Richard Michael Saputra / M26137012
05 Jan, 2025
1
TWO KEY CO NCEPTS
DI FFUSI O N CO ND UCTA NCE
H O W ELECTRO N FLO WS
CO NNECTI O N BETWEEN B - D
BA LLI S TI C CO NDUCTANCE
SUMMARY
2
DENSITY OF STATES
• 1 Energy Value  Different number of states occurs
• D, in this lecture represent the number of states / energy
(a)
(b)
(c)
Fig 1 Typical Conduction Band for (a) Atom – Atom (b) Small Devices (c) Bulk Devices
3
FERMI LEVEL
• Fermi level, denoted by μ, is an energy level above which
are all empty states (at 0K), below which are occupied
• The difference between fermi level at contacts – channel,
making the electron to flow
Fig 2 (a) Collection of discrete energy level (b) D is the density of
states
4
µ1 = µ2 = µ
E
• The difference of chemical potential is
similar to “Waterfall”
• Source  Full of electron
• Drain  Creating Virtually empty states
to be filled
• The V+ will push down the energy level,
creating a different chemical potential denoted
by μ2 (as the positive potential lowers the
energy of an electron)
D(E)
µ
µ
qV
D(E)
Incapable to contribute
the current flow, already
occupied
5
Conductance Formula :
The Number of Electrons in Channel = Moving
Electrons:
L
𝐷. 𝑞𝑉 𝐼
= 𝑥𝑡
2
𝑞
6
Conductance Formula :
𝐷. 𝑞 2
………………………………….(1)
𝐺 𝐸 =
2t
D~𝐴𝐿
Time Spent on Channel :
𝐿
………………………………………….(2)
𝑡𝐵 =
𝑉
Connecting (1) and (2) we get :
L
𝑞2 𝐷𝑉
𝐺 𝐸 =
…………………………………….(3)
2𝐿
7
Ballistic Conductance
From Heisenberg Uncertainty:
2𝑡𝐵
= ℎ ……………………….... (4)
𝐷
Connecting (1) and (4) we get:
𝑞2
𝐺 𝐸 = 𝑥 𝑀 Note that M is Positive Number
ℎ
𝑞2
= Quantum of Conductance
ℎ
Fig 5 Schrodinger Equation & Heisenberg Uncertainty
Conductance Formula :
𝐷. 𝑞 2
………………………………….(1)
𝐺 𝐸 =
2t
L
General formula of I, in diffusive current:
𝑑𝑛
………………………………….(5)
𝐼 = −𝐷
𝑑𝑧
𝑄
𝐿2
………………………………….(6)
𝑡𝑑 = =
𝐼 2𝐷
Connecting (1) and (6) we get :
𝐴 𝑞2 𝐷𝑉
𝐺 𝐸 =
…………………………………….(7)
𝐿 2 A𝐿
σ
9
In a small devices, the diffusion conductance becoming irrelevant, and seems to be exclusively
used the ballistic conductance, how to connect from small  big devices? .
2
𝐿 𝐿
𝑡= +
𝑣 2𝐷
𝐿𝑣
𝑡 = 𝑡𝐵 (1 + )
2𝐷
𝐿
𝑡 = 𝑡𝐵 (1 + )
𝜆
𝐷. 𝑞 2
𝐺 𝐸 =
2t
Only A fraction of electron go ballistic :
D/2L
T D/2L
𝐼 = 𝑇 𝐼𝐵
𝜆
𝑇=
(𝐿 + 𝜆)
Put it together and we have:
𝐺𝐵
𝐺𝐵 𝜆
𝜎𝐴
𝐺 𝐸 =
=
=
𝐿 𝐿+𝜆 𝐿+𝜆
1+
𝜆
10
•
•
•
•
To connect the new perspective, consider of a big devices as a parallel series of nano-transistors
Conductance in small devices, depends on its area, A
As L∞, the diffusive transport becoming dominant
Conductance, refers to the how easily the electron flows is driven by the transport phenomenon
and the density of state
11
06 Jan, 2025
12