Ion beam scanning control methods and systems for ion implantation uniformity

1. A method for calibrating an ion beam scanner in an ion implantation system, the method comprising:

measuring a plurality of initial current density values at a plurality of locations along a scan direction, the initial current density values individually corresponding to one of a plurality of initial voltage scan intervals and one of a corresponding plurality of initial scan time values;

creating a system of linear equations based on the measured initial current density values and the initial scan time values; and

determining a set of scan time values for the voltage scan intervals corresponding to a solution to the system of linear equations that reduces current density profile deviations.

2. The method of claim 1 , wherein the initial current density values are measured at an integer number m locations, wherein the individual current density values correspond to one of an integer number n initial voltage scan intervals, and wherein m is greater than n.

3. The method of claim 2 , wherein the m locations and the n initial voltage scan intervals do not correspond with one another.

4. The method of claim 2 , wherein creating the system of linear equations comprises:

forming a matrix A of the measured initial current density values with m rows corresponding to the m locations along the scan direction and n columns corresponding to the n initial voltage scan intervals and time values;

forming an initial time vector T 0 comprising the n initial scan time values; and

computing an initial profile vector P 0 comprising m initial current density profile values, wherein the initial profile vector P 0 =A*T 0 .

5. The method of claim 4 , wherein determining a set of scan time values comprises:

computing a profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m );

computing a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0j −P AVG for j=1 through m;

computing an inverse matrix A −1 ;

multiplying the inverse matrix A −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n scan time deviation values, wherein ΔT SOLUTION =A −1 *ΔP; and

computing a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the initial time vector T 0 , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0 .

6. The method of claim 5 , wherein the inverse matrix A −1 is computed using singular value decomposition (SVD).

7. The method of claim 6 , further comprising:

selectively truncating the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n; and

selectively truncating the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values;

wherein the initial profile vector P 0 is computed as P 0 =A T *T 0T ; and

wherein determining the set of scan time values comprises:

computing the profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m );

computing a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0J −P AVG for j=1 through m;

computing an inverse matrix A T −1 ;

multiplying the inverse matrix A T −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n′ scan time deviation values, wherein ΔT SOLUTION =A T −1 *ΔP; and

computing a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the truncated initial time vector T 0T , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0T .

8. The method of claim 4 , further comprising:

selectively truncating the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n; and

selectively truncating the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values;

wherein the initial profile vector P 0 is computed as P 0 =A T *T 0T .

9. The method of claim 1 , wherein creating the system of linear equations comprises:

forming a matrix A of the measured initial current density values with an integer number m rows corresponding to the m locations along the scan direction and an integer number n columns corresponding to n initial voltage scan intervals and time values;

forming an initial time vector T 0 comprising the n initial scan time values; and

computing an initial profile vector P 0 comprising m initial profile values, wherein the initial profile vector P 0 =A*T 0 .

10. The method of claim 9 , wherein determining a set of scan time values comprises:

computing a profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m );

computing a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0j −P AVG for j=1 through m;

computing an inverse matrix A −1 ;

multiplying the inverse matrix A −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n scan time deviation values, wherein ΔT SOLUTION =A −1 *ΔP; and

computing a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the initial time vector T 0 , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0 .

11. The method of claim 10 , wherein the inverse matrix A −1 is computed using singular value decomposition (SVD).

12. The method of claim 11 , further comprising:

selectively truncating the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n; and

selectively truncating the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values;

wherein the initial profile vector P 0 is computed as P 0 =A T *T 0T ; and

wherein determining the set of scan time values comprises:

computing the profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m );

computing a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0j −P AVG for j=1 through m;

computing an inverse matrix A T −1 ;

multiplying the inverse matrix A T −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n′ scan time deviation values, wherein ΔT SOLUTION =A T −1 *ΔP; and

computing a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the truncated initial time vector T 0T , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0T .

