❝t✉r❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t
❝❤❛r❣❡ ❝❛rr✐❡rs✳ ❆♥ ❛♣♣❧✐❡❞ ♠❛❣♥❡t✐❝ ✜❡❧❞ r❡✈❡❛❧s ♣r♦♥♦✉♥❝❡❞ ♠❛❣♥❡t♦✲♦s❝✐❧❧❛t✐♦♥s ✐♥
t❤❡ ❡❧❡❝tr♦❧✉♠✐♥❡s❝❡♥❝❡ ♦❢ t❤❡ ❢r❡❡ ❡①❝✐t♦♥ ❡♠✐ss✐♦♥ ✐♥t❡♥s✐t② ✇✐t❤ ❛ ✶✴❇✲♣❡r✐♦❞✐❝✐t②✳
❚❤✐s ❡✛❡❝t ✐s ❛s❝r✐❜❡❞ t♦ ❛ ♠♦❞✉❧❛t✐♦♥ ♦❢ t❤❡ t✉♥♥❡❧✐♥❣ ♣r♦❜❛❜✐❧✐t② r❡s✉❧t✐♥❣ ❢r♦♠ t❤❡
▲❛♥❞❛✉ q✉❛♥t✐③❛t✐♦♥ ✐♥ t❤❡ ❣r❛♣❤❡♥❡ ❡❧❡❝tr♦❞❡s✳ ❆ s❤❛r♣ ❢❡❛t✉r❡ ✐♥ t❤❡ ❞✐✛❡r❡♥t✐❛❧
❝♦♥❞✉❝t❛♥❝❡ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ❋❡r♠✐ ❧❡✈❡❧ ✐s ♣✐♥♥❡❞ ❛♥❞ ❛❧❧♦✇s ❢♦r ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡
❛❝❝❡♣t♦r ❜✐♥❞✐♥❣ ❡♥❡r❣②✳
❑❡②✇♦r❞s
❡❧❡❝tr♦❧✉♠✐♥❡s❝❡♥❝❡✱ ♠❛❣♥❡t♦✲♦s❝✐❧❧❛t✐♦♥s✱ ✈❛♥ ❞❡r ❲❛❛❧s ❤❡t❡r♦str✉❝t✉r❡s✱ t✉♥❣st❡♥ ❞✐s❡✲
❧❡♥✐❞❡✱ ❤❡①❛❣♦♥❛❧ ❜♦r♦♥ ♥✐tr✐❞❡✱ ❛❝❝❡♣t♦r
❆ ♥❡✇ st❡♣ ♦❢ ❝♦♠♣❧❡①✐t② ❤❛s r❡❝❡♥t❧② ❜❡❡♥ ✉♥❞❡rt❛❦❡♥ ✐♥ t❤❡ ✜❡❧❞ ♦❢ t✇♦✲❞✐♠❡♥s✐♦♥❛❧
❝r②st❛❧s✱ ❜② ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ♣❧❛❝✐♥❣ ❛t♦♠✐❝❛❧❧② t❤✐♥ ❧❛②❡rs ♦❢ ❞✐✛❡r❡♥t ♠❛t❡r✐❛❧s ♦♥ t♦♣ ♦❢
❡❛❝❤ ♦t❤❡r✳ ❚❤❡ r❡s✉❧t✐♥❣ st❛❝❦s ❛r❡ r❡❢❡rr❡❞ t♦ ❛s ✈❛♥ ❞❡r ❲❛❛❧s ✭✈❞❲✮ ❤❡t❡r♦str✉❝t✉r❡s✳
✶✱✷
❇❛s❡❞ ♦♥ t❤✐s ✐❞❡❛✱ ❛ ❢❡✇ ♣r♦t♦t②♣❡ ❞❡✈✐❝❡s✱ s✉❝❤ ❛s t✉♥♥❡❧✐♥❣ tr❛♥s✐st♦rs
✸✕✶✶
❛♥❞✴♦r ❧✐❣❤t✲
❡♠✐tt✐♥❣ t✉♥♥❡❧✐♥❣ ❞✐♦❞❡s✱
✶✷✕✶✻
❤❛✈❡ ❜❡❡♥ ❢❛❜r✐❝❛t❡❞ ❛♥❞ s✉❝❝❡ss❢✉❧❧② t❡st❡❞✳ ❍♦✇❡✈❡r✱ ❢✉r✲
t❤❡r ✇♦r❦ ✐s ♥❡❝❡ss❛r② ✐♥ ♦r❞❡r t♦ ❜❡tt❡r ❝❤❛r❛❝t❡r✐③❡ s✉❝❤ str✉❝t✉r❡s✱ t♦ ❧❡❛r♥ ♠♦r❡ ❛❜♦✉t
