Modeling small mosfets using ensemble devices

A method of modeling statistical variation of field effect transistors having fingers comprising:

• physically measuring characteristics of existing transistors;
• extracting a scaled simulation based on said characteristics of said existing transistors using a first model;
• physically measuring characteristics of existing pairs of matched transistors;
• extracting random dopant fluctuations from said characteristics of said existing pairs of matched transistors using a second model;
• creating synthetic single finger data using said scaled simulation and said dopant fluctuation and applying the following expressions:
• I
• 1
• =
• sxI
• e
• ,
• ⁢
• x
• =
• 48
• 48
• +
• 24
• ⁢
• βσ
• Vt
• 2
• +
• 6
• ⁢
• β
• 2
• ⁢
• σ
• Vt
• 4
• +
• β
• 3
• ⁢
• σ
• Vt
• 6
• ,
• ⅆ
• 4
• ⁢
• I
• ⅆ
• V
• t
• 4
• =
• β
• ⁢
• ⅆ
• 2
• ⁢
• I
• ⅆ
• V
• t
• 2
• ;
• ⅆ
• 6
• ⁢
• I
• ⅆ
• V
• t
• 6
• =
• β
• 2
• ⁢
• ⅆ
• 2
• ⁢
• I
• ⅆ
• V
• t
• 2
• ⁢
• ⁢
• where
• ⁢
• ⁢
• =
• ⅆ
• 2
• ⁢
• I
• ⅆ
• V
• t
• 2
• I
• and where dn represents a derivative n level, I represents current, I1 represents single finger current, Ie represents ensemble current, Vt represents threshold voltage, and σvtn represents variance of threshold voltage in which n is an integer;
• extracting a single finger model from said synthetic single finger data and said random dopant fluctuations using said first model;
• creating an ensemble model by applying the following expression:
• s
• =
• 1
• -
• (
• 2
• n
• -
• 1
• n
• 2
• )
• ,
• where s comprises a skew between a typical single device model and a typical ensemble model, and n comprises a number of devices in the ensemble;
• and
• adjusting parameters of said first model to cause said single finger model to match targets for said single finger model.