13. The method of claim 9 , further comprising:

selectively truncating the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n; and

selectively truncating the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values;

wherein the initial profile vector P 0 is computed as P 0 =A T *T 0T .

14. The method of claim 1 , wherein the plurality of locations along the scan direction are spaced from one another by a profile interval distance that is less than a lateral dimension of an ion beam.

15. A calibration system for calibrating an ion beam scanner in an ion implantation system, the calibration system comprising:

a dosimetry system operable to measure a plurality of initial current density values at a corresponding plurality of locations along a scan direction in a workpiece location of an ion implantation system; and

a control system operably coupled with the dosimetry system and a power supply associated with a beam scanner of the ion implantation system, the control system being operable to cause the scanner to scan an ion beam across the workpiece location of the ion implantation system in the scan direction one or more times according to a plurality of initial voltage scan intervals and a corresponding plurality of initial voltage scan time values such that the dosimetry system can measure a plurality of initial current density values at a plurality of locations along a scan direction in a workpiece location of an ion implantation system;

wherein the initial current density values individually correspond to one of the plurality of initial voltage scan intervals and to one of the corresponding plurality of initial scan time values; and

wherein the control system is further operable to create a system of linear equations based on the measured initial current density values and the initial scan time values, and to determine a set of scan time values for the voltage scan intervals corresponding to a solution to the system of linear equations that reduces current density profile deviations.

16. The calibration system of claim 15 , wherein the control system is operable to form a matrix A of the measured initial current density values with an integer number m rows corresponding to the m locations along the scan direction and an integer number n columns corresponding to n initial voltage scan intervals and time values, to form an initial time vector T 0 comprising the n initial scan time values, and to compute an initial profile vector P 0 comprising m initial profile values, wherein the initial profile vector P 0 =A*T 0 .

17. The calibration system of claim 16 , wherein the control system is operable to compute a profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m ), to compute a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0j −P AVG for j=1 through m, to compute an inverse matrix A −1 , to multiply the inverse matrix A −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n scan time deviation values, wherein ΔT SOLUTION =A −1 *ΔP, and to compute a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the initial time vector T 0 , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0 .

18. The calibration system of claim 17 , wherein the control system is operable to compute inverse matrix A −1 using singular value decomposition (SVD).

19. The calibration system of claim 18 , wherein the control system is further operable to selectively truncate the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n, and to selectively truncate the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values;

wherein the control system computes the initial profile vector P 0 as P 0 =A T *T 0T ; and

wherein the control system determines the set of scan time values by computing the profile average value P AVG as the average of the m initial profile values, wherein P AVG =(1/m)*(P 01 +P 02 + . . . +P 0m ), computing a profile deviation vector ΔP comprising m profile deviation values, wherein ΔP j =P 0j −P AVG for j=1 through m, computing an inverse matrix A T −1 , multiplying the inverse matrix A T −1 and the profile deviation vector ΔP to obtain a time deviation solution vector ΔT SOLUTION comprising n′ scan time deviation values, wherein ΔT SOLUTION =A T −1 *ΔP, and computing a scan time solution vector T SOLUTION as the sum of the time deviation solution vector ΔT SOLUTION and the truncated initial time vector T 0T , the scan time solution vector T SOLUTION comprising the set of scan time values corresponding to the solution to the system of linear equations that reduces current density profile deviations, wherein T SOLUTION =ΔT SOLUTION +T 0T .

20. The calibration system of claim 16 , wherein the control system is further operable to selectively truncate the matrix A by eliminating one or more columns having no non-zero entries to form a truncated matrix A T having m rows corresponding to the m locations along the scan direction and n′ columns corresponding to the n′ remaining initial voltage scan intervals and time values, wherein n′ is less than n;

wherein the control system selectively truncates the initial time vector T 0 to form a truncated initial time vector T 0T comprising n′ initial scan time values; and

wherein the control system computes the initial profile vector P 0 as P 0 =A T *T 0T .