t❤❡✐r ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s✱ ✇✐t❤ t❤❡ ❛✐♠ t♦ ♣r♦♣❡r❧② ❞❡s✐❣♥ ❞❡✈✐❝❡ ♦♣❡r❛t✐♦♥✳
❍❡r❡✱ ✇❡ ✉♥✈❡✐❧ ♥❡✇ ❢❛❝❡ts ♦❢ ❧✐❣❤t ❡♠✐tt✐♥❣ ✈❞❲ ❤❡t❡r♦str✉❝t✉r❡s✱ ✇✐t❤ r❡❢❡r❡♥❝❡ t♦ t❤❡
✐ss✉❡ ♦❢ t❤❡ ❛❧✐❣♥♠❡♥t ♦❢ ❡❧❡❝tr♦♥✐❝ ❜❛♥❞s✱ ❡✛❡❝ts ♦❢ ❈♦✉❧♦♠❜ ✐♥t❡r❛❝t✐♦♥ ❛♥❞ ❛ s✉❜t❧❡ ❜✉t
st✐❧❧ ❛❝t✐✈❡ r♦❧❡ ♦❢ t❤❡ ❣r❛♣❤❡♥❡ ❡❧❡❝tr♦❞❡s ✐♥ t❤❡s❡ ❞❡✈✐❝❡s✳ ❲❡ r❡♣♦rt ♦♥ ♦♣t♦❡❧❡❝tr♦♥✐❝
♠❡❛s✉r❡♠❡♥ts ♣❡r❢♦r♠❡❞ ♦♥ ❛ ❲❙❡
2
✲❜❛s❡❞ t✉♥♥❡❧✐♥❣ ❧✐❣❤t✲❡♠✐tt✐♥❣ ❞✐♦❞❡✳ ❚❤❡ ❞✐✛❡r❡♥t✐❛❧
t✉♥♥❡❧✐♥❣ ❝♦♥❞✉❝t❛♥❝❡ ♦❢ ♦✉r str✉❝t✉r❡ s❤♦✇s ❛ ❧❛r❣❡ ③❡r♦ ❜✐❛s ❛♥♦♠❛❧② ✭♣❡❛❦✮✱ ✇❤✐❝❤ ✇❡
❛s❝r✐❜❡ t♦ ♣✐♥♥✐♥❣ ♦❢ t❤❡ ❋❡r♠✐ ❡♥❡r❣② ❛t t❤❡ ❲❙❡
2
✐♠♣✉r✐t②✴❛❝❝❡♣t♦r ❧❡✈❡❧✳ ❆ ❝♦♥❝❡✐✈❛❜❧❡
s❝❡♥❛r✐♦ ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❜❛♥❞ ❛❧✐❣♥♠❡♥t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❜✐❛s ✈♦❧t❛❣❡ ✐s ♣r♦♣♦s❡❞✳
✷

❙tr✐❦✐♥❣❧②✱ t❤❡ ❜✐❛s✲♣♦t❡♥t✐❛❧ ♦♥s❡t ❢♦r t❤❡ ❡❧❡❝tr♦❧✉♠✐♥❡s❝❡♥❝❡ ✐s ❢♦✉♥❞ t♦ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡
❡♥❡r❣② ♦❢ t❤❡ ❢r❡❡ ❡①❝✐t♦♥ ♦❢ t❤❡ ❲❙❡
2
♠♦♥♦❧❛②❡r ✭❛♥❞ ♥♦t ✇✐t❤ t❤❡ ❡♥❡r❣② ♦❢ ❛ s✐♥❣❧❡✲♣❛rt✐❝❧❡
❜❛♥❞❣❛♣✮✳ ❚❤✐s ❢❛❝t ♣♦✐♥ts ♦✉t t❤❡ r❡❧❡✈❛♥t r♦❧❡ ♦❢ ❈♦✉❧♦♠❜ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ❡❧❡❝tr✐❝❛❧❧②
✐♥❥❡❝t❡❞ ❝❛rr✐❡rs ♦♥ t❤❡ t✉♥♥❡❧✐♥❣ ♣r♦❝❡ss❡s ✐♥ ♦✉r ❞❡✈✐❝❡✳ ❋✉rt❤❡r♠♦r❡✱ ♣r♦♥♦✉♥❝❡❞ ♠❛❣♥❡t♦✲
♦s❝✐❧❧❛t✐♦♥s ❛r❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❡❧❡❝tr♦❧✉♠✐♥❡s❝❡♥❝❡ ❡♠✐ss✐♦♥ ✐♥t❡♥s✐t② ♠❡❛s✉r❡❞ ❛s ❛ ❢✉♥❝t✐♦♥
♦❢ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❛♣♣❧✐❡❞ ♣❡r♣❡♥❞✐❝✉❧❛r❧② t♦ t❤❡ ❧❛②❡r ♣❧❛♥❡s✳ ❚❤❡s❡ ♦s❝✐❧❧❛t✐♦♥s✱ ♣❡r✐♦❞✐❝ ✇✐t❤
t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ r❡✢❡❝t t❤❡ ♠♦❞✉❧❛t✐♦♥ ♦❢ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❝❛rr✐❡r t✉♥♥❡❧✐♥❣
❛♥❞ ❛r❡ ❝❛✉s❡❞ ❜② t❤❡ ▲❛♥❞❛✉ q✉❛♥t✐③❛t✐♦♥ ♦❢ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❣r❛♣❤❡♥❡ ❡❧❡❝tr♦❞❡s✳
❲❡ st✉❞✐❡❞ ❛ ❧✐❣❤t✲❡♠✐tt✐♥❣ ❞✐♦❞❡ str✉❝t✉r❡
✶✷✱✶✸
t❤❛t ✐s ❜❛s❡❞ ♦♥ ❛ ❲❙❡
2
♠♦♥♦❧❛②❡r ❛s
t❤❡ ❛❝t✐✈❡ ♣❛rt✳ ❚❤❡ ❧❛②❡r s❡q✉❡♥❝❡ ❢♦r t❤✐s ❞❡✈✐❝❡ ✇❛s ❙✐ ✴ ❙✐❖
2
✴ ❤❇◆ ✴ ❣r❛♣❤❡♥❡ ✴ ❤❇◆ ✴
❲❙❡
2
✴ ❤❇◆ ✴ ❣r❛♣❤❡♥❡✳ ❚❤❡ ❡♠✐ss✐♦♥ ❛r❡❛ ♦❢ t❤❡ str✉❝t✉r❡ ✐s ♣r❡s❡♥t❡❞ ♦♥ t❤❡ ♠✐❝r♦s❝♦♣❡
✐♠❛❣❡ ✐♥ ❋✐❣✉r❡ ✶ ✭❛✮✳ ❋✐❣✉r❡ ✶ ✭❜✮ ❞❡♣✐❝ts ❛ s❝❤❡♠❛t✐❝ ❞r❛✇✐♥❣ ♦❢ t❤❡ ❧❛②❡r❡❞ str✉❝t✉r❡✳
❚❤❡ t✇♦ ❤❇◆ s♣❛❝❡rs t❤❛t s❡♣❛r❛t❡ t❤❡ ❲❙❡
2
♠♦♥♦❧❛②❡r ❢r♦♠ t❤❡ ❣r❛♣❤❡♥❡ ❡❧❡❝tr♦❞❡s ❛r❡
t✇♦ ❧❛②❡rs t❤✐❝❦✳ ❆ ❞❡t❛✐❧❡❞ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢❛❜r✐❝❛t✐♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❘❡❢✳ ❬✶✷❪✳
❚❤❡ ♦♣t♦❡❧❡❝tr♦♥✐❝ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ s❛♠♣❧❡ ✇❡r❡ st✉❞✐❡❞ ❜② r❡❝♦r❞✐♥❣ t❤❡ ❡❧❡❝tr♦✲
❧✉♠✐♥❡s❝❡♥❝❡ ✭❊▲✮ s✐❣♥❛❧ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❜✐❛s ✈♦❧t❛❣❡ ❛♥❞ ✐♥ ♠❛❣♥❡t✐❝ ✜❡❧❞s ✉♣ t♦ ✶✹ ❚✳
❈✉rr❡♥t✲✈♦❧t❛❣❡ ❝✉r✈❡s ✇❡r❡ ♠❡❛s✉r❡❞ t♦ st✉❞② t❤❡ t✉♥♥❡❧✐♥❣ ♣r♦❝❡ss❡s ❛♥❞ ♣❤♦t♦❧✉♠✐♥❡s✲
❝❡♥❝❡ ✭P▲✮ ♠❛♣♣✐♥❣ ♦❢ t❤❡ s❛♠♣❧❡ ✇❛s ♣❡r❢♦r♠❡❞ t♦ ❛❞❞✐t✐♦♥❛❧❧② ❝❤❛r❛❝t❡r✐③❡ t❤❡ str✉❝t✉r❡✳
❆❧❧ ♠❡❛s✉r❡♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ ❛♥ ♦♣t✐❝❛❧✲✜❜❡r✲❜❛s❡❞ ✐♥s❡rt ♣❧❛❝❡❞ ✐♥ ❛ s✉♣❡r❝♦♥✲
❞✉❝t✐♥❣ ❝♦✐❧✳ ❚❤❡ ✐♥✈❡st✐❣❛t❡❞ s❛♠♣❧❡ ✇❛s ❧♦❝❛t❡❞ ♦♥ t♦♣ ♦❢ ❛♥ ①✲②✲③ ♣✐❡③♦✲st❛❣❡ ❦❡♣t ✐♥
❤❡❧✐✉♠ ❣❛s ❛t
T = 4.2
❑✳ ❚❤❡ ❧❛s❡r ❧✐❣❤t ❢r♦♠ ❛ ❝♦♥t✐♥✉♦✉s ✇❛✈❡ ❆r✰ ❧❛s❡r ✭
λ = 514.5
♥♠✮
✇❛s ❝♦✉♣❧❡❞ t♦ ❛♥ ❡①❝✐t❛t✐♦♥ ✜❜❡r ♦❢
5 µ
♠ ❞✐❛♠❡t❡r✱ ❢♦❝✉s❡❞ ♦♥ t❤❡ s❛♠♣❧❡ ❜② ❛♥ ❛s♣❤❡r✐❝
❧❡♥s✳ ❚❤❡ s✐❣♥❛❧ ✇❛s ❞❡t❡❝t❡❞ ✇✐t❤ ❛
50 µ
♠ ❝♦r❡ ✜❜❡r ✭❝♦❧❧❡❝t✐♦♥ s♣♦t ❞✐❛♠❡t❡r ♦❢
∼ 10 µ
♠✮
❜② ❛
0.5
♠ ❧♦♥❣ ♠♦♥♦❝❤r♦♠❛t♦r ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❝❤❛r❣❡✲❝♦✉♣❧❡✲❞❡✈✐❝❡ ✭❈❈❉✮ ❝❛♠❡r❛✳ ❊❧❡❝✲
tr✐❝❛❧ ♠❡❛s✉r❡♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ✉s✐♥❣ ❛ ❑❡✐t❤❧❡② ✷✹✵✵ s♦✉r❝❡✲♠❡❛s✉r❡ ✉♥✐t✳
❆ ✈❡rt✐❝❛❧ ❝✉rr❡♥t ✇❛s ♦❜s❡r✈❡❞ ✉♣♦♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❛ ❜✐❛s ✈♦❧t❛❣❡ ✭❱
b
✮ ❜❡t✇❡❡♥ t❤❡
t✇♦ ❣r❛♣❤❡♥❡ ❡❧❡❝tr♦❞❡s✳ ❙✉❝❤ ❛ ❝❤❛r❣❡ tr❛♥s❢❡r ❢r♦♠ ♦♥❡ ❣r❛♣❤❡♥❡ ❧❛②❡r t♦ t❤❡ ♦t❤❡r ❝❛♥
✸

❋✐❣✉r❡ ✶✿ ❉❡✈✐❝❡ str✉❝t✉r❡ ❛♥❞ tr❛♥s♣♦rt ❝❤❛r❛❝t❡r✐st✐❝s✳ ✭❛✮ ▼✐❝r♦s❝♦♣❡ ✐♠❛❣❡ ♦❢ t❤❡ ❛❝t✐✈❡
♣❛rt ♦❢ t❤❡ s❛♠♣❧❡✳ ❚❤❡ ❜r♦✇♥ ❛r❡❛ ✐♥❞✐❝❛t❡s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❲❙❡
2
✢❛❦❡✳ ✭❜✮ ❙❝❤❡♠❛t✐❝
✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❤❡t❡r♦str✉❝t✉r❡ s❤♦✇♥ ✐♥ ✭❛✮✳ ❚❤❡ ❧❛②❡r s❡q✉❡♥❝❡ ❢r♦♠ ❜♦tt♦♠ t♦ t♦♣ ✐s
❤❇◆✴❣r❛♣❤❡♥❡✴❤❇◆✴❲❙❡
2
✴❤❇◆✴❣r❛♣❤❡♥❡✳ ✭❝✮ ❉✐✛❡r❡♥t✐❛❧ ❝♦♥❞✉❝t❛♥❝❡
G = dI/dV
b
✳ ❚❤❡
❝♦❧♦r❡❞ r❡❣✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s✐t✉❛t✐♦♥ s❤♦✇♥ ✐♥ ✭❢✮ ❛♥❞ ✭❣✮✳ ✭❞✮ ■♥t❡❣r❛t❡❞ ❊▲ s✐❣♥❛❧ ✐♥
❛ s♣❡❝tr❛❧ r❛♥❣❡ ❢r♦♠
1.48
❡❱ t♦
1.85
❡❱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❜✐❛s ✈♦❧t❛❣❡✳ ✭❡✮ ❙❝❤❡♠❛t✐❝ ❜❛♥❞
❞✐❛❣r❛♠ ❢♦r ③❡r♦ ❜✐❛s✳ ❚❤❡ r❡❞ ♣❛r❛❜♦❧✐❝ ❜❛♥❞s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❑✲♣♦✐♥t ♦❢ t❤❡ ❇r✐❧❧♦✉✐♥
③♦♥❡ ✐♥ ❲❙❡
2
✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❣r❡❡♥ ❧✐♥❡s ❞❡♣✐❝t ❞♦♥♦r✴❛❝❝❡♣t♦r✲❧✐❦❡ ❜❛♥❞s✳ ✭❢✮ ❙❝❤❡♠❛t✐❝
❜❛♥❞ ❞✐❛❣r❛♠ ❢♦r t❤❡ ❝❛s❡ ♦❢ ✐♥t❡r♠❡❞✐❛t❡ ❜✐❛s✳ ❍❡r❡ ❤♦❧❡s ❝❛♥ t✉♥♥❡❧ t❤r♦✉❣❤ t❤❡ ❤❇◆
❜❛rr✐❡r ✐♥t♦ ❲❙❡
2
✱ ❜✉t ❡❧❡❝tr♦♥s ❝❛♥ ♥♦t✳ ❱
h
❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✈♦❧t❛❣❡ r❡q✉✐r❡❞ t♦ ✐♥❥❡❝t
❤♦❧❡s✳ ✭❣✮ ❇❛♥❞ ❞✐❛❣r❛♠ ❢♦r ❧❛r❣❡ ❜✐❛s✳ ■♥ t❤✐s ❝❛s❡✱ ❜♦t❤ ❤♦❧❡s ❛♥❞ ❡❧❡❝tr♦♥s ❝❛♥ t✉♥♥❡❧✳
❱
e
st❛♥❞s ❢♦r t❤❡ ✈♦❧t❛❣❡ t❤r❡s❤♦❧❞ ❢♦r t❤❡ t✉♥♥❡❧✐♥❣ ♦❢ ❡❧❡❝tr♦♥s ✐♥t♦ ❡①❝✐t♦♥✐❝ st❛t❡s ♦❢
❲❙❡
2
✳ ❚❤❡ ❡①❝✐t♦♥ ✐s ❞❡♣✐❝t❡❞ s❝❤❡♠❛t✐❝❛❧❧②✳ ■♥ t❤✐s r❡❣✐♠❡ ❧✐❣❤t ❡♠✐ss✐♦♥ ✐s ♦❜s❡r✈❡❞ ❞✉❡
t♦ ❡①❝✐t♦♥ r❡❝♦♠❜✐♥❛t✐♦♥ ✐♥ t❤❡ ❲❙❡
2
♠♦♥♦❧❛②❡r✳ ❊
osc
✐♥❞✐❝❛t❡s t❤❡ ♦✛s❡t ❡①tr❛❝t❡❞ ❢r♦♠
❊q♥✳ ✷✳
✹